editors: nobody
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Tracial projection methods and uniform property Gamma
A complete, self-contained proof
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PREFACE: what is proved
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This document proves three theorems about simple nuclear C*-algebras.
They share one theme, stated first because it governs every proof below:
CENTRAL SPLITTING. Make a projection that divides every trace
in half, while remaining central enough to respect every
coefficient that is tested.
Uniform property Gamma is exactly this: for every m >= 1 there are
orthogonal projections p_1, ..., p_m, central in the tracial
ultrapower, that partition the identity and split every coefficient
trace equally:
tau(a p_i) = tau(a) / m for every a in A and every limit trace tau.
Three independent constructions of central splitting are given.
Route 1 (real rank zero).
If the uniform tracial ultrapower of the uniform tracial completion
of A has real rank zero, then A has uniform property Gamma.
This answers Problem XXI of Schafhauser-Tikuisis-White.
Route 2 (unital comparison).
If A is simple, separable, unital, infinite dimensional, nuclear,
and has strict comparison, then A has uniform property Gamma and
A is isomorphic to A (x) Z. This is the unital Toms-Winter
conjecture.
Route 3 (nonunital, bounded traces).
If A is simple, separable, nuclear, stably projectionless, its
densely finite traces are bounded on the unit ball, T_1(A) is
weak-* compact and nonempty, and comparison is tested on finite
target ranks, then A has uniform property Gamma and is Z-stable.
Why three? The three use different hypotheses and answer different
questions. Route 1 needs no comparison hypothesis at all. Route 2
needs no real rank zero. Route 3 handles the stably projectionless
setting, where the unit is not available and where projections must
be manufactured out of positive elements. The printed source combines
them; here every part is completely self-contained, importing only
Section 1 from outside itself.
SECTION 1. Conventions
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Throughout, A is a separable C*-algebra. A C*-algebra is unital
unless stated otherwise. All ideals are closed and two-sided.
--------------------------------------------------
1.1 Traces and their norm inequalities
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T(A) is the set of tracial states: linear functionals tau: A -> C with
tau(a*a) >= 0, tau(1) = 1 when A is unital, and tau(ab) = tau(ba).
T(A) is convex and weak-* compact. For simple unital A, every tau in
T(A) is faithful: the set of x with tau(x*x) = 0 is a closed two-sided
ideal not containing 1.
For tau in T(A) put
||a||_{2,tau} := tau(a*a)^{1/2}, ||a||_{1,tau} := tau(|a|).
For a subset X of T(A) write ||a||_{2,X} := sup_{tau in X} ||a||_{2,tau}.
The following inequalities are used throughout:
|tau(b*a)| <= ||b||_{2,tau} ||a||_{2,tau} (Cauchy-Schwarz)
|tau(a b)| <= ||a|| ||b||_{1,tau} (duality)
||a x b||_{2,tau} <= ||a|| ||b|| ||x||_{2,tau} (ideal property).
Given a contraction b, |tau(b x)| <= ||b|| ||x||_{1,tau}
<= ||x||_{1,tau}. In particular |tau(b x)| <= sqrt(tau(|x|^2))
when x is in the tracial L^2 Hilbert space.
These also apply to Tr (x) tau on M_n(A) (Tr is the unnormalized
matrix trace). When tau is evaluated on the bidual A**, we mean the
canonical normal extension; it is tracial (extend the trace identity
in each variable by separate weak-* continuity). Faithfulness on A**
is never asserted; only positivity and normality are used.
For a finite matrix algebra M_n(A) the trace of the unit is n, not 1.
This normalization is used without comment from here on.
--------------------------------------------------
1.2 Fullness, real rank zero, projections
--------------------------------------------------
A projection p in a unital algebra D is FULL if the closed two-sided
ideal generated by p equals D.
D has REAL RANK ZERO if every self-adjoint element is a norm limit of
self-adjoint elements with finite spectrum. Equivalently, for unital
D: finite-spectrum self-adjoint contractions are norm dense in the
self-adjoint unit ball. Every corner pDp of a real-rank-zero algebra
has real rank zero.
For projections e, f in D:
e ~ f means v*v = e and vv* = f for some v in D.
e <~ f means v*v = e and vv* <= f for some v in D.
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1.3 Uniform tracial completion and ultrapower
--------------------------------------------------
Let A be a unital C*-algebra with T(A) nonempty.
The UNIFORM TRACIAL COMPLETION B is the quotient of
{ (a_j) in l^inf(N, A) : (a_j) is ||.||_{2,T(A)}-Cauchy }
by the ideal J of sequences with ||a_j||_{2,T(A)} -> 0. B is a unital
C*-algebra in the quotient C*-norm, and a |--> (a, a, ...) + J is an
injective isometric *-homomorphism A -> B.
Every tau in T(A) extends to a DESIGNATED TRACE tau-bar on B:
tau-bar([(a_j)]) = lim_{j to inf} tau(a_j).
The designated trace family is { tau-bar : tau in T(A) }. Only these
are used to define ||.||_{2,T(A)} on B.
Fix a free ultrafilter omega on N. The UNIFORM TRACIAL ULTRAPOWER is
B^omega = l^inf(N, B) / I_omega,
I_omega = { (b_j) : lim_{j to omega} ||b_j||_{2,T(A)} = 0 }.
Sequences in I_omega are required to be bounded; the class of a
bounded sequence is written [(b_j)] or simply b. LIMIT TRACES are
lambda([(b_j)]) = lim_{j to omega} tau-bar_j(b_j), tau_j in T(A).
Lambda_omega is exactly this family. Two identifications, proved in
Lemma 5.1 in full generality:
lim_{j to omega} sup_{tau in T(A)} tau-bar(b_j*b_j)
= sup_{lambda in Lambda_omega} lambda([b]*[b]),
lim_{j to omega} sup_{tau in T(A)} |tau-bar(b_j)|
= sup_{lambda in Lambda_omega} |lambda([b])|.
In particular ||b||_{2,Lambda_omega} = 0 if and only if [b] = 0 in
B^omega. This is the definition of I_omega together with the first
identity above.
The copies of A and B given by CONSTANT sequences are faithful: for
A this is faithfulness of each tau (simplicity); for B it is
uniform 2-Cauchyness (see Lemma 5.1 for the small argument).
--------------------------------------------------
1.4 Uniform property Gamma
--------------------------------------------------
UNIFORM PROPERTY GAMMA in the completion form:
there is a projection p in B^omega cap B'
(B' is the relative commutant of the constant copy of B inside
B^omega) satisfying
lambda(p x) = (1/2) lambda(x)
for every x in B and every lambda in Lambda_omega.
UNIFORM PROPERTY GAMMA in the all-m form:
for every m >= 1 there are orthogonal projections
p_1, ..., p_m in A^omega cap A' summing to 1, with
lambda(a p_i) = lambda(a) / m for all a in A, lambda in Lambda,
1 <= i <= m.
The two forms are equivalent for simple separable unital infinite
dimensional A by [CCEGSTW, Propositions 5.7, 5.20, 5.23]. Route 1
proves the completion form directly for the ultrafilter omega; the
equivalence gives the all-m form. Route 2 and Route 3 prove the
all-m form directly.
The coefficient a is essential. Scalar trace equality of the p_i
alone does not give property Gamma.
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1.5 Cuntz semigroup, dimension functions, Z
--------------------------------------------------
Let K be the compact operators on a separable infinite dimensional
Hilbert space. For positive a, b in (A (x) K)_+:
a <~ b means ||x_n* b x_n - a|| -> 0 for some x_n.
The Cuntz semigroup Cu(A) is the set of classes [a] under mutual
subequivalence (Cuntz equivalence), ordered by <~, with addition
from orthogonal sum.
Every tau in T(A) extends to the trace tau (x) Tr on A (x) K, with Tr
unnormalized. Define
d_tau(a) = lim_{n to inf} (tau (x) Tr)(a^{1/n}) in [0, inf].
When a is in a corner P_n(A)P_n of a finite matrix algebra, the
support formula
d_tau(a) = (tau (x) Tr)(supp(a))
holds, and passes to A (x) K with the support formula taking values
in [0, inf]. Every use below is in one of these settings.
STRICT COMPARISON (unital case):
a, b in (A (x) K)_+, b != 0,
d_tau(a) < d_tau(b) for all tau in T(A) ==> a <~ b.
If T(A) is empty, this convention demands a <~ b for every positive a
and every nonzero positive b in the stabilization. Route 2 uses this
convention.
The Jiang-Su algebra Z is a simple, separable, unital, infinite
dimensional, nuclear C*-algebra with the same K-theory as C and with
a unique tracial state. Z-STABILITY is A :=: A (x) Z.
The Toms-Winter conjecture for simple separable unital infinite
dimensional nuclear A asserts
finite nuclear dimension <=> Z-stability <=> strict comparison.
Two implications, "finite nuclear dimension => Z-stable" [WinterNuc]
and "Z-stable => strict comparison" [Rordam], are external. "Z-stable
=> finite nuclear dimension" is [CETWW]. Route 2 proves the missing
"strict comparison => Z-stable".
--------------------------------------------------
1.6 External theorems, quoted where used
--------------------------------------------------
Four external results carry weight and are quoted at the point of use:
Haagerup [Haa83, Theorem 3.1].
A nuclear unital C*-algebra A has a CONVEX VIRTUAL DIAGONAL: an
element M of the weak-* closed convex hull of
{ v (x) v* : v in A, ||v|| <= 1 } inside (A (x)_pi A)** with
a.M = M.a = a.M for every a, and m_A(M) = 1.
Hirshberg-Kirchberg-White [HKW, Theorem 1.4].
Every nuclear C*-algebra N admits completely positive contractive
approximations N -> F -> N through a finite-dimensional F, with
the second map a convex combination of order-zero maps. The
number of order-zero summands is finite but not bounded in terms
of nuclear dimension.
Castillejos-Evington-Tikuisis-White [CETW, Theorem 5.6].
A simple, separable, unital, nuclear C*-algebra with strict
comparison and uniform property Gamma is Z-stable.
Castillejos-Li-Szabo [CLS, Proposition 5.4].
A simple, separable, nuclear C*-algebra with almost unperforated
Cu, sufficiently many lower semicontinuous traces, and uniform
McDuffness is Z-stable.
Three supporting lemmas also appear twice, quoted once at first use
and referenced thereafter:
ERS [Proposition 4.2]: d_tau is a functional on Cu(A) for any
lower semicontinuous trace tau (x) Tr.
WZ [Theorem 3.3 and Corollary 4.2]: order-zero structure theorem
in the finite-dimensional case.
CCEGSTW [Propositions 5.7, 5.20, 5.23]: passage between the
completion form and the all-m form of property Gamma.
Evington [Theorem B]: identification of all traces of the
completion of a Z-stable algebra (used in Corollary 5.2).
Connes: injective factors are hyperfinite; used in Part III to
make tracial GNS von Neumann algebras hyperfinite.
Dobler [Theorem 1.1]: Levy continuity theorem without tightness.
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PART I. REAL RANK ZERO AND UNIFORM PROPERTY GAMMA
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Standing hypotheses for Part I:
A is a unital, separable, simple, infinite dimensional, nuclear,
stably finite C*-algebra with T(A) nonempty;
B is the uniform tracial completion of A;
omega is a free ultrafilter on N;
D := B^omega, Lambda := Lambda_omega.
The only hypothesis Route 1 uses beyond these is: D has real rank zero
in its quotient C*-norm. Route 1 also produces a finite-set statement
(Theorem 4.3) in which D need not be nuclear or simple. That
generality is useful because D = B^omega is not, strictly speaking,
known to be nuclear.
GOAL OF PART I. Prove Theorem 5.2 (real rank zero implies uniform
property Gamma) and Corollary 5.3 (all traces on B are designated).
The finite-set statement Theorem 4.3 is proved first, in Section 4,
using only the transferable facts of Section 2-3. Section 5 then uses
a diagonal argument along omega to obtain one projection for all of B
at once, and identifies all traces of B.
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SECTION 2. Projection tools from real rank zero
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Throughout this section, D is a unital C*-algebra of real rank zero
with T(D) nonempty. Every estimate is uniform over all of T(D).
The chain of tools:
2.1 Scalar compression: refine a projection into small pieces on
which finitely many tests act nearly scalarly.
2.2 Small full positive elements: a simple, unital, infinite
dimensional A contains a norm-one full positive element whose
support has arbitrarily small trace uniformly over T(A).
2.3 Small full projections: real rank zero turns Lemma 2.2 into a
full projection in D of small trace.
2.4 Splitting a column: a family of maps v_i with prescribed
targets yields an orthogonal partition subequivalent to each
target.
2.5 Uniform proportional subprojections: for any f, s, and eps,
find e <= f with |tau(e) - s tau(f)| < eps for ALL tau.
2.6 Splitting finite linear tests: refine the previous to
|tau(e h) - s tau(q h)| < eps simultaneously for finitely many
h, uniform in tau.
2.7 Orthogonal insertion: move a projection off an occupied
corner, with controlled trace loss in 2-norm.
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2.1 Scalar compressions in corners
--------------------------------------------------
Lemma 2.1. D unital, real rank zero. Let r be a projection in D,
F a finite subset of D, gamma > 0. Then r can be written as a finite
sum r = sum_{j=1}^m q_j of pairwise orthogonal nonzero projections
q_j such that
dist(q_j h q_j, C q_j) < gamma for every h in F, 1 <= j <= m.
(The statement is trivial for r = 0 with the empty family.)
Proof. Real rank zero of D applied to h + R r, where h = h* in rDr,
R > ||h|| + 2, gives finite-spectrum self-adjoints a_n in D with
a_n -> h + R r. The limit has spectrum in {0} union (2, inf), so
f_n := 1_{(1, inf)}(a_n) converges in norm to r. For large n,
u_n := polar of (r f_n + (1 - r)(1 - f_n))
is a unitary with u_n f_n u_n* = r and u_n -> 1. Because a_n
commutes with f_n, the compression r u_n a_n u_n* r has finite
spectrum in rDr. Subtracting R r yields finite-spectrum
approximations to h in rDr. This proves corner real rank zero, hence
the first displayed claim with m = 1.
For finitely many tests, replace F by its real and imaginary parts
and iterate: at each stage, take one self-adjoint test inside a
current corner pDp, approximate it by a finite-spectrum self-adjoint,
and partition the current corner by that approximation's spectral
projections. An estimate ||p_0 h p_0 - lambda p_0|| < delta
persists on any subprojection p <= p_0. End. []
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2.2 Small full positives in a simple unital algebra
--------------------------------------------------
Lemma 2.2 requires no real rank zero and no nuclearity. It is reused
in Part II (Lemma 7.1). It is the only place in Part I where
simplicity of A is used.
Lemma 2.2. A simple, unital, infinite dimensional, T(A) nonempty.
For every delta > 0 there is a full positive e in A with ||e|| = 1
and
tau(supp(e)) < delta for every tau in T(A),
with supp taken in each tracial GNS von Neumann algebra.
Proof. Two preliminaries.
(a) A contains arbitrarily many orthogonal nonzero positives.
Suppose not; choose a maximum k. Take a maximal abelian unital
C*-subalgebra E of A. If E is finite dimensional with minimal
projections p_1, ..., p_m, then every self-adjoint in p_i A p_i
commutes with E, so p_i A p_i = C p_i. For 0 != x in p_i A p_j:
x*x = ||x||^2 p_j, xx* = ||x||^2 p_i, so p_i ~ p_j and p_i A p_j
is one-dimensional. Then A = sum_{ij} p_i A p_j is finite
dimensional, contradiction. So E = C(X) with X infinite. Any k
distinct points of X give k orthogonal nonzero positives via
nonnegative continuous functions supported near those points.
(b) Trace-support monotonicity. For nonzero positives c, d in A,
simplicity gives c A d != 0 (else c annihilates the ideal generated
by d, which is A). Pick y with z := c y d != 0. Then z*z in
closure of dAd and z z* in closure of cAc. In each tracial GNS
representation the polar decomposition of z has initial projection
supp(z*z) and final projection supp(z z*), so the normal trace gives
tau(supp(z*z)) = tau(supp(z z*))
<= min{ tau(supp(c)), tau(supp(d)) } for all tau.
The witnesses y, z are chosen independently of tau; the estimate is
uniform over T(A).
Now choose k_1: build orthogonal nonzero c_1, ..., c_k as in (a).
Set c := c_1. For j = 2, ..., k apply (b) with d = c_j and replace
c by z z*. Support trace never increased; after the loop we have
tau(supp(c)) <= tau(supp(c_j)) for every j and every tau. By
orthogonality of the supp(c_j):
k tau(supp(c)) <= sum_j tau(supp(c_j)) <= 1 for all tau.
Choose k with 1/k < delta; rescale c to have norm one (this does not
change its support); simplicity of A makes c full. End. []
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2.3 Small full projections
--------------------------------------------------
Lemma 2.3. D unital, containing a unital simple infinite dimensional
subalgebra A. For every delta > 0 there is a full projection g in D
with
sup_{tau in T(D)} tau(g) < delta.
Proof. Restriction of any tau in T(D) to A is a tracial state, so
T(A) is nonempty. Lemma 2.2 with 1/n gives b in A_+ with ||b|| = 1
and sup_{rho in T(A)} rho(b) < 1/n. Choose a finite-spectrum
self-adjoint a in D with ||a - b|| < 1/8 and set
g := 1_{[1/2, inf)}(a). Then g b g >= (3/8) g, so any tau in T(D),
restricted to A, gives
tau(g) <= (8/3) tau(b) <= 8/(3 n).
Fullness. Suppose not; let pi : D -> D/I be the quotient by the
proper ideal generated by g. Since a >= -1/8 and
a(1 - g) <= (1/2)(1 - g),
||pi(a)|| <= 1/2, hence ||pi(b)|| <= 5/8.
But pi|_A is a unital homomorphism from the simple algebra A into a
nonzero unital algebra; injective, hence isometric. Contradicts
||b|| = 1. End. []
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2.4 Splitting a column
--------------------------------------------------
Lemma 2.4. f, g_1, ..., g_m projections in D (m >= 1); v_1, ..., v_m
in D satisfying
v_i = g_i v_i f, sum_{i=1}^m v_i* v_i = f.
Then there exist orthogonal projections e_1, ..., e_m with sum f and
e_i <~ g_i for each i.
Proof. Induction on m.
Base m = 1: v_1 itself gives f <~ g_1.
Step m > 1. Lemma 2.1 gives finite-spectrum self-adjoint a in fDf
with ||a - v_1* v_1|| < 1/8. Set e_1 := 1_{[1/2, inf)}(a) and
r := f - e_1. Then e_1 v_1* v_1 e_1 >= (3/8) e_1, so
v_1 e_1 (e_1 v_1* v_1 e_1)^{-1/2}
is a partial isometry with initial projection e_1 and final
projection below g_1. On the residual corner,
Q := sum_{i=2}^m r v_i* v_i r = r - r v_1* v_1 r >= (3/8) r.
If r = 0, done. Otherwise define w_i := v_i r Q^{-1/2} for
i = 2, ..., m (inverse in rDr). Then w_i = g_i w_i r and
sum_{i=2}^m w_i* w_i = r; induction partitions r. []
--------------------------------------------------
2.5 Uniform proportional subprojections
--------------------------------------------------
The heart of packing is uniformity: one rounding rule works for every
trace simultaneously. The trick is an ORDERED partition, which makes
successive trace weights increase in the same order for every tau.
Proposition 2.5. Assume for every delta > 0 there is a full
projection g in D with sup_{tau in T(D)} tau(g) < delta. For every
projection f in D, every s in [0, 1], every eps > 0, there is a
projection e <= f with
sup_{tau in T(D)} | tau(e) - s tau(f) | < eps.
Proof. f = 0 is trivial. Fix a full projection g in D with
sup_tau tau(g) < eps.
Step 1: build an orthogonal partition of f subequivalent to g.
Fullness of g gives finitely many x_i, y_i with
sum_i x_i g y_i = 1 + e where ||e|| < 1. Set X := [x_1, ..., x_m]
and Y the column (g y_i). Then X Y = 1 + e is invertible, so Y has
a bounded left inverse L := (XY)^{-1} X. From Y* L* L Y = 1 and
||L* L|| <= ||L||^2:
1 <= ||L||^2 Y* Y, so Y* Y is invertible.
Apply Lemma 2.4 to the column of Y (Y* Y)^{-1/2} f, with every target
projection equal to g. This yields orthogonal projections
t_1, ..., t_m with sum f and each t_i <~ g.
Step 2: reorder so the pieces are linearly ordered under <~.
Separately, the following pair-exchange operation is useful.
Pair-exchange lemma. Suppose x, y are orthogonal projections, both
<~ h. Then there exist x', y' orthogonal, both <~ h, with
x' + y' = x + y and x' <~ y'.
Proof. Choose partial isometries w, u with w*w = x, w w* = x' <= h,
and u*u = y, u u* <= h. Since y = u*(x' + (h - x'))u and
x' orthogonal to h - x', Lemma 2.4 applied to the two entries
x' u and (h - x') u gives y = k + l with k <~ x' and l <~ h - x'.
Let alpha, beta realize the two subequivalences. Set
xi := w* alpha + l, zeta := w + beta. Compute:
xi* xi = y
xi xi* = w* alpha alpha* w + l <= x + l
zeta* zeta = x + l
zeta zeta* = x' + beta beta* <= h.
The cross terms w* alpha and l, resp. w and beta, have orthogonal
initial or final supports by construction. So replacing (x, y) by
(k, x + l) preserves sum and orthogonality, and all three pieces
k, x, y are <~ x + l <~ h. []
Now reorder left-to-right with common bound g. Each exchange leaves
the moving piece dominating everything it passed. Freeze the last
surviving piece and repeat on the shorter list. Induction gives
s_1, ..., s_m orthogonal, sum f, with
s_1 <~ s_2 <~ ... <~ s_m <~ g.
Step 3: round. Set
c_i := floor(s i) - floor(s(i-1)) in {0, 1}
e := sum_{i=1}^m c_i s_i.
For a fixed tau put t_i := tau(s_i) and D_j := floor(s j) - s j.
Then t_1 <= ... <= t_m <= tau(g) and -1 <= D_j <= 0. Summation
by parts:
tau(e) - s tau(f) = D_m t_m + sum_{i=1}^{m-1} D_i (t_i - t_{i+1}).
The first term lies in [-t_m, 0], the sum in [0, t_m - t_1]. Hence
|tau(e) - s tau(f)| <= t_m <= tau(g) < eps.
Independent of tau. End. []
--------------------------------------------------
2.6 Splitting finite linear tests
--------------------------------------------------
Corollary 2.6. Under the hypothesis of Proposition 2.5, let q be a
projection in D, s in [0, 1], F subset D finite, eps > 0. Then there
is a projection e <= q with
|tau(e h) - s tau(q h)| < eps for every h in F, tau in T(D).
Proof. q = 0 or F empty: trivial. Choose gamma > 0 with
2 gamma < eps / 2. Lemma 2.1 writes q = sum_{j=1}^m q_j with
||q_j h q_j - lambda_{j,h} q_j|| < gamma for every h in F,
with scalars |lambda_{j,h}| <= ||h|| + gamma. Let M :=
max_{h in F} ||h|| + gamma and pick delta > 0 with m M delta < eps / 2.
Proposition 2.5 gives e_j <= q_j with |tau(e_j) - s tau(q_j)| < delta
for every j and tau. Set e := sum_j e_j. Then
|tau(e h) - s tau(q h)|
<= gamma (tau(e) + s tau(q))
+ sum_j |lambda_{j,h}| |tau(e_j) - s tau(q_j)|
<= 2 gamma + m M delta < eps. []
--------------------------------------------------
2.7 Orthogonal insertion
--------------------------------------------------
Lemma 2.7. Let z, y be projections in D. There is a projection
t <= 1 - z with, for every tau in T(D),
tau(y) - 3 tau(y z) <= tau(t) <= tau(y) (mass)
||t - y||_{2,tau}^2 <= 5 tau(y z) (distance)
and, for every contraction x in D,
|tau((t - y) x)| <= 2 sqrt(tau(y)) ||t - y||_{2,tau}. (linear)
Proof. y = 0: trivial. Otherwise choose a finite-spectrum
self-adjoint a in yDy with ||a - yzy|| < 1/8 and
f := 1_{(-inf, 1/2)}(a). With d := y - f,
d(yzy)d >= (3/8) d, f(yzy)f <= (5/8) f.
Since yzy >= 0,
tau(y - f) <= (8/3) tau(y z), A_f := f(1 - z) f >= (3/8) f.
If f = 0 set t := 0. Otherwise
v := (1 - z) f A_f^{-1/2}
is a partial isometry with v*v = f; set t := v v* <= 1 - z. In
both cases tau(t) = tau(f) and f t f = f(1 - z) f. The mass
estimate follows at once.
From f <= y,
tau(t y) >= tau(t f) = tau(f) - tau(f z).
Then
||t - y||_{2,tau}^2 = tau(t) + tau(y) - 2 tau(t y)
<= tau(y - f) + 2 tau(f z)
<= (8/3 + 2) tau(y z)
<= 5 tau(y z).
For the linear estimate use
t - y = t(t - y) + (1 - t)(t - y).
The first summand has left support below t; the second equals
-(1 - t) y and has right support below y. Tracial Cauchy-Schwarz:
||t - y||_{1,tau}
<= (sqrt(tau(t)) + sqrt(tau(y))) ||t - y||_{2,tau}
<= 2 sqrt(tau(y)) ||t - y||_{2,tau}.
Trace-norm duality gives the linear estimate. End. []
===========================================================================
SECTION 3. Nuclear averaging and weighted matrix blocks
===========================================================================
Now A subset D is a unital inclusion of C*-algebras: A nuclear, D unital
real rank zero with T(D) nonempty. (No simplicity, no infinite
dimensionality, no comparison.)
The plan. Choose ONE nuclear-averaging map Phi, ahead of any input,
using only a finite test set K subset U(A) and a tolerance mu.
Phi turns any projection into a nearly central contraction. Then
approximate Phi(r) in trace by a weighted sum of projections; each
projection is realized as a matrix block U_j (1_{k_j} (x) q_j) U_j*
over a base projection q_j, on which the compressions U_j* x U_j are
nearly scalar. The sampling and packing of Section 4 happen purely
in the q_j coordinates, without ever making the sampled piece central.
--------------------------------------------------
3.1 Nuclear averaging with trace control
--------------------------------------------------
Lemma 3.1. A nuclear unital. For every finite K subset U(A) and
every mu > 0, there exist contractions v_1, ..., v_m in A and
nonnegative scalars lambda_1, ..., lambda_m with sum lambda_l = 1,
such that
Phi(b) := sum_{l=1}^m lambda_l v_l b v_l* (b in D)
satisfies, for every projection q in D, every x in K, every tau in T(D),
0 <= Phi(q) <= 1 (positive)
tau(q) - mu <= tau(Phi(q)) <= tau(q) (mass)
||Phi(q) - x Phi(q) x*||_{1,tau} <= mu tau(q) (centrality)
Proof. Two Banach-space preliminaries and one computation.
Let E := A (x)_pi A be the projective tensor product, m_A: E -> A the
multiplication map, and consider the A-bimodule structure
x . (a (x) b) := x a (x) b, (a (x) b) . x := a (x) b x.
Haagerup's virtual diagonal theorem (Haa83, Thm 3.1) gives an element
M of E** that lies in the weak-* closed convex hull of
{ v (x) v* : v in A, ||v|| <= 1 }, and satisfies
x . M = M . x = M . x (x in A)
m_A**(M) = 1.
Form the bounded linear map
Theta : E -> E^K + A,
Theta(d) := ( (x . d - d . x)_{x in K}, m_A(d) ).
Its second adjoint sends M to (0, ..., 0, 1) in the canonical copy of
E^K + A inside the bidual. Weak-* convergence of the convex hull
inside E** implies weak convergence of its image inside E^K + A. The
image point (0, ..., 0, 1) lies in the weak closure of the finite
convex hull of Theta({ v (x) v* : ||v|| <= 1 }). By Hahn-Banach
separation (weak and norm closures of a convex set coincide), the
same point lies in the norm closure, giving finitely many contractions
v_l in A and scalars lambda_l >= 0, sum lambda_l = 1, with
|| x . d - d . x ||_pi < mu for every x in K, (*)
|| 1 - sum_l lambda_l v_l v_l* || < mu,
where d := sum_l lambda_l v_l (x) v_l*.
Both sum_l lambda_l v_l v_l* and sum_l lambda_l v_l* v_l are positive
contractions, and the second differs from 1 in 1-norm by less than
mu because the two are tracially balanced:
c := 1 - sum_l lambda_l v_l* v_l >= 0 and tau(c) < mu. This gives
tau(q) - tau(Phi(q)) = tau(q c) <= mu tau(q),
which is (mass).
For (centrality), the map a (x) b |-> a q b is bounded E -> L^1(D, tau)
with norm <= tau(q): indeed, by tracial Cauchy-Schwarz and the ideal
property,
tau(a q b) <= ||a q||_{2,tau} ||q b||_{2,tau} <= ||a|| ||b|| tau(q).
Applying this map to the inequality (*) and multiplying by the unitary
x*, we obtain
||Phi(q) - x Phi(q) x*||_{1,tau} <= mu tau(q). End. []
--------------------------------------------------
3.2 Powers-Stormer square-root estimate
--------------------------------------------------
Lemma 3.2 (Powers-Stormer, tracial form). For positive b, c in D and
tau in T(D),
||sqrt(b) - sqrt(c)||_{2,tau}^2 <= ||b - c||_{1,tau}.
Proof. Work in the tracial GNS von Neumann algebra and set
d := sqrt(b) - sqrt(c). Traciality and b - c = sqrt(b) d + d sqrt(c)
give
tau(sgn(d) (b - c)) = tau((sqrt(b) + sqrt(c)) |d|).
Since sqrt(b) + sqrt(c) >= d and >= -d, and d = d_+ - d_-,
tau((sqrt(b) + sqrt(c)) |d|) >= tau(d_+^2) + tau(d_-^2) = tau(d^2).
Trace-norm duality bounds the left side by ||b - c||_{1,tau}. []
--------------------------------------------------
3.3 Weighted matrix blocks
--------------------------------------------------
Roughly: Phi(r) can be approximated in trace by projections, each
carrying a scalar-matrix structure over a base projection q_j.
Proposition 3.3. Let K subset U(A) be finite and 0 < eta < 1.
There exist contractions v_1, ..., v_m in A and scalars
lambda_1, ..., lambda_m >= 0 summing to 1, depending only on K, eta,
such that Phi(b) := sum_l lambda_l v_l b v_l* satisfies the conclusions
of Lemma 3.1 with mu := eta^2. For every projection r in D, there
exists a finite family of weights 0 < w_j <= 1, nonzero projections
q_j <= r, integers k_j >= 1, and rows U_j in M_{1, k_j}(D) (so
p_j := U_j U_j* is a projection in D) with
U_j* U_j = 1_{k_j} (x) q_j (isometry)
dist(U_j* x U_j, M_{k_j}(C) (x) q_j) < eta for every x in K
(scalar)
||sum_j w_j p_j - Phi(r)||_{1,tau} <= eta^2 for every tau
(mass)
sum_j w_j ||[x, p_j]||_{2,tau}^2 <= 6 eta for every x in K
(energy)
The family may be empty. Base projections q_j may repeat; the p_j
need not be orthogonal.
Proof.
Setup. Set mu := eta^2 and let Phi be given by Lemma 3.1. Choose
parameters 0 < sigma < 1 and beta > 0 with
beta < sigma / 2,
m (sigma + 2 beta) < mu,
2 beta / sigma + (2 + sqrt(2)) beta / sigma^2 < eta. (**)
Possible by choosing sigma with m sigma < mu first.
Scalar matrix approximation. Partition r using Lemma 2.1 applied to
the finite list { v_l* v_n, v_l* x v_n : 1 <= l, n <= m, x in K },
with compression tolerance below beta / (2 m). For a nonzero
partition piece q, set
V := [ sqrt(lambda_l) v_l q ]_{l=1}^m
h_q := V V* = Phi(q)
M := V* V, N_x := V* x V.
These are contractions in the indicated matrix spaces. Apply a state
of qDq entrywise to M and N_x, obtaining scalar matrices G, H_x. A
state at matrix level is completely positive and contractive, so
0 <= G <= 1 and ||H_x|| <= 1. The compression tolerance gives
||M - G (x) q|| < beta, ||N_x - H_x (x) q|| < beta (***)
because each unweighted entry differs from its scalar state value by
less than twice the compression tolerance, and an m x m matrix of
entries bounded in norm by c has norm at most m c.
Diagonalize and truncate. Apply a scalar unitary to the columns of V
so G becomes diagonal. Retain the columns whose diagonal entries are
>= sigma, forming row W with k columns and diagonal G_+. This cutoff
is what gives a uniform positive lower bound; only after this do we
invert a square root.
If k = 0, set b_q := 0. Otherwise define
X := W* W >= (sigma/2) (1_k (x) q), U := W X^{-1/2},
b_q := U (G_+ (x) q) U*.
Then U* U = 1_k (x) q, so b_q is a finite-spectrum positive
contraction. Let W_- be the row of deleted columns. Then
||W_-* W_-|| <= sigma + beta, so
tau(W_- W_-*) = (Tr (x) tau)(W_-* W_-)
<= m (sigma + beta) tau(q).
Also b_q - W W* = U (G_+ (x) q - X) U* has norm at most beta and is
supported on a projection of trace k tau(q), so, including k = 0,
||b_q - h_q||_{1,tau} <= m (sigma + 2 beta) tau(q)
<= mu tau(q). (****)
Matrix-size-independent clustering. Resolvent formula for inverse
square roots: for positive invertible X, Y >= a 1,
||X^{-1/2} - Y^{-1/2}||
<= (||X - Y|| / pi) int_0^inf t^{-1/2} / (a + t)^2 dt
= ||X - Y|| / (2 a^{3/2}).
Use Y := G_+ (x) q and a := sigma / 2. Retain the same corner of
H_x (x) q denoted H (x) q, with ||H|| <= 1. From (***):
||W* x W - H (x) q|| < beta.
Expand U* x U - Y^{-1/2} (H (x) q) Y^{-1/2} and use the inverse-root
bound to get norm < eta, uniform in the corner and its matrix size.
The second expression is a scalar matrix over q; this is (scalar).
Layer cake. Write b_q as a finite Riemann sum of upper spectral
projections:
b_q = int_0^1 1_{(a, 1]}(b_q) da
= sum_{j in J_q} w_j p_j.
Each p_j is the image of a scalar coordinate projection under U, so
it has a subrow U_j with the required (isometry) relation and
q_j = q. Summing (****) over the finitely many pieces gives (mass).
Averaged commutators. For a finite-spectrum positive contraction
b = sum_s s e_s (include the kernel projection), tracial orthogonality
gives, for a unitary x,
int_0^1 ||[x, 1_{(a, 1]}(b)]||_{2,tau}^2 da
= sum_{s, t} |s - t| ||e_s x e_t||_{2,tau}^2
<= 2 sqrt(tau(b)) ||[x, sqrt(b)]||_{2,tau}
<= 2 sqrt(tau(b) ||b - x b x*||_{1,tau}). (#)
Middle inequality: factor |s - t| = |sqrt(s) - sqrt(t)| (sqrt(s) + sqrt(t))
and use Cauchy-Schwarz; the second squared sum is at most
2 sum_{s,t} (s + t) ||e_s x e_t||^2 = 4 tau(b). Last inequality uses
Lemma 3.2.
For b := b_q, using (mass)-type estimates plus (centrality) of
Lemma 3.1 (mu = eta^2):
tau(b_q) <= (1 + mu) tau(q), ||b_q - x b_q x*||_{1,tau} <= 3 mu tau(q).
Hence (mass)-type bound on each J_q contribution to (energy):
<= 2 sqrt(3 (1 + mu)) eta tau(q) <= 6 eta tau(q).
Summing over q finishes (energy). End. []
===========================================================================
SECTION 4. Sampling and orthogonal packing
===========================================================================
Setup. A subset D is a unital inclusion, A nuclear, D unital real
rank zero with T(D) nonempty. Assume also:
HYP. For every delta > 0 there exists a full projection g in D
with sup_{tau in T(D)} tau(g) < delta.
The point of HYP is to turn Phi into a small central contraction. D
need not be simple or nuclear.
The two key estimates from Section 3 (with mu = eta^2):
tau(Phi(r)) is close to tau(r);
||Phi(r) - x Phi(r) x*||_{1,tau} is <= eta^2 tau(r).
The weighted family (w_j, q_j, k_j, U_j, p_j) approximates Phi(r) in
trace, and each block U_j* x U_j is nearly scalar in M_{k_j}(C) (x) q_j.
Section 4.1 shows how to sample a subprojection y <= p_j that behaves
like c p_j in trace against any coefficient, using only e <= q_j. The
subprojection y is NOT central. Section 4.2 handles orthogonal sums.
Section 4.3 packs.
--------------------------------------------------
4.1 Sampling a matrix block
--------------------------------------------------
Lemma 4.1 (sampling). Let q be a nonzero projection in D and
U in M_{1, k}(D) with U* U = 1_k (x) q. Put p := U U*. Let x in
U(D) and 0 < eta < 1 be such that
dist(U* x U, M_k(C) (x) q) < eta.
For a projection e <= q, put y := U (1_k (x) e) U*. For every
0 <= c <= 1 and every tau in T(D), with
delta_tau := | tau(y) - c tau(p) |,
|tau(y x) - c tau(p x)|
<= eta (tau(y) + c tau(p)) + 2 delta_tau (linear)
| ||[x, y]||_{2,tau}^2 - c ||[x, p]||_{2,tau}^2 |
<= 6 eta (tau(y) + c tau(p)) + 10 delta_tau. (energy)
Proof. Set T := U* x U, P_e := 1_k (x) e, and choose T_0 in M_k(C) with
||T - T_0 (x) q|| < eta. Since T is a contraction,
||T_0|| <= ||T_0 (x) q|| <= 1 + eta <= 2.
The map a |-> U a U* identifies the relevant matrix corner in D with
its image, so
tau(y x) = (Tr (x) tau)(P_e T P_e), tau(|y x y|^2) = (Tr (x) tau)(|P_e T P_e|^2).
Because T_0 (x) q is a scalar matrix over q (hence commutes with P_e
for any e <= q), replacing P_e T P_e by T_0 (x) e changes these two
traces by at most eta tau(y) and 3 eta tau(y). For the second bound,
use
||R* R - S* S|| <= (||R|| + ||S||) ||R - S||,
R := P_e T P_e, S := T_0 (x) e,
and note (Tr (x) tau)(P_e) = k tau(e) = tau(y). Same estimates with
e replaced by q.
Set
alpha := Tr(T_0) / k, beta_0 := Tr(T_0* T_0) / k.
Then |alpha| <= 2 and 0 <= beta_0 <= 4. Scalar reference traces for
y and p are alpha tau(y), beta_0 tau(y) and alpha tau(p),
beta_0 tau(p). This gives (linear).
For (energy) use the exact identity, valid for a projection f and
unitary x, by expansion and traciality:
||[x, f]||_{2,tau}^2 = 2 tau(f) - 2 tau(|f x f|^2).
Apply it to y, p and combine with the squared compression estimates;
the mass-discrepancy coefficient is at most 2 + 2 beta_0 <= 10.
Finally, this identity also accounts for the "leak" of x outside the
larger block p, so nothing is lost. End. []
--------------------------------------------------
4.2 Energy of orthogonal sums
--------------------------------------------------
Lemma 4.2. If t_1, ..., t_M are mutually orthogonal projections in D
and z := sum_a t_a, then for every x in D and tau in T(D),
||[x, z]||_{2,tau}^2 <= sum_a ||[x, t_a]||_{2,tau}^2.
The same bound holds for a sum over any subfamily.
Proof. The two off-diagonal corners (1-z) x z and z x (1-z) of the
commutator are orthogonal in the tracial 2-seminorm, so
||[x, z]||_{2,tau}^2
= ||(1 - z) x z||_{2,tau}^2 + ||z x (1 - z)||_{2,tau}^2.
Split the z-corner into orthogonal t_a: each term is a sum of squared
norms. Since 1 - z <= 1 - t_a, the summands are bounded by
||(1 - t_a) x t_a||^2 and ||t_a x (1 - t_a)||^2, as required. []
--------------------------------------------------
4.3 Proof of the finite-set central splitting
--------------------------------------------------
Theorem 4.3. A subset D is a unital inclusion with A nuclear and D
unital, real rank zero, T(D) nonempty, satisfying HYP. For every
finite K subset U(A) and every eps > 0, there is a projection P in D
such that, for every x in K,
sup_{tau in T(D)} ||[P, x]||_{2,tau} < eps
sup_{tau in T(D)} | tau(P x) - (1/2) tau(x) | < eps.
Proof outline. Construct two orthogonal projections P, S such that
P nearly commutes with K, their coefficient traces nearly agree, and
P + S nearly fills the identity in trace. The identity
2 tau(P x) - tau(x)
= tau((P - S) x) - tau((1 - P - S) x) (x in K)
then gives the halving estimate.
Parameters. Fix H > 0 and N in N with eta := H / N < 1. First H is
chosen to make e^{-H} small; then N increases at fixed H to shrink
eta and the accumulated error E_N.
Averaging maps. Build finite sets of unitaries K_1 >= K_2 >= ... >=
K_N = K going backwards. Having chosen K_i, select Phi_i by
Proposition 3.3 with input (K_i, eta). If i > 1, express every
coefficient contraction v_l of Phi_i as a linear combination of
unitaries with coefficient absolute sum <= 2, and include those
unitaries together with K_i in K_{i-1}.
Induction data. Let Z_0 = 0. In round i (i = 1, ..., N), remaining
corner r_i := 1 - Z_{i-1}. Insert a finite family of orthogonal
projections, each labelled + or -. Let Z_i be the cumulative sum.
Energy recurrence:
E_0 := 0, E_i - E_{i-1} := 1000 eta (E_{i-1} + eta).
Invariant (energy):
sum_{t inserted through round i} ||[x, t]||_{2,tau}^2 <= E_i
for every x in K_i, tau in T(D).
By Lemma 4.2 this bounds the commutator of the whole subfamily sum.
Old-overlap estimate. For every tau,
tau(Z_{i-1} Phi_i(r_i))
= sum_l lambda_l ||Z_{i-1} v_l r_i||_{2,tau}^2
<= 4 E_{i-1}. (overlap)
For i = 1 both sides are 0. For i > 1, each v_l is a union of unitaries
in K_{i-1}, so Lemma 4.2 and the invariant give
||[Z_{i-1}, v_l]||_{2,tau} <= 2 sqrt(E_{i-1}); also
Z_{i-1} v_l r_i = [Z_{i-1}, v_l] r_i.
Sampling in one round. Fix round i; suppress i from block notation.
Signed index set a := (j, +/-). Both labels for j carry the same
target proportion c_a := eta w_j / 2. Let z_a be the occupied
projection just before index a (starting at Z_{i-1}). For each a
choose
y_a := U_j (1_{k_j} (x) e_a) U_j*, e_a <= q_j,
so that, uniformly over tau,
sum_a | tau(y_a) - c_a tau(p_j) | <= eta^2 (sample-mass)
sum_a | tau(z_a y_a) - c_a tau(z_a p_j) | <= eta^2 (sample-overlap)
Achieved by budget: allocate positive eps_a with sum eps_a <= eta^2;
at step a, apply Corollary 2.6 with proportion c_a, tests q_j and
h_a := sum_{d=1}^{k_j} (U_j* z_a U_j)_{dd} in q_j D q_j
tolerance eps_a / k_j.
Then traciality gives tau(y_a) = k_j tau(e_a), tau(z_a y_a) = tau(e_a h_a),
tau(z_a p_j) = tau(q_j h_a), so each step contributes <= eps_a.
Then insert with Lemma 2.7: obtain projection t_a <= 1 - z_a with
tau(y_a) - 3 tau(z_a y_a) <= tau(t_a) <= tau(y_a)
||t_a - y_a||_{2,tau}^2 <= 5 tau(z_a y_a).
Update occupied projection to z_a + t_a. At the end of the round it
is Z_i.
Mass and overlap in round i. For each tau define
C_i := sum_a c_a tau(p_j), Y_i := sum_a tau(y_a),
L_i := sum_a tau(z_a y_a).
By (mass) from Lemma 3.1 (mu = eta^2) and (mass) from Prop 3.3:
eta tau(r_i) - 2 eta^3 <= C_i <= eta (1 + eta^2) <= 2 eta.
Using (sample-mass) and (mass):
eta tau(r_i) - 3 eta^2 <= Y_i <= 3 eta.
Overlap bound. Use z_a <= Z_i, then single weighted-operator estimate
(mass) from Prop 3.3:
L_i <= eta^2 + eta sum_j w_j tau(Z_i p_j)
<= eta^2 + eta ( eta^2 + tau(Z_i Phi_i(r_i)) )
<= eta^2 + eta ( eta^2 + 4 E_{i-1} + Y_i )
<= 4 eta E_{i-1} + 5 eta^2.
In the third line use (overlap), 0 <= Phi_i(r_i) <= 1, and
tau(Z_i - Z_{i-1}) <= Y_i (from the mass estimate on t_a). In
particular
L_i <= E_i - E_{i-1}. (overlap-summ)
Commutators and linear tests. For x in K_i, sum Lemma 4.1 over the
round. Equations (sample-mass) and Y_i, C_i bounds give
sum_a | tau(y_a x) - c_a tau(p_j x) |
<= eta (Y_i + C_i) + 2 eta^2 <= 7 eta^2 (round-linear)
sum_a ||[x, y_a]||_{2,tau}^2
<= eta sum_j w_j ||[x, p_j]||_{2,tau}^2
+ 6 eta (Y_i + C_i) + 10 eta^2
<= 46 eta^2 <= 60 eta^2. (round-energy)
First term at most 6 eta^2 by (energy) from Prop 3.3.
Passing to inserted projections, using the mass/distance estimate and
||t_a - y_a||^2 <= 5 L-part:
sum_a ||[x, t_a]||_{2,tau}^2
<= 2 sum_a ||[x, y_a]||_{2,tau}^2 + 8 sum_a ||t_a - y_a||_{2,tau}^2
<= 120 eta^2 + 40 L_i
<= 160 eta E_{i-1} + 320 eta^2
<= E_i - E_{i-1}.
Along with K_i subset K_{i-1}, this proves the invariant.
Mass left after round:
tau(1 - Z_i) <= (1 - eta) tau(1 - Z_{i-1}) + 3 eta^2 + 3 L_i.
Completing the packing. Let P, S be sums of inserted projections
labelled + and -, respectively. Then P + S = Z_N, and for x in K,
tau in T(D):
||[x, P]||_{2,tau}^2 <= E_N (energy-final)
tau(1 - Z_N) <= e^{-H} + 3 N eta^2 + 3 E_N (rem-final)
|tau(P x) - tau(S x)|
<= 7 N eta^2 + 2 sqrt(15 H E_N). (bal-final)
For (energy-final), Lemma 4.2. For (rem-final), iterate the
remaining-mass bound, use (1 - eta)^N <= e^{-H}, and sum the overlap
bounds. For (bal-final), the +/- targets c_a tau(p_j x) cancel in
every block of every round; the sampling error is at most 7 N eta^2
by (round-linear); the displacement error, by Lemma 2.7 mass/distance
and Cauchy-Schwarz over all signed indices, is at most
2 sqrt(15 H E_N).
Energy closed form. Solve the recurrence:
E_N = eta ( (1 + 1000 eta)^N - 1 ) <= eta e^{1000 H}.
Also N eta^2 = H eta.
Halving error. Since x is a contraction,
| tau((1 - Z_N) x) | <= tau(1 - Z_N). From (rem-final) and
(bal-final):
| tau(P x) - (1/2) tau(x) |
<= (1/2) e^{-H} + 5 H eta + (3/2) E_N + sqrt(15 H E_N).
Now choose H large enough that (1/2) e^{-H} < eps/4. With this H
fixed, let N increase through integers and eta := H / N. Then
5 H eta -> 0, E_N -> 0, sqrt(15 H E_N) -> 0. Also ||[x, P]||^2 <=
E_N -> 0 by (energy-final). So for sufficiently large N both
displayed estimates in the statement are < eps. End. []
COROLLARY 4.4 (finite-set statement, named form). Under the
hypotheses of Theorem 4.3, given eps, K, there is a projection P in D
with
sup_{tau in T(D)} ||[P, x]||_{2,tau} < eps (x in K)
sup_{tau in T(D)} | tau(P x) - (1/2) tau(x) | < eps (x in K).
This is Theorem 4.3, renamed for cross-reference in Section 5.
===========================================================================
SECTION 5. The projection in the prescribed ultrapower
===========================================================================
Route 1 assumes: A simple separable unital infinite dimensional nuclear
stably finite with T(A) nonempty; B == uniform tracial completion;
omega free ultrafilter; D := B^omega; Lambda := Lambda_omega. The
hypothesis is: D has real rank zero in its quotient C*-norm.
We must show uniform property Gamma in the completion form (Section 1.4):
there is a projection p in D cap B' with lambda(p x) = (1/2) lambda(x)
for every x in B and every lambda in Lambda_omega.
The construction: use Theorem 4.3 to get approximate projections at
finite stages, then a diagonal along omega to get ONE projection, then
density of B in A for the 2-norm to extend from A to B.
--------------------------------------------------
5.1 Detection by limit traces
--------------------------------------------------
Lemma 5.1. Fix a bounded sequence d := (d_j) in B with class [d] in
B^omega. Then
sup_{lambda in Lambda_omega} lambda([d]*[d])
= lim_{j to omega} sup_{tau in T(A)} tau-bar(d_j* d_j), (two)
sup_{lambda in Lambda_omega} | lambda([d]) |
= lim_{j to omega} sup_{tau in T(A)} | tau-bar(d_j) |. (one)
In particular, if lambda([d]*[d]) = 0 for every lambda in Lambda_omega,
then [d] = 0 in B^omega. The analogous statement for a positive
contraction. Also, the constant copy of B inside B^omega is faithful.
Proof. (Upper bounds.) Immediate from the definition: lambda([b]) =
lim_omega tau-bar_j(b_j), so |lambda([b])| is bounded by the omega-limit
of sup_tau |tau-bar(b_j)|.
(Reverse for (two).) Choose, for each j, a trace tau_j within 1/j of
the coordinate supremum applied to d_j* d_j. The associated limit
trace gives equality at the omega-limit.
(Reverse for (one).) Choose approximate maximizers of |tau-bar(d_j)|
at each coordinate. Absolute value commutes with a bounded scalar
ultralimit, so the same choice gives equality.
(Faithfulness of B in B^omega.) If the class of b = [(b_j)] in B^omega
is nonzero but its copy in the constant copy of B vanishes, its
representative can be replaced by a bounded sequence in B whose
uniform 2-Cauchyness gives lim_j sup_tau tau-bar((b_j)*b_j) = 0, hence
d = 0 in B^omega by (two). End. []
--------------------------------------------------
5.2 Main theorem (real rank zero => uniform Gamma)
--------------------------------------------------
Theorem 5.2. Let A, B, omega, D, Lambda be as at the top of Section
5. If D has real rank zero in its quotient C*-norm, then there is
a projection p in D cap B' such that
lambda(p x) = (1/2) lambda(x) for every x in B, lambda in Lambda.
In particular A has uniform property Gamma.
Proof. Step 1: verify the hypotheses of Theorem 4.3 for the pair
A subset D. Since A is simple, infinite dimensional, unital, its
restriction of any tau in T(D) to A is a tracial state, so T(A) is
nonempty. Lemma 2.3 applies to A subset D: for every delta > 0 there
is a full projection g in D with sup_{tau in T(D)} tau(g) < delta.
Along with real rank zero of D, Theorem 4.3 applies.
Step 2: finite-set approximations. Choose increasing finite subsets
F_n subset U(A) with union dense in U(A), each containing 1. By
Theorem 4.3 there are projections P_n in D with
||[P_n, x]||_{2,tau} < 1/n, |tau(P_n x) - (1/2) tau(x)| < 1/n
for every x in F_n and every tau in T(D). (finite-stage)
Step 3: lift and diagonalize. Lift P_n to a positive contraction
sequence (c_{n, j})_j in B: first lift self-adjointly, then clip the
real line to [0, 1] by continuous functional calculus. Since P_n is a
projection,
lim_{j to omega} ||c_{n, j}^2 - c_{n, j}||_{2,T(A)} = 0.
Fix n. Lemma 5.1 applied to the sequence j |-> c_{n, j}^2 - c_{n, j},
to the sequences j |-> [c_{n, j}, x] for each x in F_n, and to the
sequences j |-> tau-bar(c_{n, j} x) - (1/2) tau-bar(x), gives an
omega-large set of coordinates j on which
||c_{n, j}^2 - c_{n, j}||_{2,T(A)} < 1/n
||[c_{n, j}, x]||_{2,T(A)} < 2/n (x in F_n)
sup_{tau in T(A)} | tau-bar(c_{n, j} x) - (1/2) tau-bar(x) | < 2/n
(x in F_n).
There are finitely many conditions; their omega-large sets intersect
in an omega-large set, nonempty. Choose one coordinate j(n) in the
intersection, and put b_n := c_{n, j(n)}.
Step 4: assemble. (b_n) is a bounded sequence of positive
contractions in B. Its projection defect, and every test error for
eventually included x, converge to zero in the ordinary sense. Hence
p := [(b_n)] in B^omega is a projection. Since every x in U(A) is
eventually in F_n, p commutes with every such x. And
lambda(p x) = (1/2) lambda(x) for every x in U(A), lambda in Lambda.
Unitary linear span gives the same for every x in A. No further
ultrapower is taken; the original omega is retained.
Step 5: extend to B. A is dense in B for the designated uniform
2-norm. Indeed, for x = [(a_j)] in B (class of an operator-bounded
uniform 2-Cauchy sequence), the estimate
||x - a_m||_{2,T(A)}^2
= sup_tau tau((a_j - a_m)*(a_j - a_m)) |_{j -> inf}
-> 0
holds as m -> inf, by uniform 2-Cauchyness. So A-dense follows.
Given x in B, choose a in A with ||x - a||_{2,T(A)} < delta. For
every lambda in Lambda, multiplication by the contraction p and
tracial Cauchy-Schwarz give
||[p, x]||_{2,lambda} <= 2 delta, | lambda(p x) - (1/2) lambda(x) |
<= (3/2) delta.
Let delta -> 0. The first gives [p, x] = 0 in B^omega, hence in D,
by Lemma 5.1 together with the uniform 2-norm detecting zero. The
second gives the trace identity. So p lies in D cap B' and satisfies
the completion form of uniform property Gamma. End. []
--------------------------------------------------
5.3 Traces on the completion
--------------------------------------------------
Corollary 5.3. Under the hypotheses of Theorem 5.2, every tracial
state of B is the designated extension of a unique trace in T(A).
Restriction is an affine homeomorphism T(B) -> T(A), inverse
tau |-> tau-bar, and
|sigma(b)| <= ||b||_{2,T(A)} for b in B, sigma in T(B).
Proof. Set X_B := { tau-bar : tau in T(A) }. By CCEGSTW, Prop 3.23
(ii)-(iv), X_B is a compact face of T(B), and the uniform 2-norm is
complete on the unit ball of B. So (B, X_B) is a factorial tracially
complete pair in the terminology of that paper. Theorem 5.2 and
CCEGSTW Prop 5.20 give property Gamma for this pair, and then
CCEGSTW Thm 1.4 gives complemented partitions of unity. Evington
[Theorem B] then gives T(B) = X_B, provided (B, X_B) is of type II_1:
each designated GNS von Neumann algebra must contain no nonzero type I
central summand.
For tau in T(A), the finite von Neumann algebra pi_tau(A)'' has no
nonzero type I central summand: otherwise there would be a nonzero
homogeneous summand M_n(Z) with Z abelian; composing the
representation of A into this summand with a character of Z gives a
unital finite dimensional representation of A; simplicity would make
it injective; contradicting infinite dimensionality. Thus every
pi_tau(A)'' is type II_1. CCEGSTW Prop 3.23(vi) gives the same for
the completions, so (B, X_B) is type II_1 and Evington's trace theorem
applies.
Then restriction T(B) -> T(A) is a continuous affine bijection
between compact Hausdorff spaces, hence a homeomorphism. Finally,
for sigma = tau-bar and b in B, tracial Cauchy-Schwarz gives
|sigma(b)| <= tau-bar(b* b)^{1/2} <= ||b||_{2,T(A)}. End. []
This completes Route 1.
===========================================================================
PART II. STRICT COMPARISON AND JIANG-SU ABSORPTION
===========================================================================
Standing hypotheses for Part II:
A is simple, separable, unital, infinite dimensional, nuclear;
T(A) nonempty (except in the traceless case treated at the end of
Section 10);
A has strict comparison in the sense of Definition 1.5;
omega is a free ultrafilter on N;
D := A^omega, the uniform tracial ultrapower of A (NOT the
completion);
Lambda := Lambda_omega, the family of limit traces on D.
GOAL. Show A has uniform property Gamma in the all-m form, and A is
Z-stable. The D here is A^omega, not B^omega. This is exactly the
algebra mentioned in Section 1.4's all-m form.
The route:
- Section 6: setup. D is unital, real rank zero is NOT assumed;
strict comparison is packaged into the form of CETW. The
diagonal principle for approximations in D is proved.
- Section 7: with strict comparison + separability, build projections
in D with prescribed spectral support and trace-zero symmetries
in projection corners.
- Section 8: turn these symmetries into coefficient-centered
selfadjoint unitaries.
- Section 9: use nuclearity (HKW) to average finitely many centered
symmetries into a single observable with bounded central moments,
and clip it.
- Section 10: adjoin these observables successively to build
commuting rows with a common Gaussian limit; use equal-measure
Gaussian intervals to split traces.
===========================================================================
SECTION 6. Comparison and the tracial ultrapower
===========================================================================
This section fixes the ultrapower D := A^omega, its limit traces, and
the diagonal principle that turns approximate coefficient tests into
exact identities.
--------------------------------------------------
6.1 Hereditary subalgebras and support
--------------------------------------------------
For a C*-algebra E, write E_+ for its positive cone, E_sa for its
selfadjoint part, and for b in E_+ set
Her(b) := closure of b E b.
Let (b - t)_+ denote the positive part of b - t 1 (computed in
E-tilde if E is nonunital). Spectral projections, including
supp(b) := 1_{(0, inf)}(b), are taken in the indicated bidual or von
Neumann algebra. All tensor products are minimal.
--------------------------------------------------
6.2 Cuntz comparison and dimension functions
--------------------------------------------------
Recall Cuntz subequivalence a <~ b in (A (x) K)_+ and the dimension
function d_tau(a) := lim_n (tau (x) Tr)(a^{1/n}). Both from Section 1.
STRICT COMPARISON (unital case, Def 1.5):
a, b in (A (x) K)_+ with b != 0,
d_tau(a) < d_tau(b) for all tau in T(A) ==> a <~ b.
If T(A) = empty, the convention demands a <~ b for every positive a
and every nonzero positive b in the stabilization.
The COMMUTED form of the CETW convention. In CETW Section 5, strict
comparison is packaged functionally: an element phi of the set
F(Cu(A)) of additive, order-preserving, supremum-preserving maps
Cu(A) -> [0, inf] vanishes at 0. The functional form says
[u] <= [v] if phi([u]) <= phi([v]) for all phi in F(Cu(A)),
and phi([u]) < phi([v]) whenever 0 < phi([v]) < inf.
The following lemma bridges this to our point-based convention.
Lemma 6.1 (compatibility of conventions). A simple unital with
T(A) nonempty and strict comparison (Definition 1.5) implies the
functional form just stated.
Proof. Every tau in T(A) gives a functional [a] |-> d_tau(a) on Cu(A)
by ERS Prop 4.2 (applied to the lower semicontinuous trace tau (x) Tr).
Only this direction of the trace-functional correspondence is used.
Let u, v in (A (x) K)_+ satisfy the functional inequalities. If v = 0,
any tau in T(A) gives d_tau(u) = 0, and the faithful stabilized trace
then gives u = 0, so u <~ v.
So suppose v != 0. Faithfulness of tau (x) Tr on the stabilized
algebra gives d_tau(v) > 0 for every tau in T(A). The functional
hypothesis gives d_tau(u) < d_tau(v) whenever d_tau(v) < inf.
Fix s > 0. Compress u by a large finite-rank corner projection to
obtain c in M_k(A) with ||u - c|| < s/2. In the tracial GNS von
Neumann representation for any tau, put P := 1_{(s, inf)}(u). Then
P c P >= (s/2) P, so the polar decomposition of c^{1/2} P has initial
projection P and final projection dominated by supp(c). Taking the
normal semifinite trace:
d_tau((u - s)_+) = tau(P) <= tau(supp(c)) <= k.
Same k for every tau. So if d_tau(v) = inf then
d_tau((u - s)_+) < d_tau(v) as well. If d_tau(v) < inf, monotonicity
d_tau((u - s)_+) <= d_tau(u) < d_tau(v) closes the case.
Now Definition 1.5 yields (u - s)_+ <~ v for every s > 0. For each
n choose comparison witnesses y_n with ||y_n* v y_n - (u - 1/n)_+|| <
1/n. Since ||u - (u - 1/n)_+|| <= 1/n, these witnesses prove u <~ v.
End. []
--------------------------------------------------
6.3 The uniform tracial ultrapower D = A^omega
--------------------------------------------------
Fix omega. For tau in T(A) put ||x||_{2, tau} := tau(x* x)^{1/2}; for
a set X subset T(A) use ||x||_{2, X} := sup_{tau in X} ||x||_{2, tau}.
Define
I := { (a_l) in l^inf(N, A) : lim_{l to omega} ||a_l||_{2, T(A)} = 0 }
D := l^inf(N, A) / I.
The tracial inequalities xy, yx, x* all obey the correct 1-Lipschitz
bound in ||.||_{2, tau}; so I is a closed two-sided *-ideal and D is a
unital C*-algebra. Faithfulness of every tau makes a |-> (a, a, ...)
injective, so we identify A with its image in D.
The LIMIT TRACES Lambda are:
(a_l) + I |-> lim_{l to omega} tau_l(a_l), tau_l in T(A).
Cauchy-Schwarz makes the formula independent of representatives;
positivity, traciality, and normalization follow coordinatewise. For
a bounded lift (a_l) of x in D:
||x||_{2, Lambda} = lim_{l to omega} ||a_l||_{2, T(A)}, (two-norm)
sup_{tau in Lambda} |tau(x)|
= lim_{l to omega} sup_{sigma in T(A)} |sigma(a_l)|. (linear-norm)
Upper bounds are from the definition. For the reverse, choose at
each coordinate a trace within 1/l of the supremum; take limits. In
particular ||x||_{2, Lambda} = 0 implies x = 0 in D. Only these
limits from Lambda are used; no description of other traces on D is
attempted.
For tau in Lambda, when tau is evaluated on D** we mean the normal
extension. It is tracial (separate continuity). Neither this
extension nor the extension from A to A** is asserted faithful on
the whole bidual. All uses below need only positivity and normality.
--------------------------------------------------
6.4 Tracial diagonal principle
--------------------------------------------------
Lemma 6.2 (tracial diagonal). Fix finitely many variables
x_1, ..., x_m. Each variable is constrained to the closed unit ball
of D, to its selfadjoint part, or to its positive part. Let
(P_k)_{k >= 1} be countably many *-polynomials in these variables with
parameters in D. For each k choose one of the tests
t_k(x) := ||P_k(x)||_{2, Lambda} or
t_k(x) := sup_{tau in Lambda} |tau(P_k(x))|,
and a constant c_k >= 0. Suppose that for every n >= 1 and every
eps > 0 there is an admissible tuple x in D^m with
t_k(x) < c_k + eps for 1 <= k <= n.
Then there is an admissible tuple x with t_k(x) <= c_k for every k.
In particular, if every c_k = 0, all the prescribed tracial equations
hold exactly. A polynomial with zero uniform 2-norm is zero in D.
Proof. STEP A (LIFTS). A positive contraction has a positive
contraction lift; a selfadjoint contraction has a selfadjoint
contraction lift; a contraction x with ||x|| <= R has a lift in the
R-ball. For the last, clip a general lift a by a f(a* a) with
f(t) := min(1, R^2 / t) on (R^2, inf) and 1 below; this does not
change the class in D.
STEP B (COUNTABLE PARAMETERS). Only countably many parameters occur
in the polynomials. Fix one bounded representative sequence for
each, once and for all. No multiplicative compatibility between the
chosen lifts is needed.
STEP C (COORDINATE TESTS). Given a lifted admissible tuple, let
t_{k, l} be the coordinate test obtained by evaluating the lifted
polynomial at coordinate l in A and replacing Lambda by T(A):
t_k(x) = lim_{l to omega} t_{k, l}.
Each coordinate sequence is bounded: it involves only finitely many
bounded parameters and variables in fixed balls.
STEP D (DIAGONAL). For each n choose an admissible trial tuple
x^(n) with t_k(x^(n)) < c_k + 1/(2 n) for k <= n, and lifts
x^(n)_{i, l}. The set
E_n := { l : t_{k, l}(x^(n)_{1, l}, ..., x^(n)_{m, l})
< c_k + 1/n for all k <= n }
belongs to omega. Let
F_n := { l >= n } cap E_1 cap ... cap E_n.
These are decreasing members of omega with empty intersection. For
each l that belongs to some F_n, let r(l) := max{ n : l in F_n } (a
finite integer, at most l), and set x_{i, l} := x^(r(l))_{i, l}. For
l not in any F_n, set x_{i, l} := 0.
STEP E (CONCLUSION). These sequences remain in their fixed balls.
If n >= k and l in F_n, then r(l) >= n and l in E_{r(l)}, so
t_{k, l}(x_{1, l}, ..., x_{m, l}) < c_k + 1 / r(l) <= c_k + 1/n.
Taking the omega-limit and letting n -> inf proves the claim. The
final "zero uniform 2-norm is zero in D" clause is (two-norm). End. []
--------------------------------------------------
6.5 Where separability is used
--------------------------------------------------
When a condition must hold for every element of A, apply it first on
a countable norm-dense subset of the appropriate ball. On fixed
balls, polynomial evaluation is uniformly norm continuous in each
coefficient, and both test seminorms are bounded by the operator norm.
For example:
||[x, a] - [x, b]||_{2, Lambda} <= 2 ||x|| ||a - b||
sup_{tau in Lambda} |tau((a - b) x)| <= ||a - b|| ||x||.
So the identities and inequalities extend from the dense subset to
every coefficient by continuity. In particular a vanishing uniform
2-norm commutator gives exact commutation in D.
--------------------------------------------------
6.6 The absorption criterion
--------------------------------------------------
Definition 6.3 (uniform property Gamma, all-m form). For simple
separable unital A with T(A) nonempty, uniform property Gamma asks:
for every m >= 1 there are projections p_1, ..., p_m in D cap A'
satisfying
sum_{i=1}^m p_i = 1, tau(a p_i) = tau(a) / m
for every a in A, tau in Lambda, 1 <= i <= m.
This is [CETW, Definition 2.1, Equation (2.1)], in the uniform
tracial ultrapower notation used here.
Theorem 6.4 (CETW absorption). Let A be simple separable unital
nuclear with T(A) nonempty, strict comparison in the functional
form, and uniform property Gamma. Then A :=: A (x) Z.
This is [CETW, Theorem 5.6, (ii) => (i)]. Lemma 6.1 makes the strict
comparison of Theorem 10.1 match the functional form. We shall
establish uniform property Gamma in Sections 7-10.
===========================================================================
SECTION 7. Rank approximation and projection supply
===========================================================================
Standing for this section: A simple unital infinite dimensional with
T(A) nonempty (Sections 7.1 uses only this). Sections 7.2-7.4 add
strict comparison in the sense of Definition 1.5, and use D = A^omega
with limit traces Lambda.
The route:
7.1 Ranks approximate trace values. For b in A_+ and delta > 0
there is a finite-matrix positive v with
tau(b) - delta < d_tau(v) <= tau(b) for every tau. Uses only
simplicity, unitality, infinite dimensionality, T(A) nonempty.
7.2 Almost projections in Hereditary Subalgebras. Comparison
places such v <~ b and then spectral cutdowns give a positive
contraction p_0 in Her(b) close to a projection with trace
close to tau(b).
7.3 Projection supply in D. Pass lifts to D: with exact spectral
support under any given positive contraction h, and with trace
tau(p) = tau((h - eta)_+).
7.4 Trace-zero symmetries in every projection corner. Immediate
from 7.3 with h = q and eta = 1/2.
--------------------------------------------------
7.1 Rank approximation of trace values
--------------------------------------------------
The rank of a positive element is d_tau(a) = (tau (x) Tr)(supp(a)).
The first task is to replace a general tau(b) by the rank of a
finite-matrix positive, keeping everything simultaneously across T(A).
Proposition 7.1. A simple unital infinite dimensional, T(A) nonempty.
For b in A_+ and delta > 0 there exist an integer L >= 1 and a positive
v in M_L(A) such that
tau(b) - delta < d_tau(v) <= tau(b) for every tau in T(A).
No bound on L is asserted, and no comparison or nuclearity is used.
Proof. Take e from Lemma 2.2 (thin full positive) with d_tau(e) < delta
for every tau in T(A).
Step 1: an exact finite identity from the fullness of e. Choose
u_i, v_i in A, 1 <= i <= n, with s := sum_i u_i e v_i invertible,
||1 - s|| < 1. Form
U := (u_1 e^{1/2}, ..., u_n e^{1/2}),
V := (e^{1/2} v_1, ..., e^{1/2} v_n)^T.
Then s = U V, so y := V* V = sum_i v_i* e v_i satisfies
||s^{-1}||^{-2} 1 <= s* s = V* U* U V <= ||U||^2 y,
hence y is invertible. Setting x_i := v_i y^{-1/2}, we get the exact
identity
sum_{i=1}^n x_i* e x_i = 1. (frame)
Step 2: transfer the trace. Define
c := sum_{i=1}^n e^{1/2} x_i b x_i* e^{1/2} in Her(e)_+.
Traciality and (frame) give, for every tau in T(A),
tau(c) = sum_i tau(b x_i* e x_i) = tau(b),
d_tau(c) <= d_tau(e) < delta. (transfer)
Step 3: integer cuts. Choose L >= max{1, ||c||} and
v := direct sum_{j=1}^L (c - j)_+ in M_L(A)_+.
For 0 <= s <= ||c||, N_L(s) := sum_{j=1}^L 1_{(j, inf)}(s) satisfies
0 <= s - N_L(s) <= 1_{(0, inf)}(s). (count)
(For nonintegral positive s the count is floor(s); at a positive
integer s it is s - 1; at s = 0 both sides are 0.) Applying normal
functional calculus and the trace, using the unnormalized matrix trace:
d_tau(v) = sum_j tau(1_{(j, inf)}(c)),
0 <= tau(c) - d_tau(v) <= d_tau(c) < delta.
Together with (transfer), tau(b) - delta < d_tau(v) <= tau(b). End. []
--------------------------------------------------
7.2 Almost projections in Hereditary Subalgebras
--------------------------------------------------
Standing now: A simple unital infinite dimensional with T(A) nonempty
and strict comparison (Def 1.5).
Lemma 7.2. For b in A_+ with ||b|| < 1 and delta > 0, there is a
positive contraction p_0 in Her(b) such that, for every tau in T(A),
tau(b) - 2 delta < tau(p_0) <= tau(b),
0 <= tau(p_0 - p_0^2) < 2 delta.
Proof. b = 0: take p_0 = 0. Otherwise take v in M_L(A)_+ from
Prop 7.1. Faithfulness of each tau and the support formula give
0 < d_tau(b), d_tau(v) <= tau(b) <= ||b|| d_tau(b) < d_tau(b).
Strict comparison in A (x) K gives v <~ b.
Step 1: a single uniform t > 0 works. The function
t |-> d_tau((v - 3 t)_+) is lower semicontinuous in tau (it is the
supremum over m in N of the continuous functions
tau |-> (tau (x) Tr)(min{1, m (v - 3 t)_+})). The open sets
U_t := { tau in T(A) : d_tau((v - 3 t)_+) > tau(b) - 2 delta }
increase as t decreases and, by Prop 7.1, cover T(A). Compactness of
T(A) gives a finite subcover; the smallest of its cut parameters t
gives
d_tau((v - 3 t)_+) > tau(b) - 2 delta for every tau. (uniform-cut)
Step 2: comparison witnesses and rank transfer. Regard b as supported
in the first corner of A (x) K and v in the first L corners; let P_1,
P_L be the corresponding corner projections. From v <~ b choose X in
A (x) K with ||X* b X - v|| < t and set
x := P_1 X P_L in M_{1, L}(A),
w := x* b x = P_L X* b X P_L in M_L(A), ||w - v|| < t.
Put alpha := ||w - v||, gamma := t - alpha > 0. In M_L(A)** put
E := 1_{(3 t, inf)}(v), F := 1_{(t, inf)}(w). The order inequalities
(w - 2 t)_+ >= w - 2 t 1 and w >= v - alpha 1 give
E (w - 2 t)_+ E >= E (v - (2 t + alpha) 1) E >= gamma E,
F v F >= F (w - alpha 1) F >= gamma F.
The following observation is used twice. Let a be positive in a von
Neumann algebra and Q a projection with Q a Q >= gamma Q for gamma > 0.
Then the polar decomposition of a^{1/2} Q has initial projection Q
(absorbs through Q a Q >= gamma Q) and final projection at most
supp(a) (its range is inside the range of a^{1/2}). Applying that
observation to the two displayed inequalities, and taking normal
traces, yields
d_tau((v - 3 t)_+) <= d_tau((w - 2 t)_+),
d_tau((w - t)_+) <= d_tau(v). (rank-xfer)
The constant gamma is independent of tau.
Step 3: switch to a hereditary subalgebra. Set a := b^{1/2} x in
M_{1, L}(A) and w' := a a* = b^{1/2} x x* b^{1/2} in Her(b)_+. In the
bidual of the linking matrix algebra, the polar decomposition a = u |a|
identifies the positive spectral projections of a* a = w and aa* = w':
1_{(s, inf)}(w') = u 1_{(s, inf)}(w) u* (s > 0).
Hence w and w' have the same cut ranks (tau (x) Tr on the w-corner;
tau on the w'-corner).
Step 4: choose the algebra. Take continuous g: [0, inf) -> [0, 1]
with g = 0 on [0, t] and g = 1 on [2 t, inf), and put p_0 := g(w').
Then p_0 in Her(b)_+ (because g(0) = 0). Scalar functional calculus:
1_{(2 t, inf)} <= g <= 1_{(t, inf)},
0 <= g - g^2 <= 1_{(t, inf)} - 1_{(2 t, inf)}.
Taking traces and using equality of cut ranks:
d_tau((w - 2 t)_+) <= tau(p_0) <= d_tau((w - t)_+),
tau(p_0 - p_0^2) <= d_tau((w - t)_+) - d_tau((w - 2 t)_+).
Combine with (rank-xfer) and (uniform-cut) to get the stated
inequalities. v = 0 is included (the rank approximation then gives
tau(b) < delta for all tau, and x = 0 is a witness). End. []
--------------------------------------------------
7.3 Projection supply with exact spectral support
--------------------------------------------------
Now D := A^omega with limit traces Lambda. Strict comparison continues.
Proposition 7.3. For a positive contraction h in D and 0 < eta < 1,
there is a projection p in D with
p <= 1_{[eta, 1]}(h) in D**, tau(p) = tau((h - eta)_+)
for every tau in Lambda. (supply)
Also, with k_eta(s) := min(s / eta, 1) on [0, 1],
k_eta(h) p = p = p k_eta(h) in D. (cutoff)
Proof. Choose positive contractive lifts (h_l)_l of h. For
b_l := (h_l - eta)_+ in A_+, ||b_l|| <= 1 - eta < 1. Lemma 7.2 with
delta := 1/l gives positive contractions p_l in Her(b_l) with
sup_{tau in T(A)} | tau(p_l) - tau(b_l) | <= 2/l,
sup_{tau in T(A)} tau(p_l - p_l^2) <= 2/l.
Let p be the class of (p_l) in D. Since p_l is a positive contraction,
0 <= (p_l - p_l^2)^2 <= p_l - p_l^2,
so the projection defect has vanishing uniform 2-norm along omega;
p = p* = p^2 in D. For an arbitrary lambda = lim_omega tau_l on
represented sequences, the first inequality gives
lambda(p) = lim_{l to omega} tau_l(b_l) = lambda((h - eta)_+).
For the support assertion, functional calculus in A** gives
supp(b_l) = 1_{(eta, 1]}(h_l), k_eta(h_l) supp(b_l) = supp(b_l).
Every element of Her(b_l) is supported on both sides by supp(b_l),
so k_eta(h_l) p_l = p_l exactly at every l. Passing to the quotient
and taking adjoints gives (cutoff).
Finally, represent D** faithfully on a Hilbert space. The identity
(1 - k_eta(h)) p = 0 says the range of p lies in the kernel of
1 - k_eta(h), whose kernel projection is
1_{{1}}(k_eta(h)) = 1_{[eta, 1]}(h).
So p has the asserted order bound in D**. This uses an exact identity
in D and did not require any normal trace on D** to be faithful. End. []
--------------------------------------------------
7.4 Trace-zero symmetries in projection corners
--------------------------------------------------
Corollary 7.4. For every projection q in D there is a selfadjoint
r in qDq with
r^2 = q, tau(r) = 0 for every tau in Lambda.
Proof. Apply Prop 7.3 to h = q and eta = 1/2. It gives a projection
p <= q with tau(p) = tau(q)/2 for every tau in Lambda. Set
r := 2 p - q in qDq. Then r = r*, and r^2 = 4 p - 4 p + q = q, and
tau(r) = 2 tau(p) - tau(q) = 0. q = 0 is trivially included. End. []
===========================================================================
SECTION 8. Scalar compressions and coefficient centering
===========================================================================
Still with A simple unital infinite dimensional with strict comparison,
T(A) nonempty, D = A^omega, Lambda limit traces.
The plan. Use Proposition 7.3 to partition a projection into blocks on
which finitely many prescribed coefficients have nearly scalar
compressions. In each block, Corollary 7.4 supplies a trace-zero
symmetry. Combining, one selfadjoint unitary centered in trace against
any prescribed separable coefficient algebra. All estimates are
uniform over Lambda.
--------------------------------------------------
8.1 Finite scalar compressions
--------------------------------------------------
Lemma 8.1. Let q be a projection in D, x_1, ..., x_n selfadjoint
contractions in D, and eps, delta > 0. There exist pairwise orthogonal
projections p_1, ..., p_L <= q and scalars mu_{s, j} in [-1, 1] with
||p_s x_j p_s - mu_{s, j} p_s|| <= eps (1 <= j <= n, 1 <= s <= L)
tau(q - sum_{s=1}^L p_s) <= delta for every tau in Lambda.
The family may be empty. No commutation of p_s with x_j is asserted.
Proof. Assume 0 < eps, delta < 1.
One coefficient x, one corner. Pick continuous f_1, ..., f_m: [-1,1] ->
[0,1] and scalars lambda_1, ..., lambda_m in [-1, 1] with
sum_j f_j = 1, supp(f_j) subset { |t - lambda_j| <= eps }.
(Piecewise linear hat functions on a fine enough subdivision.)
For a projection v in D, let f^v(v x v) denote functional calculus in
vDv (unit v; zero if v = 0). If P(t) = sum_k a_k t^k, then
P^v(v x v) = a_0 v + sum_{k>=1} a_k (v x v)^k.
Uniform modulus for changing corner. Suppose v <= u are projections
and tau in Lambda, t := tau(u - v). Since u - v is a projection:
||u - v||_{2, tau} = sqrt(t), ||u x u - v x v||_{2, tau} <= 2 sqrt(t).
The second inequality writes the difference as
(u - v) x u + v x (u - v) and uses the tracial 2-norm inequalities.
Telescope powers (all contractive):
||(u x u)^k - (v x v)^k||_{2, tau} <= 2 k sqrt(t) (k >= 1).
Therefore, with |tau(a)| <= ||a||_{2, tau} and the corner-polynomial
identity,
| tau(P^u(u x u)) - tau(P^v(v x v)) |
<= C_P sqrt(t), C_P := |a_0| + 2 sum_{k>=1} k |a_k|.
The |a_0| term records the change of corner units. For f continuous
and P within a in sup-norm:
| tau(f^u(u x u)) - tau(f^v(v x v)) |
<= 2 a + C_P sqrt(tau(u - v)). (modulus)
Independent of x, tau, and the two projections.
Choose for each j a polynomial P_j with ||f_j - P_j||_infinity <=
delta / (16 m), and set
C := max{1, C_{P_1}, ..., C_{P_m}},
alpha := delta / (4 m),
kappa := min{ delta / (4 m), (delta / (8 m C))^2 } > 0.
Then for v <= u with tau(u - v) < kappa:
| tau(f_j^u(u x u)) - tau(f_j^v(v x v)) | <= delta / (4 m)
for every j. (uniform)
One run. Starting from a residual projection u, put v_0 := u and
successively
y_j := f_j^{v_{j-1}}(v_{j-1} x v_{j-1}) in v_{j-1} D v_{j-1}.
By Prop 7.3 with cutoff alpha, get a_j in D with
a_j <= 1_{[alpha, 1]}(y_j) <= supp(y_j) <= v_{j-1},
tau(a_j) = tau((y_j - alpha)_+) >= tau(y_j) - alpha (tau in Lambda).
(The trace estimate: 0 <= y_j - (y_j - alpha)_+ <= alpha v_{j-1}.)
Set v_j := v_{j-1} - a_j. The removed a_j are mutually orthogonal.
The compression estimate. In the bidual corner v_{j-1} D** v_{j-1},
supp(y_j) lies under the spectral projection of v_{j-1} x v_{j-1} for
[lambda_j - eps, lambda_j + eps] cap [-1, 1]. Compressing the
corresponding spectral order bounds by a_j:
-eps a_j <= a_j x a_j - lambda_j a_j <= eps a_j,
||a_j x a_j - lambda_j a_j|| <= eps.
This is a norm estimate in D; no commutation with x is required.
One run removes a fixed positive trace mass. Claim: for every tau in
Lambda,
tau(u) > delta ==> tau(u - v_m) >= kappa. (progress)
Suppose tau(u) > delta and tau(u - v_m) < kappa. The partition identity
in the initial corner gives sum_j tau(f_j^u(u x u)) = tau(u) > delta, so
some j has tau(f_j^u(u x u)) > delta / m. Because v_{j-1} <= u and
tau(u - v_{j-1}) < kappa, the (uniform) bound applies at step j:
tau(a_j) >= tau(y_j) - alpha
> delta / m - delta / (4 m) - delta / (4 m) = delta / (2 m) > kappa,
contradicting a_j <= u - v_m. The witnessing index may vary with tau,
but every run processes all m indices, and the constant kappa is fixed
in advance.
Repeating. Start at q and perform K := floor(1 / kappa) + 1 runs, each
on the previous residual. Collect all removed projections as
p_1, ..., p_L, let w := q - sum_s p_s be the final residual. If some
tau in Lambda had tau(w) > delta, the residual at the start of every
run would have trace > delta at that tau, and (progress) would give
tau(q - w) >= K kappa > 1,
impossible. So tau(w) <= delta for every tau in Lambda simultaneously.
Finite construction; no compactness of Lambda is used. Every removed
projection retains its compression estimate, proving the one-coefficient
case.
Finite family. Starting with a single block q, iterate over j. At
stage j begin with L_{j-1} current blocks; apply the one-coefficient
case inside each with compression tolerance eps and trace-loss
tolerance delta / (n max{1, L_{j-1}}). Total trace lost at that stage
is <= delta / n at every trace. Compression bounds survive: if
p <= p_0 and ||p_0 x_k p_0 - mu p_0|| <= eps, then
p x_k p - mu p = p (p_0 x_k p_0 - mu p_0) p, so ||p x_k p - mu p||
<= eps. There are finitely many blocks at every stage. Summing over
n stages gives (scalar-compressions). End. []
--------------------------------------------------
8.2 Centered symmetries against finitely many coefficients
--------------------------------------------------
Lemma 8.2. For a projection q in D, a finite set H subset D, and
zeta > 0, there is r = r* in qDq with
r^2 = q, sup_{tau in Lambda} | tau(b r) | < zeta (b in H).
Proof. q = 0 or H = empty: trivial. Else put M := max{1, max_b ||b||}
and choose 0 < theta <= min{1/4, zeta / (8 M)}. Apply Lemma 8.1 with
both tolerances theta, to the real and imaginary parts of every b / M
(b in H). It gives orthogonal p_s <= q, a residual w := q - sum_s p_s,
and complex scalars nu_{s, b} with
||p_s b p_s - nu_{s, b} p_s|| <= 2 M theta,
tau(w) <= theta for every tau in Lambda.
Cor 7.4 supplies r_s = r_s* in p_s D p_s with r_s^2 = p_s,
tau(r_s) = 0. Set
r := w + sum_s r_s.
Orthogonality of the supporting p_s gives r = r* in qDq and
r^2 = w + sum_s p_s = q. For every tau in Lambda, using that the
restriction of tau to p_s D p_s has norm tau(p_s):
| tau(b r_s) | = | tau((p_s b p_s - nu_{s, b} p_s) r_s) | <= 2 M theta tau(p_s),
| tau(b w) | = | tau(w b w) | <= M tau(w) = M theta (or fewer, for w).
Summing, using sum_s tau(p_s) <= 1:
| tau(b r) | <= 2 M theta sum_s tau(p_s) + M theta
<= 3 M theta <= 3 zeta / 8 < zeta.
End. []
--------------------------------------------------
8.3 Centered symmetries against finite-dim commutants
--------------------------------------------------
Proposition 8.3. Let E subset D be a finite dimensional C*-subalgebra
(possibly nonunital), and S subset D a unital norm separable C*-
subalgebra. Then there is a selfadjoint unitary r in D cap E' with
tau(s r) = 0 for every s in S, tau in Lambda. (centered)
Proof. Adjoining 1_D to E adds only the scalar summand C (1_D - e)
when e := 1_E != 1_D (and nothing when E = 0); it preserves finite
dimensionality. It suffices to work with the resulting unital
finite-dimensional algebra. Choose matrix units
{ e^(l)_{ij} : 1 <= i, j <= n_l }, 1 <= l <= a,
sum_{l, i} e^(l)_{ii} = 1_D.
Finitely many coefficient tests. Fix a finite H subset S and xi > 0,
let N_0 := sum_l n_l. In each block, apply Lemma 8.2 inside the corner
e^(l)_{11} D e^(l)_{11} to the finite family
H_l := { e^(l)_{1 i} s e^(l)_{i 1} : s in H, 1 <= i <= n_l },
with tolerance xi / N_0. Let u_l := the resulting corner symmetry.
Put
R := sum_{l = 1}^a sum_{i = 1}^{n_l} e^(l)_{i 1} u_l e^(l)_{1 i}.
Every summand is selfadjoint, and the matrix-unit relations give
R^2 = 1_D, R e^(l)_{jk} = e^(l)_{j 1} u_l e^(l)_{1 k} = e^(l)_{jk} R.
Terms from distinct blocks have orthogonal supports. So R is a
selfadjoint unitary commuting with E. For every s in H and tau in
Lambda, cyclicity:
tau(sR) = sum_{l} sum_i tau(e^(l)_{1 i} s e^(l)_{i 1} u_l),
| tau(sR) | < N_0 xi / N_0 = xi.
Diagonal assembly. Choose a norm-dense sequence (s_k)_{k >= 1} in the
unit ball of S and apply Lemma 6.2 to one variable in the selfadjoint
unit ball of D, with the tests
||r^2 - 1_D||_{2, Lambda} = 0,
||[r, e^(l)_{ij}]||_{2, Lambda} = 0 (all (l), (i, j)),
sup_{tau in Lambda} | tau(s_k r) | = 0 (k >= 1).
Every finite list is satisfiable to arbitrary precision by the
unitaries R just constructed (the square and commutation relations
already hold exactly for those trials). Lemma 6.2 gives r = r* in D
with all tests. Since the uniform 2-norm detects zero in D, r^2 = 1_D
and [r, e^(l)_{ij}] = 0 in D. So r is a selfadjoint unitary in D cap E'.
Finally, for s in S, choose k with ||s - s_k|| < xi:
| tau((s - s_k) r) | <= ||s - s_k|| ||r|| < xi.
So tau(s r) = 0 for the entire unit ball of S, and by scaling for all
of S. End. []
Summary. Proposition 8.3 supplies EXACT commutation with E and EXACT
coefficient centering over S. It does not assert commutation with S.
Section 9 exploits this gap: choose symmetries successively, each
centered against the algebra generated by its predecessors.
===========================================================================
SECTION 9. Nuclear averaging and bounded conditional moments
===========================================================================
Throughout: A simple separable unital infinite dimensional nuclear with
strict comparison and T(A) nonempty. D := A^omega, Lambda := limit
traces.
Goal. Assemble from many centered symmetries ONE selfadjoint z in
D cap A' with the following central moments:
tau(b z) = 0, tau(b z^2) = tau(b), tau(z^4) <= 3
for every b in A and tau in Lambda, and with small commutators
[z, a] for a in a prescribed finite set E. This z feeds into the
row-construction of Section 10. Three steps:
9.1-9.2 Replace a CP approximation by a homomorphism, without losing
too much trace.
9.3 Average finitely many centered symmetries into z.
9.4 Clip to a fixed norm radius.
9.5-9.6 Transfer local assertions in A to separable coefficient
subalgebras of D.
--------------------------------------------------
9.1 HKW approximation (quoted)
--------------------------------------------------
Theorem 9.1 (HKW). Let N be a nuclear C*-algebra, K subset N finite,
eps > 0. There are a finite-dimensional C*-algebra F and CP
contractive maps N -> F -> N such that ||phi psi(x) - x|| < eps for
every x in K, and phi = sum_{i=1}^M beta_i phi_i with beta_i >= 0,
sum beta_i = 1, and each phi_i of order zero.
The number M is finite for the chosen approximation, but is not
bounded independently of K, eps. This is where the CHAIN of
centered symmetries in Prop 9.3 acquires a length that depends on
nuclearity; the norm of z in that proposition grows with M.
We record without proof the following consequence: the approximation
may be written sum_i beta_i phi_i psi_i with the OUTGOING map
phi_i: F_i -> N of order zero. Simply take F_i := F, psi_i := psi.
--------------------------------------------------
9.2 Replacing a CP order-zero map by a homomorphism on a projection
--------------------------------------------------
Recall the finite-dimensional case of the order-zero structure
theorem [WZ, Theorem 3.3 and Corollary 4.2]: for CP contractive
order-zero phi: F -> D with h := phi(1_F), there is a supporting
*-homomorphism pi in the bidual D** with
phi(x) = h pi(x) = pi(x) h, h commutes with pi(F), pi(1_F) = supp(h).
Lemma 9.2 (homomorphism replacement). F finite dimensional, phi: F -> D
CP contractive order zero, h := phi(1_F). For 0 < eta < 1, there is a
possibly nonunital *-homomorphism rho: F -> D such that, writing
P := rho(1_F), for every tau in Lambda,
tau(P) >= tau(h) - eta,
||phi(c) - rho(c)||_{2, tau}^2 <= 4 (1 - tau(h)) + 2 eta
for every c in F with ||c|| <= 1.
Proof.
Support in D**. Put q := supp(h) in D**. Since for any projection e
in F the positives phi(e) and phi(1_F - e) are orthogonal, phi(e)
commutes with their sum h. Projections span F linearly, so h commutes
with phi(F). For t > 0 put
pi_t(x) := (h + t 1)^{-1/2} phi(x) (h + t 1)^{-1/2}
= (h + t 1)^{-1} phi(x).
This is CP contractive, since pi_t(1_F) = h(h + t 1)^{-1} <= 1. For a
projection e with orthogonality of phi(e) and phi(1_F - e),
pi_t(e) = phi(e) (phi(e) + t 1)^{-1} -> supp(phi(e))
strongly as t -> 0. Because projections span F linearly, pi_t has a
pointwise strong limit pi: F -> D**, CP, pi(1_F) = q, and pi takes
projections to projections with range in qD**q. As a UCP map into
that corner, pi is multiplicative: equality in the Schwarz inequality
at a projection e (Schwarz applied to e + z x for all z in C) forces
pi(e x) = pi(e) pi(x) and pi(x e) = pi(x) pi(e); taking linear
combinations gives multiplicativity. Also
h pi_t(x) = phi(x) - t pi_t(x) -> phi(x) (norm),
so h pi(x) = phi(x); commutation with h passes to the strong limit:
phi(x) = h pi(x) = pi(x) h, pi(1_F) = supp(h) = q. (oz-supp)
The zero map is included with q = 0.
Replacing phi by a homomorphism on a projection. Write
F = direct sum_alpha M_{n_alpha} and choose matrix units
e^(alpha)_{st}. Work in one block; omit the superscript.
Set a := phi(e_{11}). Prop 7.3 with eta (the same eta) gives a
projection p in D with
p <= 1_{[eta, 1]}(a) <= pi(e_{11}),
tau(p) = tau((a - eta)_+) for every tau in Lambda.
The last inequality follows from pi(e_{11}) = supp(a). Define on [0, 1]:
k(s) := min(s / eta, 1),
g(s) := k(s) / s on (0, 1], g(0) := 1 / eta.
Both are continuous, k(s) = s g(s). The spectral containment of p
gives k(a) p = p k(a) = p. Then the elements
v_s := pi(e_{s1}) k(a) = phi(e_{s1}) g(a) in D
are in D, though pi(e_{s1}) was only in D**. (The equality uses
pi(e_{s1}) a = phi(e_{s1}) from (oz-supp).) Define
rho(e_{st}) := v_s p v_t* = pi(e_{s1}) p pi(e_{1t}) in D.
Since p <= pi(e_{11}), multiplication in matrix-unit fashion gives
rho(e_{st}) rho(e_{uv}) = delta_{tu} rho(e_{sv}),
rho(e_{st})* = rho(e_{ts}).
Different blocks have orthogonal products. So these formulas extend
to a *-homomorphism rho: F -> D. Put
P := rho(1_F), P <= q, P pi(x) = pi(x) P = rho(x) (x in F).
This uses just support containment of p; it does not require p to
commute with a, nor P with h.
Unit-trace loss. Fix tau in Lambda, use the normal extension to D**.
In each block, scalar functional calculus gives
0 <= a - (a - eta)_+ <= eta pi(e_{11}),
0 <= tau(a) - tau(p) <= eta tau(pi(e_{11})).
Traciality and the partial isometries pi(e_{s1}) imply that
phi(e_{ss}), rho(e_{ss}), and pi(e_{ss}) have the same traces as a, p,
and pi(e_{11}), respectively. Summing over all diagonal entries of
all blocks:
0 <= tau(h) - tau(P) <= eta tau(q) <= eta. (unit-loss)
The loss is measured by tau(q), so no matrix-size factor appears.
Coefficient error. For a contraction c in F, (oz-supp) and
commutation with pi(c) give
||phi(c) - pi(c)||_{2, tau}^2 <= tau((q - h)^2) <= tau(q - h)
<= 1 - tau(h),
||rho(c) - pi(c)||_{2, tau}^2 <= tau(q - P) <= 1 - tau(h) + eta.
Here q - h is a positive contraction and q - P a projection, and both
commute with pi(c). Squared triangle inequality finishes. End. []
--------------------------------------------------
9.3 An observable with exact coefficient moments
--------------------------------------------------
Proposition 9.3. Let E subset A be a finite set of selfadjoint
contractions, and 0 < eta < 1. Then there is a selfadjoint z in D
such that, for every tau in Lambda,
||[z, a]||_{2, tau}^2 <= 72 eta (commutators)
tau(b z) = 0, tau(b z^2) = tau(b) (b in A) (coefficient)
tau(z^4) <= 3. (fourth)
Construction: z = sum_{i=1}^M sqrt(beta_i) r_i with r_i selfadjoint
unitaries, beta_i >= 0, sum beta_i = 1. In fact
tau(z^4) <= 3 - 2 sum_i beta_i^2. No norm bound on z independent of
M is asserted.
Proof. Apply Theorem 9.1 to
{1_A} union E union { a^2 : a in E }
with error eta. In the finite-family notation, put T_i := phi_i psi_i,
T_0 := sum_i beta_i T_i; then ||T_0(x) - x|| < eta on this set. Regard
outgoing maps as into D and set h_i := phi_i(1_{F_i}). Fix one tau in
Lambda throughout the weighted estimates.
Unit defect. From positivity and contractivity, T_i(1_A) <= h_i <= 1,
so
sum_i beta_i (1 - tau(h_i)) <= tau(1 - T_0(1_A)) < eta. (unit-def)
One-step error. For a in E, Schwarz for T_i gives T_i(a)^2 <=
T_i(a^2). Expansion and traciality:
sum_i beta_i ||a - T_i(a)||_{2, tau}^2
<= tau(a^2) - 2 tau(a T_0(a)) + tau(T_0(a^2))
<= 3 eta. (three)
Each trace term is bounded by the norm of its argument, using
||a|| <= 1.
Replace phi_i by a homomorphism. Lemma 9.2, with the same eta,
gives rho_i: F_i -> D for each i. Using (unit-def) and
Lemma 9.2:
sum_i beta_i ||T_i(a) - rho_i psi_i(a)||_{2, tau}^2
<= 4 sum_i beta_i (1 - tau(h_i)) + 2 eta <= 6 eta. (six)
Second triangle inequality with (three) and (six):
sum_i beta_i ||a - rho_i psi_i(a)||_{2, tau}^2
<= 2 (3 eta) + 2 (6 eta) = 18 eta. (eighteen)
Successive centering. Choose symmetries r_i one at a time. At step i
apply Prop 8.3 with
E_i := rho_i(F_i), S_i := C^*(A, r_1, ..., r_{i-1}).
S_i is unital and norm separable; E_i is finite dimensional and may
be nonunital. Get
r_i = r_i*, r_i^2 = 1, r_i in E_i',
tau(s r_i) = 0 for every s in S_i, tau in Lambda. (centering)
Because r_i commutes with rho_i psi_i(a) and multiplying by a unitary
preserves the tracial 2-seminorm,
||[r_i, a]||_{2, tau} <= 2 ||a - rho_i psi_i(a)||_{2, tau}.
Thus (eighteen) gives
sum_i beta_i ||[r_i, a]||_{2, tau}^2 <= 72 eta (a in E). (seventy-two)
Every weighted estimate was obtained at the SAME fixed tau; the
constants do not depend on tau.
Cancel cross terms. Set z := sum_i sqrt(beta_i) r_i. If a word
contains r_j exactly once and other letters lie in S_j, cyclicity
writes its tau as tau(s r_j) for some s in S_j; it vanishes. For
i < j, each of the four terms of [r_i, a]* [r_j, a] has this form
(a, r_i in S_j); the reverse mixed term is its adjoint. So
||[z, a]||_{2, tau}^2 = sum_i beta_i ||[r_i, a]||_{2, tau}^2 <= 72 eta.
For b in A: the same argument gives tau(b r_i) = 0 and tau(b r_i r_j) = 0
for i != j. Diagonal terms use r_i^2 = 1, sum beta_i = 1:
tau(b z) = 0, tau(b z^2) = tau(b). (coeff)
Nothing in this requires the r_i to commute.
Fourth moment. Put z_i := sum_{j <= i} sqrt(beta_j) r_j (so z_0 = 0).
At step i set X := z_{i-1}, r := r_i, c := sqrt(beta_i). Expand
tau((X + c r)^4) and group only after the trace:
tau((X + c r)^4) = tau(X^4) + 4 c tau(X^3 r)
+ c^2 (4 tau(X^2) + 2 tau(X r X r))
+ 4 c^3 tau(X r) + c^4.
Bookkeeping: the four words with one r are cyclic permutations of
X^3 r; the four with three r reduce by cyclicity and r^2 = 1 to X r; of
the six with two r, four have trace tau(X^2), and two are alternating.
Terms with one or three r vanish by (centering). Tracial
Cauchy-Schwarz:
| tau(X r X r) | <= ||X r||_{2, tau} ||r X||_{2, tau} = tau(X^2).
And tau(X^2) = sum_{j < i} beta_j by (coeff) applied to the partial
sum. Hence
tau(z_i^4) <= tau(z_{i-1}^4) + 6 beta_i sum_{j < i} beta_j + beta_i^2.
Summing over i:
tau(z^4) <= 6 sum_{j < i} beta_i beta_j + sum_i beta_i^2
= 3 (sum beta_i)^2 - 2 sum_i beta_i^2 = 3 - 2 sum beta_i^2 <= 3. []
Finally, ||z|| <= sum sqrt(beta_i) <= sqrt(M) - not uniform in M.
This is exactly why Section 9.4 clips z.
--------------------------------------------------
9.4 Tracial clipping
--------------------------------------------------
Lemma 9.4 (clipping). z = z* in D, R > 0, d := g_R(z) with
g_R(s) := max(-R, min(s, R)). For every a in D, tau in Lambda:
||[d, a]||_{2, tau} <= ||[z, a]||_{2, tau} (comm)
and for every contraction b in D:
| tau(b (d - z)) | <= R^{-3} tau(z^4), (tail)
| tau(b (d^2 - z^2)) | <= R^{-2} tau(z^4), (tail2)
tau(d^4) <= tau(z^4). (fourth)
In particular, if z is from Prop 9.3, then ||d|| <= R and, for every
tau in Lambda:
||[d, a]||_{2, tau} <= sqrt(72 eta) (a in E)
| tau(b d) | <= 3 / R^3 (b in A, ||b|| <= 1)
| tau(b (d^2 - 1)) | <= 3 / R^2 (b in A, ||b|| <= 1)
tau(d^4) <= 3.
Proof. Fix tau and let L^2(D, tau) be the tracial L^2 Hilbert space.
Left and right multiplication by z give commuting bounded selfadjoint
operators L_z and R_z; continuous functional calculus carries g_R to
L_d and R_d. Let E_z be their joint spectral measure, and mu_a the
finite positive measure with mu_a(U) := < E_z(U) a, a >. Since g_R is
1-Lipschitz,
||[d, a]||_{2, tau}^2
= int_{R^2} |g_R(s) - g_R(t)|^2 dmu_a(s, t)
<= int_{R^2} |s - t|^2 dmu_a(s, t)
= ||[z, a]||_{2, tau}^2.
This proves (comm).
Scalar inequalities. For real s,
|s - g_R(s)| <= |s|^4 / R^3,
|s^2 - g_R(s)^2| <= |s|^4 / R^2,
|g_R(s)|^4 <= |s|^4.
(Check s <= R and s > R separately.) Functional calculus gives (tail)
and (fourth) after multiplying by a contraction b. The detail: for
contractive b,
| tau(b x) | <= tau(|x|) (norm-contractive coefficient),
which follows from the polar decomposition x = u|x| in D**, with y :=
|x|^{1/2} and c := b u:
|tau(b x)| = |tau(y c y)| <= tau(y^2)^{1/2} tau(y c* c y)^{1/2}
<= tau(y^2) = tau(|x|).
So (tail) and (fourth) hold without a commutation assumption on b.
Combining with Prop 9.3 gives the four displayed inequalities. End. []
--------------------------------------------------
9.5 Local bounded moments in A
--------------------------------------------------
Prop 9.5. Fix R >= 1. For every finite E subset A of selfadjoint
contractions, finite H subset A of contractions, and sigma > 0, there
is a selfadjoint d in A with ||d|| <= R such that, simultaneously,
||[d, a]||_{2, T(A)} < sigma (a in E)
sup_{tau in T(A)} | tau(b d) | <= 3 / R^3 + sigma
(b in H)
sup_{tau in T(A)} | tau(b (d^2 - 1)) | <= 3 / R^2 + sigma
(b in H)
sup_{tau in T(A)} tau(d^4) <= 3 + sigma.
Proof. Choose 0 < eta < 1 so sqrt(72 eta) < sigma / 2. Props 9.3
and 9.4 give d-bar = d-bar* in D, ||d-bar|| <= R, satisfying
(clipped). Choose a selfadjoint lift of d-bar to l^inf(N, A) and
apply g_R at every coordinate; the resulting lift (d_l) is still a
lift of d-bar and is selfadjoint with ||d_l|| <= R at every l. By
Lemma 6.3 and the trace-test identity in Lemma 6.2,
lim_{l to omega} ||[d_l, a]||_{2, T(A)} = ||[d-bar, a]||_{2, Lambda} < sigma/2,
lim_{l to omega} sup_{tau in T(A)} |tau(b d_l)| = sup_{tau in Lambda} |tau(b d-bar)| <= 3/R^3,
lim_{l to omega} sup_{tau in T(A)} |tau(b(d_l^2 - 1))| <= 3/R^2,
lim_{l to omega} sup_{tau in T(A)} tau(d_l^4) <= 3.
There are only finitely many tests (E, H, one fourth moment). For
each test, the desired bound in the statement holds on a set in omega,
using the strict margin sigma/2 on commutators and positive margin
sigma on moments. Their finite intersection is nonempty and in omega.
Any coordinate in it supplies the required d. End. []
Note. For fixed R, the "residual" coefficient errors are exactly
3 / R^3 and 3 / R^2; only the extra error sigma is arbitrarily small.
The norm R is independent of the nuclear approximation and of M.
This is the feature Section 9.6 needs.
--------------------------------------------------
9.6 Transfer to separable coefficient algebras in D
--------------------------------------------------
Prop 9.6. Let C subset D be a unital norm separable C*-subalgebra,
and fix R >= 1. There is a selfadjoint d in D cap C' with ||d|| <= R
such that, for every b in C with ||b|| <= 1 and every tau in Lambda,
| tau(b d) | <= 3 / R^3,
| tau(b (d^2 - 1)) | <= 3 / R^2,
tau(d^4) <= 3.
Proof. Choose norm-dense sequences (c_k) in the unit ball of C,
(x_k) in its selfadjoint unit ball, and separately chosen contraction
and selfadjoint-contraction lifts c_{k, l}, x_{k, l} in A. No
multiplicative lifting of C is needed.
At coordinate l, apply Prop 9.5 to
H_l := { c_{1, l}, ..., c_{l, l} },
E_l := { x_{1, l}, ..., x_{l, l} },
with tolerance 1/l and the same fixed R at every coordinate. Get
d_l = d_l* in A, ||d_l|| <= R, with
||[d_l, x_{k, l}]||_{2, T(A)} < 1/l
sup_{rho in T(A)} | rho(c_{k, l} d_l) | <= 3/R^3 + 1/l
sup_{rho in T(A)} | rho(c_{k, l} (d_l^2 - 1)) | <= 3/R^2 + 1/l
sup_{rho in T(A)} rho(d_l^4) <= 3 + 1/l
for every k <= l. Since Prop 9.5 permits arbitrary finite subsets
of A, the dependence on l is harmless. The common bound R makes
(d_l) a bounded sequence; let d in D be its class.
For a fixed k, the displayed estimates hold for all l >= k. Taking
ultralimits along omega (using Lemma 6.3 for commutators and Lemma 6.2
for the moment tests),
||[d, x_k]||_{2, Lambda} = 0,
| tau(c_k d) | <= 3 / R^3,
| tau(c_k (d^2 - 1)) | <= 3 / R^2,
tau(d^4) <= 3
for every tau in Lambda. Since the uniform 2-norm detects zero in D,
[d, x_k] = 0; norm continuity and linearity give d in C'. Continuity
bounds
||[d, x - y]||_{2, Lambda} <= 2 R ||x - y||,
|tau((b - c) d)| <= R ||b - c||,
|tau((b - c)(d^2 - 1))| <= (R^2 + 1) ||b - c||
extend the two coefficient bounds from (c_k)_k to the entire unit ball
of C, uniformly over Lambda. Nuclearity was used in obtaining the
LOCAL assertion in A (Prop 9.5); this transfer places no nuclearity
hypothesis on C. End. []
Summary. With Prop 9.6, we can now, for any separable unital C subset
D, find a selfadjoint d in D cap C' with controlled first, second, and
fourth central coefficient moments. Section 10 uses this to build
commuting rows.
===========================================================================
SECTION 10. A conditional central limit argument and uniform Gamma
===========================================================================
Standing: A simple separable unital infinite dimensional nuclear with
strict comparison and T(A) nonempty (until 10.4, where the traceless
case is treated). D := A^omega, Lambda := limit traces.
Plan.
10.1 Build, for each N, a commuting row d_{N,1}, ..., d_{N,N} in
D cap A' with controlled central coefficient moments.
10.2 Show that the sum s_N := N^{-1/2} sum_j d_{N,j} has a Gaussian
limit in every coefficient trace, uniformly over Lambda and A.
10.3 Cut the Gaussian at m equal-measure quantiles. Use the
diagonal principle to extract exact central projections.
10.4 Conclude uniform property Gamma and Z-stability (unital case),
then the traceless case.
--------------------------------------------------
10.1 A conditional central limit construction
--------------------------------------------------
Proposition 10.1. For every N >= 1, there exist pairwise commuting
selfadjoint elements d_{N,1}, ..., d_{N,N} in D cap A', each of norm at
most N (that is ||d_{N,j}|| <= N), with the following property. Put
C_{N,j} := C^*(A, d_{N,1}, ..., d_{N,j-1}),
s_N := N^{-1/2} sum_{k=1}^N d_{N,k}.
For every contraction b in C_{N,j} and every tau in Lambda:
| tau(b d_{N,j}) | <= 3 / N^3,
| tau(b (d_{N,j}^2 - 1)) | <= 3 / N^2,
tau(d_{N,j}^4) <= 3. (row-moments)
For every FIXED t in R,
sup_{tau in Lambda, a in A, ||a|| <= 1}
| tau(a e^{i t s_N}) - e^{-t^2 / 2} tau(a) | -> 0 (char-funcs)
as N -> inf.
Proof. Fix N. At step j, apply Prop 9.6 with C := C_{N,j} and R := N.
C_{N,j} is unital and norm separable (A is separable; only finitely
many elements have been adjoined). Prop 9.6 gives d_{N,j} in
D cap C_{N,j}', hence in D cap A', with ||d_{N,j}|| <= N, satisfying
(row-moments). (Commutation with A is inherited from C_{N,j} cont aining
A; commutation with previous d_{N,k} from d_{N,j} in C_{N,j}'.) So the
row is commuting. This constructs each row separately.
Now the Gaussian limit. Fix t in R, tau in Lambda, contraction a in A.
Put
P_0 := 1, P_j := prod_{k=1}^j e^{i t d_{N,k} / sqrt(N)},
u_j := tau(a P_j), q_N := 1 - t^2 / (2 N).
At step j, the coefficient b := a P_{j-1} belongs to C_{N,j} and has
norm <= 1. Write d := d_{N,j}. Scalar Taylor with integral remainder:
| e^{ix} - 1 - ix + x^2 / 2 | <= |x|^3 / 6 (x in R).
Functional calculus followed by the tracial inequality
| tau(b y) | <= tau(|y|) for ||b|| <= 1 (from 9.4 (comm)'s
polar-decomposition argument) gives
| tau(b e^{i t d / sqrt(N)}) - tau(b) - (i t / sqrt(N)) tau(b d)
+ (t^2 / (2 N)) tau(b d^2) |
<= |t|^3 / (6 N^{3/2}) tau(|d|^3)
<= 3^{3/4} |t|^3 / (6 N^{3/2}) (taylor)
The last inequality is Holder's inequality for the probability measure
associated with d at tau, together with tau(|d|^3) <= tau(d^4)^{3/4}
and (row-moments). Inserting the two conditional moment bounds from
(row-moments) into (taylor), for every 1 <= j <= N:
|u_j - q_N u_{j-1}| <= eps_N(t), where (rec)
eps_N(t) := 3^{3/4} |t|^3 / (6 N^{3/2}) + 3 |t| / N^{7/2}
+ 3 t^2 / (2 N^3).
Every bound here holds uniformly over tau in Lambda and contractions
a in A.
Telescope. For N >= t^2 / 4, |q_N| <= 1, so
u_N - q_N^N u_0 = sum_{j=1}^N q_N^{N-j} (u_j - q_N u_{j-1}),
|u_N - q_N^N tau(a)| <= N eps_N(t).
Since the elements in a row commute, P_N = e^{i t s_N}. Hence
sup_{tau in Lambda, ||a|| <= 1}
| tau(a e^{i t s_N}) - e^{-t^2 / 2} tau(a) |
<= 3^{3/4} |t|^3 / (6 sqrt(N)) + 3 |t| / N^{5/2}
+ 3 t^2 / (2 N^2)
+ | (1 - t^2 / (2 N))^N - e^{-t^2 / 2} |
-> 0. (clt)
This proves (char-funcs). Coefficients a P_{j-1} are exactly why the
moment bounds against the whole earlier algebra C_{N,j} were needed.
No independence of the d_{N,j}, nor equality of their distributions,
is assumed. End. []
Note. Each s_N is a SINGLE element of D cap A'. The available bound
||s_N|| <= N^{3/2} is sufficient; (s_N)_N is not used as a
representative sequence in an ultrapower.
--------------------------------------------------
10.2 Uniform weak convergence
--------------------------------------------------
Let gamma be the standard Gaussian probability measure on R, so
dgamma(x) = (1 / sqrt(2 pi)) e^{-x^2 / 2} dx,
int_R e^{i t x} dgamma(x) = e^{-t^2 / 2}.
Lemma 10.2. For every FIXED positive contraction a in A and every
FIXED bounded continuous f: R -> C,
lim_{N to inf} sup_{tau in Lambda}
| tau(a f(s_N)) - tau(a) int_R f dgamma | = 0. (weak-conv)
Proof. Suppose the conclusion fails. Then ||f||_inf > 0 (else nothing
to prove), and there are delta > 0, integers N_k -> inf, and traces
tau_k in Lambda such that
| tau_k(a f(s_{N_k})) - tau_k(a) int f dgamma | >= delta. (fail)
The positive functional x |-> tau_k(a^{1/2} x a^{1/2}) has norm
tau_k(a) and agrees with x |-> tau_k(a x) by traciality. So the left
side of (fail) is at most 2 ||f||_inf tau_k(a). In particular
tau_k(a) >= c := delta / (2 ||f||_inf) > 0.
On the compact spectrum of s_{N_k}, define a probability measure mu_k
on R by
int_R h dmu_k := tau_k(a h(s_{N_k})) / tau_k(a),
for every bounded continuous h: R -> C.
By (char-funcs), for every fixed t in R:
| int e^{i t x} dmu_k(x) - e^{-t^2 / 2} |
<= (1 / c) sup_{tau in Lambda}
| tau(a e^{i t s_{N_k}}) - e^{-t^2 / 2} tau(a) |
-> 0.
Dobler's Levy continuity theorem [Dobler, Thm 1.1] gives mu_k -> gamma
weakly: the limiting characteristic function e^{-t^2/2} is that of a
probability measure and is continuous at zero. So int f dmu_k ->
int f dgamma. Multiplying by 0 < tau_k(a) <= 1 contradicts (fail).
The argument allows the testing trace to vary with k, so it proves the
asserted uniformity. It also accommodates the varying spectra of
s_{N_k}; no common spectral bound is needed for the continuity theorem.
End. []
--------------------------------------------------
10.3 Uniform property Gamma via Gaussian partitions
--------------------------------------------------
Theorem 10.3 (uniform property Gamma). For every integer m >= 1 there
exist pairwise orthogonal projections p_1, ..., p_m in D cap A' with
sum_{i=1}^m p_i = 1, tau(a p_i) = tau(a) / m
for every a in A, tau in Lambda, 1 <= i <= m.
In particular A has uniform property Gamma.
Proof. m = 1 is trivial (p_1 = 1). So assume m >= 2.
Gaussian quantiles. Choose -inf = q_0 < q_1 < ... < q_{m-1} < q_m =
+inf with gamma(I_i) = 1 / m for each interval I_i = (q_{i-1}, q_i).
(Assignment of the finite endpoints is immaterial; gamma has no atoms.)
Fix a finite H subset of positive contractions in A and eps > 0. Choose
theta > 0 with
2 theta < eps, 17 theta / 16 < eps^2.
Choose pairwise disjoint open intervals about the finitely many
quantiles q_1, ..., q_{m-1} whose union U satisfies gamma(U) < theta.
Interpolate linearly across U to get continuous f_1, ..., f_m: R -> [0, 1]
with
sum_i f_i = 1, f_i = 1_{I_i} on R minus U.
Near q_i, only f_i and f_{i+1} vary: one descends from 1 to 0, the
other its complement. Two scalar estimates:
| int f_i dgamma - 1/m | < theta, (mass-err)
int (f_i - f_i^2)^2 dgamma < theta / 16. (proj-err)
The second uses (u - u^2)^2 <= 1/16 for 0 <= u <= 1, and the integrand
vanishes outside U.
Approximate the split. Keep f_1, ..., f_m fixed. Apply Lemma 10.2 to
every pair (a, f_i) for a in H, and to (1, (f_i - f_i^2)^2) for every
i. For all N sufficiently large, every one of these finitely many
convergence errors is < theta. Set v_i := f_i(s_N). These are
positive contractions in D cap A', and sum_i v_i = 1. Combine
(mass-err), (proj-err), and (weak-conv):
sup_{tau in Lambda} | tau(a v_i) - tau(a)/m | < 2 theta < eps,
||v_i - v_i^2||_{2, Lambda}^2
= sup_{tau in Lambda} tau((f_i - f_i^2)^2(s_N))
< 17 theta / 16 < eps^2.
So any finite list of trace-splitting and projection tests can be met
arbitrarily accurately by m positive contractions that already commute
with A and sum to one.
Diagonal. Choose a countable norm-dense family (a_k)_{k >= 1} in the
positive unit ball of A. Apply Lemma 6.2 to the m variables each
constrained to the positive contraction ball of D, with the countably
many tests
||p_i^2 - p_i||_{2, Lambda} = 0, (1 <= i <= m)
|| sum_{i=1}^m p_i - 1 ||_{2, Lambda} = 0,
||[p_i, a_k]||_{2, Lambda} = 0, (all i, k)
sup_{tau in Lambda} | tau(a_k p_i - a_k / m) | = 0, (all i, k).
The preceding finite construction verifies the hypothesis of Lemma 6.2.
The sum-to-one equation is included among the tests, regardless of the
chosen positive contraction lifts of any trial tuple. Lemma 6.2 gives
m positive contractions p_i satisfying every test.
Interpretation. Since the uniform 2-norm detects zero in D, the p_i
are projections; they commute with A (extending from (a_k)_k by
continuity); and sum to 1. Pairwise orthogonality: for each i,
0 = p_i(1 - p_i) p_i = sum_{j != i} p_i p_j p_i
= sum_{j != i} (p_j p_i)* (p_j p_i),
positivity forces p_j p_i = 0 for j != i. Trace equations extend from
the dense (a_k)_k to positive contractions by continuity, and by
scaling and linearity to all of A. So (p_i)_i witnesses uniform
property Gamma in the all-m form. End. []
--------------------------------------------------
10.4 Z-stability and the traceless case
--------------------------------------------------
Theorem 10.4 (main unital comparison theorem). A simple, separable,
unital, infinite dimensional, nuclear, with strict comparison (Def 1.5).
Then A :=: A (x) Z. If T(A) is nonempty, the proof constructs uniform
property Gamma directly.
Proof, case T(A) nonempty. Theorem 10.3 gives uniform property Gamma.
Lemma 6.1 converts the comparison convention. Theorem 6.4 (CETW) then
gives A :=: A (x) Z.
Proof, case T(A) = empty. The comparison convention says: EVERY
positive in A (x) K is Cuntz-subequivalent to EVERY nonzero positive in
A (x) K. We show A is purely infinite, then invoke Kirby-Phillips.
Step 1: every nonzero Her subalgebra contains an infinite projection.
Given 0 != b in A_+, apply the comparison convention to b as target and
1 as source: 1 <~ b in (A (x) K)_+. Comparison witnesses can be taken
in A: put everything in the corner e := 1_A (x) e_{11} of A (x) K, and
compress every stabilized witness z_n to x_n := e z_n e. Then
x_n* b x_n = e z_n* b z_n e -> e,
which is x_n* b x_n -> 1 after identifying the corner with A. Choose
x in A making t := x* b x invertible, and put
w := b^{1/2} x t^{-1/2}.
Then w* w = 1, and
p := w w* = b^{1/2} x t^{-1} x* b^{1/2} in Her(b)
is a projection Murray-von Neumann equivalent to 1.
Step 2: the identity 1 is infinite. There are orthogonal nonzero
b, c in A: for a selfadjoint element h not scalar, take continuous
functions vanishing near two distinct points of sp(h) and use them.
For THIS b and c, the above projection p in Her(b) annihilates c
(because p in Her(b) and c orthogonal to b), so p != 1. Thus 1 is
infinite.
Step 3: pure infiniteness. For any nonzero Her subalgebra E of A,
choose 0 != h in E_+ and repeat Step 1 with b := h to get a projection
p_h in Her(h) subset E equivalent to 1. This p_h is infinite inside E:
an isometry w_h with w_h* w_h = 1 and w_h w_h* = p_h identifies A with
p_h A p_h subset E, and carries a proper subprojection of 1 (equivalent
to 1) to a proper subprojection of p_h (equivalent to p_h). So A is
purely infinite (hereditary-subalgebra definition).
Step 4: Z-stability from pure infiniteness. Kirchberg-Phillips
[KP, Theorem 3.14] gives A (x) O_inf :=: A. O_inf is strongly
self-absorbing [TW, Examples 1.14(ii)], and Winter's theorem
[WinterSSA, Theorem 3.1] gives O_inf (x) Z :=: O_inf. Associativity
of the minimal tensor product:
A (x) Z :=: (A (x) O_inf) (x) Z
:=: A (x) (O_inf (x) Z)
:=: A (x) O_inf :=: A.
No Universal Coefficient Theorem hypothesis is used. Both cases are
complete. End. []
This completes Part II.
===========================================================================
PART III. NONUNITAL ALGEBRAS WITH BOUNDED TRACES
===========================================================================
Standing hypotheses for Part III:
A is a nonzero simple, separable, nuclear C*-algebra;
A (x) K has no nonzero projections (stably projectionless);
every trace in T_lsc(A) is bounded on the positive unit ball;
the set T_1(A) := { tau in T_lsc(A) : ||tau|| = 1 } is
nonempty and weak-* compact;
comparison hypothesis: for a, b in (A (x) K)_+, b != 0,
d_tau(a) < 1 for every tau in T_lsc(A) with d_tau(b) = 1
==> a <~ b. (comparison)
omega is a free ultrafilter on N.
Set T := T_1(A). Every NONZERO densely finite trace is a positive
scalar multiple of a unique element of T. Simplicity makes T
faithful. Projectionlessness implies that A is non-elementary and has
no nonzero finite dimensional quotient.
Notation. ||x||_{2, sigma} := sigma(x* x)^{1/2}, with the subscript a
set of traces meaning supremum. Tr is the unnormalized matrix trace.
Her(b) := closure of b C b in the ambient C*-algebra C.
GOAL OF PART III. Show that A has uniform property Gamma and is
Z-stable. In this setting there is no natural unit and no projection
supply, so the strategy splits into four stages:
S1 (Section 11) Build D := A^omega. Show D is unital, identify
its limit traces Lambda, and verify density/lifting facts.
S2 (Section 12) Given a target trace function on Lambda and an
ambient "support", select a projection p in D with h p = p
and tau(p) prescribed. Add a one-sided cost constraint.
S3 (Sections 13-14) Build arbitrarily small central splittings
with errors of order lambda^{3/2}, then amplify to any
proportion c in (0, 1) and then to exact projections.
S4 (Section 15) Transfer to nonunital Z-stability.
===========================================================================
SECTION 11. The ultrapower D := A^omega and preliminary inputs
===========================================================================
--------------------------------------------------
11.1 The unital tracial quotient
--------------------------------------------------
Define
D := A^omega := l^inf(A) / I_T,
I_T := { (a_n) in l^inf(A) : lim_{n to omega} ||a_n||_{2, T} = 0 }.
The tracial inequalities make I_T a closed two-sided *-ideal; D is a
C*-algebra. We regard A as a subalgebra of D through constant
sequences; simplicity and faithfulness of members of T make this
injective.
Lemma 11.1 (unit). D is unital. Lambda := the limit traces on D of
the form lambda([(a_n)]) := lim_omega sigma_n(a_n) with sigma_n in T
consists of tracial states, and every lambda in Lambda restricts to an
element of T. Moreover,
||[(a_n)]||_{2, Lambda} = lim_{n to omega} ||a_n||_{2, T},
and this norm detects zero in D (i.e., ||u||_{2, Lambda} = 0 implies
u = 0).
Proof. Choose an increasing sequential contractive approximate unit
(u_n) for A. The continuous functions sigma |-> sigma(u_n) increase
to 1 on the compact Hausdorff space T; by Dini's theorem they converge
UNIFORMLY to 1. In a tracial GNS representation, for a bounded
sequence (a_n),
||(1 - u_n) a_n||_{2, sigma}^2
<= ||a_n||^2 sigma((1 - u_n)^2) <= ||a_n||^2 (1 - sigma(u_n)),
and the same estimate holds with a_n on the left. Because uniform
convergence of sigma |-> sigma(u_n) holds, (u_n) is a bounded sequence
in A representing a class in D that acts as a left and right identity
for A:
[(u_n)] [(a_n)] = [(a_n)], [(a_n)] [(u_n)] = [(a_n)].
So [(u_n)] = 1_D, D is unital. Under every lambda in Lambda the class
1_D takes the value 1 by uniform convergence of sigma_n(u_n) on T;
restriction to A has norm one by the same argument plus normalization
of the sigma_n. The norm identity is by choosing at each coordinate a
trace within 1/n of the supremum; its "detects zero" clause is the
definition of I_T. End. []
From now on D is unital and A subset D is a (nonunital) inclusion with
A dense in D in the (Lambda) 2-norm along omega (Lemma 11.1).
Consequence (uniformity of moments). Bounds on finitely many
continuous expressions in the LIMIT-trace values of a BOUNDED tuple
of elements of D extend, up to arbitrary positive slack, to uniform
coordinate bounds along omega. Specifically: if some (finite) list of
continuous functions of the limit-trace values is known to be bounded
by c, then for any eps > 0 the corresponding coordinate functions are
bounded by c + eps on an omega-large set. This is the contrapositive
of the definition of limit trace.
Order-zero lifts. A completely positive map is ORDER ZERO when it
preserves orthogonality of positive elements. All order-zero maps
below are contractions. Matrix order-zero maps lift across arbitrary
C*-quotients [CETW, Proposition 1.8]. A homomorphism from a finite
direct sum of matrix algebras into a quotient therefore has an
order-zero lift (this is standard; see [CETW, proof of Lemma 4.4]).
--------------------------------------------------
11.2 GNS von Neumann algebras and hyperfiniteness
--------------------------------------------------
For sigma in T, set M_sigma := pi_sigma(A)'' (the tracial GNS von
Neumann algebra). Kaplansky density gives a SURJECTION
l^inf(A) -> M_sigma^omega. (kaplansky)
Surjectivity. Approximate the n-th entry of a bounded sequence in
M_sigma by an A-element in the same ball to within 2-norm 1/n; also
for the unit of M_sigma.
Consequence: the inclusion of any finite dimensional unital subalgebra
of M_sigma into M_sigma^omega has a coordinatewise order-zero lift
to A.
Nuclearity + Connes. Connes' injectivity theorem makes M_sigma
hyperfinite, with separable predual, for every sigma in T (including
nonextreme traces); see also [CETW, proof of Lemma 4.5]. Notation:
for a positive b in an ambient C*-algebra C, Her(b) := closure of b C b.
===========================================================================
SECTION 12. Selecting projections with a prescribed trace
===========================================================================
--------------------------------------------------
12.1 Rank approximation by finite matrices
--------------------------------------------------
Given g in A_+ and xi > 0, produce a positive k in a finite matrix
algebra over A whose trace UNDER EACH sigma in T is within xi of
sigma(g), and whose rank is only a small excess over its trace.
Lemma 12.1 (rank). For g in A_+ and xi > 0, there is a positive
contraction k in some M_l(A) with, for every sigma in T,
| sigma(k) - sigma(g) | < xi,
0 <= d_sigma(k) - sigma(k) < xi.
Proof. Step 1: find 0 != b in A_+ with d_sigma(b) < xi for all
sigma in T. Take an extreme point sigma of the compact convex T; its
tracial GNS M_sigma is a factor, faithful. If M_sigma were finite
dimensional then M_sigma = M_n (C) for some n, and (by faithfulness)
A would be finite dimensional, contradiction. So M_sigma is a
II_1 factor, and contains unital copies of M_l for every l. By
(kaplansky) and order-zero lifting, there is a nonzero order-zero
map theta: M_l -> A for every l. The images theta(e_{ii}) of the
diagonal matrix units are orthogonal projections in A that are
equally weighted in every tracial representation (equivalent). Their
sum has trace <= 1, so each has trace <= 1/l. Set b := theta(e_{11}).
Take l > 1/xi to ensure d_sigma(b) <= 1/l < xi.
Step 2: transfer the trace to Her(b). By simplicity, the ideal
generated by b is all of A. Approximate g^{1/2} in that ideal by
U D_b V with U a row, V a column, and D_b := diag(b, ..., b). Then
(U D_b V)* (U D_b V) is within xi of g in the appropriate trace sense.
Set
C := (D_b U* U D_b)^{1/2} in M_n(Her(b)), v := C V.
Each entry v_j belongs to closure of b A. Therefore
c := sum_j v_j v_j* in Her(b)_+, with sigma(c) = sigma(v* v) by
traciality, and (from the approximation) | sigma(c) - sigma(g) | < xi.
Step 3: integer cuts. Choose N >= ||c||. Let f_j(s) := min(1,
max(0, s - j + 1)) on [0, inf]. Set k := direct sum_{j=1}^N f_j(c) in
M_N(A).
On the spectrum of c the functions sum to s, while
0 <= sum_{j=1}^N 1_{{f_j(s) > 0}} - s <= 1_{{s > 0}}.
So d_sigma(k) - sigma(k) <= d_sigma(c) <= d_sigma(b) < xi and
sigma(k) = sigma(c). End. []
--------------------------------------------------
12.2 Projection supply with exact support
--------------------------------------------------
The comparison hypothesis in this section enters only through finding
a comparison-witness inside a prescribed hereditary subalgebra.
Section 12.2 patches together pieces of a projection using a "small
auxiliary matrix corner" to satisfy a strict rank comparison at each
stage.
Proposition 12.2 (selection). Let h, z, a be positive contractions in
D with h z = z and tau(a) <= tau(z) for every tau in Lambda. Then
there is a projection p in D with
h p = p, tau(p) = tau(a) for every tau in Lambda.
Proof. The construction proceeds coordinate-by-coordinate along
omega. Choose positive contraction lifts h_n, z_n, a_n in A. Set
alpha_n := sup_{sigma in T} ( sigma(a_n) - sigma(z_n) )_+,
beta_n := ||(1 - h_n) z_n||_{2, T}.
Both quantities vanish along omega (from h z = z and tau(a) <= tau(z)).
Choose slow-decay 0 < delta_n < 1 with beta_n / delta_n -> 0 along
omega, and put
gamma_n := alpha_n + beta_n / delta_n + delta_n,
H_n := (h_n - 1 + delta_n)_+.
H_n is a positive contraction in A with
supp(H_n) = 1_{[1 - delta_n, 1]}(h_n).
In a tracial representation, with Q := supp(H_n), the scalar bound
delta_n (1 - Q) <= 1 - h_n holds, so, applying Cauchy-Schwarz,
sigma(z_n) <= sigma(Q) + sigma((1 - Q) z_n) <= d_sigma(H_n) + beta_n / delta_n.
Therefore sigma(a_n) <= d_sigma(H_n) + gamma_n for every sigma in T.
Rank approximation. Lemma 12.1 with input a_n and tolerance gamma_n
gives k_n in M_l(A). A second call, with input 8 gamma_n u (u a fixed
positive contraction in A with inf_T sigma(u) > 7/8), gives c_n with
5 gamma_n <= d_sigma(c_n) <= 10 gamma_n for every sigma.
Thus, in DISJOINT finite matrix blocks, via additivity of d_sigma,
d_sigma(k_n) <= d_sigma(H_n) + 3 gamma_n < d_sigma(H_n + c_n).
The target b := H_n + c_n is nonzero; all ranks on both sides are
finite at every sigma (k_n finite matrix; c_n finite matrix; H_n in
A). The comparison hypothesis in (comparison) gives k_n <~ b.
Cutdown with trace control. Write b := H_n + c_n. Choose x with
||x* b x - k_n|| < eta_n, where 0 < eta_n -> 0 along omega (Cuntz
comparison permits arbitrary positive tolerance), and set
d := (k_n - 1/2)_+^{1/2}, y := b^{1/2} x d.
Then |y|^2 >= d^2/4, so in the polar decomposition y = v |y| in the
bidual, the support of d lies under v* v (from d^2 <= 4 |y|^2) and v d
belongs to the closed right ideal supported by b. Set
w := sqrt(2) v d, w* w = 2 (k_n - 1/2)_+,
r := w w* in Her(b), 0 <= r <= 1.
For f(s) := 2 max(s - 1/2, 0) on [0, 1]:
0 <= s - f(s) <= 1_{{s > 0}} - s,
0 <= f(s) - f(s)^2 <= 1_{{s > 0}} - f(s).
Traciality transfers these estimates from w* w = f(k_n) to
r_n := w w*, so | sigma(r_n) - sigma(a_n) | -> 0 and
sigma(r_n - r_n^2) -> 0 uniformly in sigma (both along omega).
Compress to the H-block. The bidual r_n controlled sigma(r_n) and
sigma(r_n - r_n^2), but r_n itself is in the bidual of the ambient
matrix algebra, and the class (r_n) in l^inf(A) is not available.
The positive contraction in A is instead
Y_n := 2 y y* (in P_H M_N(A) P_H identified with A),
which satisfies 0 <= Y_n <= 1 because ||y* y|| = ||d x* b x d||
<= (1/2) ||x* b x|| <= 1/2. Its trace mirrors sigma(r_n):
sigma(Y_n) = 2 sigma(d x* b x d)
= 2 sigma(d k_n d) + 2 sigma(d (x* b x - k_n) d)
= 2 sigma((k_n - 1/2)_+ k_n) + O(eta_n),
which equals sigma(a_n) up to O(sqrt(gamma_n)) along omega
(projection-nearness of k_n bounds sigma(k_n - (2k_n - 1)_+ k_n)
by O(sqrt(gamma_n)) using the small-mass argument on
[(1/2) + eps, 1 - eps] with eps := sqrt(gamma_n)).
Projection defect. Comparing Y_n to the bidual r_n,
Y_n - r_n = 2 v (d k_n d - d^2 + d (x* b x - k_n) d) v* =: e_n,
and sigma(|e_n|) <= 2 sigma((1 - k_n)(k_n - 1/2)_+) + 2 eta_n -> 0
along omega; since ||r_n|| <= 1,
|sigma(Y_n - Y_n^2) - sigma(r_n - r_n^2)| <= 4 sigma(|e_n|) -> 0,
so sigma(Y_n - Y_n^2) -> 0 also. Compressing to the H-block:
p_n := P_H Y_n P_H in A,
and the complementary corner is supported by c_n, so
sigma((1 - P_H) Y_n (1 - P_H)) <= 2 sigma(c_n) <= 20 gamma_n,
|tr((1 - P_H) Y_n P_H)| <= ||c_n||_1^{1/2} ||2 x d^2 x*||
<= O(sqrt(gamma_n)).
Hence |sigma(p_n) - sigma(a_n)| -> 0 uniformly in sigma. The defect
identity
p_n - p_n^2 = P_H (Y_n - Y_n^2) P_H + P_H Y_n (1 - P_H) Y_n P_H,
with tr(P_H Y_n (1 - P_H) Y_n P_H) <= O(sqrt(gamma_n)), gives
sigma(p_n - p_n^2) <= sigma(Y_n - Y_n^2) + O(sqrt(gamma_n)) -> 0.
h-coupling. On the H-block, 1 - h_n <= delta_n (operator order,
from H_n = (h_n - 1 + delta_n)_+). So
||(1 - h_n) p_n|| <= delta_n -> 0 along omega.
Conclusion. p := [(p_n)] in D satisfies the properties: as a
projection, its positive-contraction defects have vanishing trace,
hence vanishing uniform 2-norm; h p = p, tau(p) = tau(a) on Lambda.
End. []
--------------------------------------------------
12.3 Corner cost selection
--------------------------------------------------
Lemma 12.3. Let q be a projection in D, Y in (qDq)_+, 0 < t < 1,
eps > 0. Then there is a projection r <= q with, for every tau in
Lambda,
tau(r) = t tau(q), (cost-trace)
tau(r Y) <= t ( tau(Y) + eps tau(q) ). (cost-Y)
Proof. If q = 0, take r := 0. If Y = 0, apply Prop 12.2 with
(h, z, a) := (q, q, t q). Scaling and tolerance reduction give
0 <= Y <= q, so all functions below are in the unital corner qDq.
Choose a large integer m. Let F_0 := 0, F_m := 1. For 1 <= j < m,
define F_j as: equal 1 up to (j - 1/2)/m, linear down to 0 at j/m,
zero beyond. Define G_j (1 <= j < m) as: equal 1 up to j/m, linear
down to 0 by (j + 1/3)/m, zero beyond; G_m := 1. Then
G_j F_j = F_j, F_{j+1} G_j = G_j. (nest)
Construct orthogonal r_j inductively:
G_j r_j = r_j, tau(r_j) = t tau(F_j - F_{j-1}). (step)
If R := sum_{i < j} r_i, then F_j R = G_j R = R and
tau(R) = t tau(F_{j-1}). The triple
h := G_j - R, z := F_j - R, a := t (F_j - F_{j-1})
consists of positive contractions with h z = z and
tau(z) - tau(a) = (1 - t) tau(F_j) >= 0. Prop 12.2 gives a
projection orthogonal to R, below q, fixed by G_j.
Trace. r := sum_j r_j has trace t tau(q) (telescoping).
Spectral-support transfer. Each r_jYr_j <= ((j + 1/3)/m) r_j, while
F_j - F_{j-1} is supported on [(j - 3/2)/m, 1] (roughly). The gap
between the two thresholds is at most 2/m, so
tau(r Y) <= t sum_j ((j + 1/3)/m) tau(F_j - F_{j-1})
<= t ( tau(Y) + (2/m) tau(q) ).
Take 2/m < eps. End. []
Corollary 12.4 (matrix-summand cost). Let E subset D be unital finite
dimensional, Y in D_+, 0 < t < 1, eps > 0. Then there is a projection
p in D cap E' with, for every tau in Lambda,
tau(p d) = t tau(d) (d in E),
tau(p Y) <= t ( tau(Y) + eps ).
Proof. Apply Lemma 12.3 to each matrix summand M_n of E: in the corner
e_{11} D e_{11} of the summand, take Y' := n^{-1} sum_j e_{1j} Y e_{j1}
and use the resulting projection v; replicate it along the diagonal via
matrix units to give a projection p_M in D cap M_n'. Add up over the
summands. The unit of sum over summands equals 1_D. p is a projection
in D cap E', and tau(p d) = t tau(d) for d in E by proportionality of
each replicated piece. The cost (cost-Y) scales by n per summand,
giving the single t tau(Y) + t eps. End. []
Summary. In this nonunital setting we can:
* build small projections inside any hereditary subalgebra (Prop 12.2);
* prescribe one-sided cost, in any corner (Lemma 12.3);
* do so in a finite dimensional commutant (Cor 12.4).
Section 13 assembles these into small central splittings with error
O(lambda^{3/2}).
===========================================================================
SECTION 13. Small central splittings from finite models
===========================================================================
Standing: A, T, D, Lambda as at the top of Part III.
Goal. Construct a positive contraction e in A that approximately
splits every coefficient trace in a small proportion lambda, with
trace and projection defects of order lambda^{3/2}. This is the
"small-splitting engine" that Section 14 will amplify.
Notation. F subset A finite selfadjoint contractions; a "model" below
is a finite-dimensional unital subalgebra of D together with controlled
approximations of every x in F.
--------------------------------------------------
13.1 The small-splitting statement
--------------------------------------------------
Proposition 13.1. Let 0 < lambda < 1, F a finite set of selfadjoint
contractions in A, rho > 0. Then there is a positive contraction e
in A such that, for every sigma in T and every x in F,
||[e, x]||_{2, sigma} < rho, (comm)
| sigma(e x) - lambda sigma(x) | < lambda^{3/2} + rho, (linear)
sigma(e - e^2) < lambda^{3/2} + rho. (proj)
The error lambda^{3/2} is intrinsic to this construction; Section 14
accumulates O(1/lambda) of them to get O(sqrt(lambda)) error, then
diagonalizes.
--------------------------------------------------
13.2 Averaged finite-dimensional models
--------------------------------------------------
The first half of the proof: approximate the tuple (x)_{x in F} by
models. The number of models may grow with the accuracy, but only
their WEIGHTED total error enters the construction.
Proposition 13.2. Let F subset A be a finite set of selfadjoint
contractions and Delta > 0. Then there exist unital finite dimensional
subalgebras D_i subset D, selfadjoint contractions x^(i) in D_i for
each x in F, and weights w_i > 0 with sum_i w_i = 1, such that with
E_i := sum_{x in F} (x - x^(i))^2,
one has
sum_i w_i tau(E_i) < Delta for every tau in Lambda. (weighted)
Proof.
Parametrize the candidate models. Consider all finite dimensional
unital E^0, order-zero maps theta: E^0 -> A, and selfadjoint
contractions b_x in E^0 (one per x in F). For such a triple (E^0,
theta, (b_x)), define
L(sigma) := 1 - sigma(theta(1))
+ sum_{x in F} sigma((x - theta(b_x))^2) (sigma in T).
These are nonnegative continuous affine functions on the compact
convex T.
At each sigma, L(sigma) can be made arbitrarily small. Hyperfiniteness
of M_sigma supplies a unital finite dimensional subalgebra of M_sigma
approximating the tuple (x) in the tracial L^2 norm by selfadjoint
contractions (Kaplansky density reduces to bounded elements). The
inclusion M_l -> M_sigma^omega has an order-zero lift to A by the
argument in 11.2. Taking one coordinate of a lift that is close to
the tuple makes L(sigma) small including the unit-defect term
1 - sigma(theta(1)).
Convex combinations have small supremum. Fix kappa > 0. Suppose for
contradiction that every finite convex combination f of the functions
L has max_{sigma in T} f(sigma) >= kappa. Then the convex hull of
the set {L} is disjoint from the open convex set
O := { g in C(T, R) : max_T g < kappa }.
By Hahn-Banach separation there is a nonzero bounded linear functional
phi on C(T, R) with phi <= phi_on_O on the hull of {L}. Since O is
closed under adding ARBITRARY negative functions, phi is positive:
if not, adding a strongly negative function would push < phi, f >
arbitrarily far down avoiding the separation inequality. So phi
normalizes to a probability measure mu on T.
Barycenter. Let sigma_mu be the barycenter of mu in compact convex T.
Affinity of each L gives L(sigma_mu) = int L dmu >= phi(f)/||phi||
>= kappa. But L(sigma_mu) can be made arbitrarily small by choosing
the model at sigma_mu (previous paragraph). Contradiction. Hence for
kappa sufficiently small there is a finite convex combination of the
L with sup < kappa.
Denote the resulting data by weights w_i, algebras E^0_i, order-zero
maps theta_i: E^0_i -> A, elements b_{i, x} in E^0_i.
Replacing order-zero maps by homomorphisms on projections. Now view
each theta_i in D (rather than A). Write theta_i = h_i pi_i with
h_i := theta_i(1) in D and pi_i a supporting homomorphism in D**
(Section 9.2). Define continuous functions
f(s) := (2 s - 1)_+, g(s) := min(1, 2 s) on [0, 1].
Then g f = f. Apply Prop 12.2 to the triple
( g(theta_i(e_{11})), f(theta_i(e_{11})), f(theta_i(e_{11})) )
in the e_{11}-corner of a matrix summand of E^0_i. Get a projection
v_i in D with
g(theta_i(e_{11})) v_i = v_i,
tau(v_i) = tau(f(theta_i(e_{11}))) for every tau in Lambda.
Note v_i lies under pi_i(e_{11}) because f <= 1_{[1/2, 1]} is supported
on the support of theta_i(e_{11}) <= pi_i(e_{11}).
The elements pi_i(e_{j 1}) g(theta_i(e_{11})) = pi_i(e_{j 1}) g(h_i)
belong to D (using the structure theorem (oz-supp) and the functional
calculus containment pi(e_{j1})f(h) in D); hence pi_i(e_{j 1}) v_i is
in D. Replicating v_i along each diagonal and summing over the summands
gives a projection Q_i in D commuting with pi_i(E^0_i) and satisfying
Q_i pi_i(d) in D for d in E^0_i. Traciality and the replication give
tau(Q_i) = tau(f(h_i)) for every tau in Lambda.
The model in D. Set
D_i := C (1 - Q_i) + Q_i pi_i(E^0_i),
x^(i) := Q_i pi_i(b_{i, x}).
This is a unital finite dimensional subalgebra of D; zero image
summands are simply absent. No commutation of Q_i with h_i is
required.
The triangle estimate. For a contraction c in E^0_i,
tracial L^2 contractivity (Q_i is a projection, x |-> Q_i x Q_i is a
contraction, and 1 - f(s) <= 2(1 - s)) gives
|| theta_i(c) - Q_i pi_i(c) ||_{2, tau}
<= || 1 - h_i ||_{2, tau} + || 1 - Q_i ||_{2, tau}
<= (1 + sqrt(2)) tau(1 - h_i)^{1/2}.
Set m := |F| and C_0 := (1 + sqrt(2))^2. Combining:
tau(E_i) <= 2 sum_{x in F} tau((x - theta_i(b_{i, x}))^2)
+ 2 m C_0 tau(1 - theta_i(1)).
Averaging the above with weights w_i and using the convex choice of
(w_i):
sum_i w_i tau(E_i) <= (2 + 2 m C_0) kappa.
Taking kappa < Delta / (2 + 2 m C_0) closes the proof. End. []
--------------------------------------------------
13.3 Truncation and combined cost
--------------------------------------------------
Each model gets one projection, commuting with that model, with cost
and coefficient error controlled together. The projection sum need
not be a contraction; a block-matrix truncation controls both.
Lemma 13.3 (truncation). Let p_1, ..., p_N be projections in a unital
C*-algebra with a tracial state tau. Put
s := sum_{i=1}^N p_i, e := min(s, 1),
lambda := sum_{i=1}^N tau(p_i),
R := sum_{i=1}^N tau(p_i sum_{j < i} p_j).
For every selfadjoint x in the algebra:
||[e, x]||_{2, tau}
<= 2 ( 2 sum_{i=1}^N ||[p_i, x]||_{2, tau}^2 )^{1/2}, (comm)
tau(s - e), tau(e - e^2) <= sqrt(2 lambda R). (trace)
Proof. Block matrix representation. Use the unnormalized matrix
trace and set
W := (p_1, ..., p_N),
V := [[0, W], [W*, 0]],
Z := diag(x, ..., x).
Then l(V) := min(V^2, 1) has first diagonal corner e. The function
l is 2-Lipschitz on the reals. By finite-dimensional scalar functional
calculus on the pair (L_V, R_V) (left and right multiplication by V,
two commuting bounded selfadjoints on the tracial L^2 space),
||[l(V), Z]||_2 <= 2 ||[V, Z]||_2. The squared norm on the right
before the factor 4 is 2 sum_i ||[p_i, x]||_{2, tau}^2. Compressing
the left side to the first diagonal corner gives (comm).
Trace and projection defect. Let S be the support of s in the tracial
von Neumann representation. The row W is supported on the right by
diag(p_1, ..., p_N), so the polar decomposition of W gives
tau(S) <= lambda. Since tau(s^2) = lambda + 2 R,
tau((s - S)^2) = 2 R - lambda + tau(S) <= 2 R.
Both s - e and e - e^2 are positive and supported under S, and are
bounded above by |s - S|. Cauchy-Schwarz:
tau(|s - S|) <= tau(S)^{1/2} tau((s - S)^2)^{1/2} <= sqrt(2 lambda R).
Also 0 <= e - e^2 <= |s - S| by the same reasoning (e - e^2 = e(1 - e)
is supported under S AND bounded by |s - S| componentwise). This
gives (trace). End. []
The key point: neither bound in Lemma 13.3 depends on the NUMBER of
models N, only on lambda and R.
--------------------------------------------------
13.4 Proof of Proposition 13.1
--------------------------------------------------
We work inside D. Fix K > 1 and set Delta := K^{-2}. Apply Prop
13.2 to get models D_i, elements x^(i) in D_i, weights w_i.
Choose projections p_i recursively. Put s_0 := 0, and at step i set
t_i := lambda w_i,
Y_i := s_{i - 1} + K E_i,
s_j := sum_{i' <= j} p_{i'},
and apply Corollary 12.4 (matrix-summand cost) to the unital
finite-dimensional algebra D_i subset D, the positive Y_i, the
proportion t_i, and the tolerance K^{-1}. This produces p_i in
D cap D_i' with, for every tau in Lambda,
tau(p_i d) = t_i tau(d) (d in D_i), (cost-comm)
tau(p_i Y_i) <= t_i ( tau(Y_i) + K^{-1} ). (cost-Y)
In particular tau(p_i) = t_i and tau(p_i x^(i)) = t_i tau(x^(i)).
Energy and overlap. For each tau in Lambda, set
R_tau := sum_i tau(p_i s_{i-1}), H_tau := sum_i tau(p_i E_i).
Both are nonnegative. By (cost-Y) and the weighted bound from
Prop 13.2:
R_tau + K H_tau
<= sum_i t_i ( tau(s_{i-1}) + K tau(E_i) + K^{-1} )
<= lambda^2 / 2 + lambda / K + lambda / K
= lambda^2 / 2 + 2 lambda / K. (joint)
Here we used tau(s_{i-1}) = sum_{j < i} t_j, sum_{j < i} w_j <= 1,
sum_i w_i tau(E_i) < Delta = K^{-2}, and sum_i t_i K^{-1} = lambda/K.
In particular
H_tau <= lambda^2 / (2 K) + 2 lambda / K^2, (H-bound)
uniformly in tau and independent of the number of models.
Coefficient commutator. Since [p_i, x^(i)] = 0 by (cost-comm),
[p_i, x] = [p_i, x - x^(i)] for every x in F and every i.
The elementary bound below, applied with p := p_i and y := x - x^(i)
and summed over i, produces the coefficient-commutator estimate.
A useful elementary bound: for any projection p and selfadjoint
y = y* in a tracial algebra,
||[p, y]||_{2, tau}^2 <= 2 tau(p y^2). (comm-elem)
Indeed, [p, y] = p y (1 - p) - (1 - p) y p, and the two summands
occupy orthogonal corner pairs in the tracial L^2 space, so
||[p, y]||_{2, tau}^2 = 2 ||p y (1 - p)||_{2, tau}^2
= 2 tau(y p y (1 - p)) <= 2 tau(y p y) <= 2 tau(p y^2),
using traciality, positivity of 1 - p, and lastly
tau(y p y) = ||p^{1/2} y p^{1/2}||_{2, tau}^2 <= ||p^{1/2} y||^2
= tau(p y^2)
(contractivity of right multiplication by p^{1/2} in tracial
2-seminorm). Applied with p := p_i and y := x - x^(i) (selfadjoint
contraction), and using [p_i, x^(i)] = 0:
sum_i ||[p_i, x]||_{2, tau}^2
= sum_i ||[p_i, x - x^(i)]||_{2, tau}^2
<= 2 sum_i tau(p_i (x - x^(i))^2)
<= 2 H_tau.
Linear coefficient testing. The positive functional z |-> tau(p_i z)
has norm t_i, so Cauchy-Schwarz gives
|tau(p_i(x - x^(i)))| <= sqrt(t_i) sqrt(tau(p_i (x - x^(i))^2)).
Summing over i and x in F and using scalar Cauchy-Schwarz:
sum_{i, x} |tau(p_i(x - x^(i)))|
<= sqrt(lambda) sqrt(sum_{i, x} w_i tau(p_i (x - x^(i))^2))
= sqrt(lambda H_tau).
Combining with (cost-comm) at the second coefficient x = x^(i) and the
weighted bound on models:
| tau(s x) - lambda tau(x) |
<= sqrt(lambda H_tau) + lambda / K. (sum-moment)
Truncate. Let e_D := min(s, 1). Lemma 13.3, with x selfadjoint,
gives
||[e_D, x]||_{2, tau} <= 4 sqrt(H_tau),
tau(s - e_D), tau(e_D - e_D^2) <= sqrt(2 lambda R_tau).
Replacing s by e_D in (sum-moment) costs tau(s - e_D) at each tau. So
| tau(e_D x) - lambda tau(x) |
<= sqrt(lambda H_tau) + lambda / K + sqrt(2 lambda R_tau).
(e_D-moment)
Error estimates as K -> inf. From (joint),
R_tau <= lambda^2/2 + 2 lambda/K and
H_tau <= lambda^2/(2K) + 2 lambda/K^2. Hence
sqrt(lambda H_tau) -> 0 (K -> inf)
sqrt(2 lambda R_tau) -> lambda^{3/2} (K -> inf)
lambda / K -> 0.
Only the R_tau term contributes to the constant; the H_tau term is
arbitrarily small. Consequently (e_D-moment) yields
|tau(e_D x) - lambda tau(x)| <= lambda^{3/2} + eps' with eps' -> 0 as
K -> inf; similarly (comm) yields ||[e_D, x]||_{2, tau} <= eps'' with
eps'' -> 0 (in fact 4 sqrt(H_tau) -> 0); and (proj) is bounded by
sqrt(2 lambda R_tau) -> lambda^{3/2}.
Choose K large enough that the resulting errors have slack smaller
than rho.
Pass from D to A. Take a positive contraction lift of e_D in
l^inf(A). Apply continuous clipping to [0, 1] at every coordinate
(preserving the class in D). The resulting sequence is a bounded
lift of e_D by positive contractions. Along omega-large sets its
tracial 2-norm commutator with x, its coefficient tau(e x), and its
projection defect tau(e - e^2) all converge to the corresponding
limits (Lemma 11.1 and the trace-test identity). Take one coordinate
in the intersection of finitely many omega-large sets (there are only
finitely many tests, one per x in F). The resulting single positive
contraction e in A satisfies (comm), (linear), (proj) with slack
strictly less than rho. End. []
===========================================================================
SECTION 14. Amplification to uniform property Gamma
===========================================================================
Section 13 produced splittings with intrinsic error lambda^{3/2}. Fix
a target proportion c in (0, 1). We repeat the small construction
O(1/lambda) times, each on the remaining complement, so accumulated
errors scale as O(sqrt(lambda)). Letting lambda -> 0 and diagonalizing
along omega then produces exact projections.
--------------------------------------------------
14.1 Relative small-splitting
--------------------------------------------------
Lemma 14.1 (relative small-splitting). For every separable unital
C*-subalgebra E subset D and every 0 < lambda < 1, there is a positive
contraction e in D cap E' such that, for every tau in Lambda,
| tau(e x) - lambda tau(x) | <= lambda^{3/2} (x in E_sa, ||x|| <= 1),
tau(e - e^2) <= lambda^{3/2}.
Proof. Fix a norm-dense sequence (x_k)_{k >= 1} in the self-adjoint
unit ball of E; write x_{k, n} in l^inf(A) for a self-adjoint
contraction representative. At coordinate n, apply Prop 13.1
(small-splitting) to the finite set F_n := { x_{1, n}, ..., x_{n, n} },
with target lambda and slack rho := 1/n. This gives a positive
contraction e_n in A such that, for every x in F_n and every sigma in T,
||[e_n, x]||_{2, sigma} < 1/n,
| sigma(e_n x) - lambda sigma(x) | < lambda^{3/2} + 1/n,
sigma(e_n - e_n^2) < lambda^{3/2} + 1/n.
The sequence (e_n)_n is bounded (all entries are positive
contractions), so it defines a class e in D. For a fixed k, the
displayed estimates hold for all n >= k. Taking lim_omega (using
Lemma 11.1 for trace comparison and 2-norm detection):
||[e, x_k]||_{2, Lambda} = 0,
| tau(e x_k) - lambda tau(x_k) | <= lambda^{3/2} (tau in Lambda),
tau(e - e^2) <= lambda^{3/2}.
Commutators with the dense sequence (x_k)_k of zero uniform 2-norm
give exact commutation, so e in E'. Norm continuity extends the
trace inequality from the dense self-adjoint unit ball to the whole
ball, and scaling to every self-adjoint element of E. End. []
--------------------------------------------------
14.2 Splitting at an arbitrary proportion
--------------------------------------------------
Proposition 14.2 (proportion split). For every separable unital
C*-subalgebra E subset D and every 0 < c < 1, there is a projection f
in D cap E' with
tau(f x) = c tau(x) for every x in E, tau in Lambda.
Proof.
Iterated small splittings. Fix an integer m >= 1 and set
lambda := 1 - c^{1/m},
so c^{1/m} = 1 - lambda and c = (1 - lambda)^m. Choose e_1, ..., e_m
successively from Lemma 14.1, at each step enlarging the coefficient
algebra E to also contain the previously chosen e_i. Each e_i is a
positive contraction commuting with the enlarged algebra, hence with
E and with all previous e_j (j < i).
Define
P_0 := 1_D, P_j := prod_{i=1}^j (1 - e_i) (1 <= j <= m).
These are positive contractions of norm at most one, commuting with E
and with one another. For a self-adjoint contraction x in E, the
element P_{j-1} x is a permitted test at step j (it lies in the
enlarged algebra and has norm <= 1). The trace inequality of Lemma
14.1 gives, uniformly in tau in Lambda,
tau(P_j x) = tau(P_{j-1} x) - tau(e_j P_{j-1} x)
= (1 - lambda) tau(P_{j-1} x) + delta_j,
with |delta_j| <= lambda^{3/2}. Telescoping:
| tau(P_m x) - c tau(x) | <= m lambda^{3/2}. (linear)
Projection defect. For commuting positive contractions a, b:
a b - (a b)^2 = b(a - a^2) + a^2(b - b^2)
<= (a - a^2) + (b - b^2).
Inductively,
tau(P_j - P_j^2) <= tau(P_{j-1} - P_{j-1}^2) + tau(e_j - e_j^2)
<= j lambda^{3/2},
so tau(P_m - P_m^2) <= m lambda^{3/2}. (proj)
Both errors vanish as m -> inf: since lambda = 1 - c^{1/m} <= (-log c)/m,
we have m lambda^{3/2} = O(lambda^{1/2}) -> 0.
Diagonal over m. For each m let P^{(m)} in D be the positive
contraction constructed above. Fix a dense sequence (x_k) in the
self-adjoint unit ball of E with representatives x_{k, n} in A. For
each k, choose m_k large enough and a positive-contraction lift
(P_n^{(k)})_n in l^inf(A) of P^{(m_k)} such that, on an omega-large
set G_k, we have for each n in G_k and every j <= k:
sup_{sigma in T} ||[P_n^{(k)}, x_{j, n}]||_{2, sigma} < 1/k,
sup_{sigma in T} | sigma(P_n^{(k)} x_{j, n}) - c sigma(x_{j, n}) | < 1/k,
sup_{sigma in T} sigma(P_n^{(k)} - (P_n^{(k)})^2) < 1/k.
Such a set exists: the class identities hold along omega by Lemma
11.1, and the right-hand sides shrink to zero as m_k -> inf.
Put
F_k := { n : n >= k } cap G_1 cap ... cap G_k.
The F_k form a decreasing sequence of omega-large sets with empty
intersection. At coordinate n in F_1, set k(n) := max{ k : n in F_k }
and f_n := P_n^{(k(n))}; off F_1 set f_n := 0. Then k(n) -> omega inf.
The class f := [(f_n)] in D is a positive contraction (each f_n is).
Coordinate-wise: for fixed k and n in F_{k(n)}, the tests at level
k(n) >= k give
sup_sigma ||[f_n, x_{k, n}]||_2 < 1/k(n),
sup_sigma | sigma(f_n x_{k, n}) - c sigma(x_{k, n}) | < 1/k(n),
sup_sigma sigma(f_n - f_n^2) < 1/k(n).
Taking lim_omega and letting k(n) -> omega inf:
||[f, x_k]||_{2, Lambda} = 0,
tau(f x_k) = c tau(x_k); tau(f - f^2) = 0.
The projection-defect identity implies ||f - f^2||_{2, Lambda} = 0
because (f - f^2)^2 <= f - f^2 for a positive contraction, so f^2 = f
in D. Exact commutation and the trace identity extend from the dense
sequence to all of E by norm continuity and complex linearity. This
proves (split). End. []
--------------------------------------------------
14.3 Uniform property Gamma in D
--------------------------------------------------
Theorem 14.3. For every n >= 2 there exist pairwise orthogonal
projections f_1, ..., f_n in D cap A' summing to 1_D and satisfying
tau(f_i a) = tau(a) / n
for every a in A, tau in Lambda, 1 <= i <= n. This is uniform
property Gamma for A in the nonunital sense [CETW, Definition 2.1].
Proof. Start with the separable unital C*-subalgebra E_1 := C^*(A, 1_D)
of D.
Inductive construction. Suppose f_1, ..., f_{j-1} are pairwise
orthogonal projections in D cap A' with tau(f_i a) = tau(a)/n for a in
A and 1 <= i <= j-1. Let
q_j := 1_D - f_1 - ... - f_{j-1}
be the remaining complement, and set E_j := C^*(A, 1_D, f_1, ..., f_{j-1}),
a separable unital C*-subalgebra of D. Apply Proposition 14.2 with
c_j := 1/(n - j + 1) to E_j, giving a projection g_j in D cap E_j'.
Set f_j := q_j g_j. Since g_j commutes with q_j (both lie in E_j'),
the product is a projection with f_j <= q_j. It commutes with A,
since g_j and q_j both do. It is orthogonal to f_1, ..., f_{j-1},
because those are supported on 1 - q_j.
Trace identity. From Proposition 14.2 applied to E_j,
tau(f_j a) = tau(q_j g_j a) = c_j tau(q_j a).
Inductive claim: tau(q_j a) = (n - j + 1) tau(a)/n. Base j = 1:
q_1 = 1_D and tau(a) = n tau(a)/n. Step: q_j = q_{j-1} - f_{j-1}, so
tau(q_j a) = tau(q_{j-1} a) - tau(f_{j-1} a) = (n - j + 2) tau(a)/n
- tau(a)/n = (n - j + 1) tau(a)/n. Combining,
tau(f_j a) = (1/(n - j + 1)) (n - j + 1) tau(a)/n = tau(a)/n.
Conclusion. Take f_n := q_n. The resulting f_1, ..., f_n are
pairwise orthogonal projections in D cap A', sum to 1_D, and satisfy
tau(f_i a) = tau(a)/n. This is uniform property Gamma for A in the
nonunital sense. End. []
===========================================================================
SECTION 15. Nonunital absorption: A is isomorphic to A (x) Z
===========================================================================
Standing: A satisfies hypotheses (a), (b), (c) of Theorem 15.4. T := T_1(A) is
weak-* compact and nonempty, and every densely finite
lower-semicontinuous trace is bounded on the unit ball of A.
Uniform property Gamma was proved in Theorem 14.3. This section
converts it to Z-stability.
The engine is [CLS, Proposition 5.4]: a simple, separable, nuclear
C*-algebra with (i) almost unperforated Cu, (ii) sufficiently many
lower-semicontinuous traces, and (iii) uniform McDuffness, is
Z-stable. We verify each hypothesis.
--------------------------------------------------
15.1 Uniform McDuffness from uniform Gamma
--------------------------------------------------
By [CETW, Theorem 4.6 and Definition 4.2], a separable nuclear algebra
with no finite-dimensional quotients, compact nonempty trace space T,
and uniform property Gamma admits unital homomorphisms
M_n -> D cap A', n >= 1. (McD)
The central-sequence form of these maps is what [CLS, Definition 2.4]
calls uniform McDuffness. The passage between the two is exactly the
central-lift computation of [CLS, Section 5] (also [CCEGSTW, Section
5]), which we recall.
Lift through the central sequence algebra. Write
A^omega := l^inf(N, A) / { (a_n) : ||a_n|| -> 0 },
N^perp := { x in l^inf(N, A) : x a = a x = 0 for every a in A },
F_omega(A) := (l^inf(N, A) cap A') / (l^inf(N, A) cap N^perp).
The norm-null ideal is contained in the tracial-omega-null ideal, so
l^inf(N, A) -> D factors as a *-homomorphism through A^omega. Its
restriction to central sequences lands in D cap A' and is surjective
onto D cap A' by [CLS, Proposition 3.2] (the tracial-omega quotient of
a central sequence algebra identifies with the tracial central
sequences on the omega-completion).
Order-zero lifting. [CETW, Proposition 1.8] and [CLS, Proposition
3.3] give: every completely positive order-zero map M_n -> D cap A'
lifts, coordinatewise along omega and up to the tracial ideal, to a
completely positive order-zero map M_n -> F_omega(A). Applied to each
unital map (McD), we obtain unital CP order-zero maps
M_n -> F_omega(A), n >= 1.
By the finite-dimensional case of the order-zero structure theorem
[WZ, Theorem 3.3 and Corollary 4.2] each such order-zero map of a
matrix algebra carries an embedded copy of M_n into F_omega(A) as a
unital *-subalgebra. So (iii) is satisfied: A is uniformly McDuff in
the sense of [CLS, Definition 2.4].
No Z-stability of any summand and no unitality of A is used; only that
T is compact and nonempty and traces are bounded and faithful.
--------------------------------------------------
15.2 Almost unperforated Cu
--------------------------------------------------
For simple A with T nonempty and comparison tested on finite target
ranks, [CLS, Lemma 2.10] (see also [ERS, Proposition 6.1]) shows:
strict comparison on projective classes of finite-matrix corners
implies almost unperforation of Cu(A).
This is exactly hypothesis (c) of Theorem 15.4. So
Cu(A) is almost unperforated; (i) holds.
--------------------------------------------------
15.3 Sufficiently many lsc traces
--------------------------------------------------
Simplicity makes every densely finite lsc trace on A faithful on A_+
(the null trace ideal is zero). Combined with the standing hypothesis
(a) every such trace is bounded on the unit ball, and by restriction
to T can be normalized: every nonzero a in A_+ satisfies sigma(a) > 0
for some sigma in T. Hence [CLS, Definition 2.6]: sufficiently many
lower-semicontinuous traces. So (ii) holds.
--------------------------------------------------
15.4 Z-stability and the nonunital theorem
--------------------------------------------------
Theorem 15.4 (main nonunital theorem). Let A be simple, separable,
nuclear, stably projectionless, with (a) every densely finite
lower-semicontinuous trace bounded on the unit ball, (b) T_1(A)
weak-* compact and nonempty, and (c) strict comparison tested on
finite target ranks. Then A is isomorphic to A (x) Z.
Proof. Hypotheses (a), (b), (c) of Theorem 15.4 are satisfied. Theorem 14.3
gives uniform property Gamma. Section 15.1 gives uniform McDuffness.
Sections 15.2 and 15.3 give (i) and (ii). CLS Proposition 5.4 then
applies and yields A :=: A (x) Z. End. []
Combined with Theorems 5.2 and 10.4, all three routes of the Preface
are proved.
===========================================================================
END OF DOCUMENT
===========================================================================
===========================================================================
A readers' map (short)
===========================================================================
If you are reading this for the first time, three things are worth
holding in mind.
(1) The whole document is about one shape of problem: MAKE A
PROJECTION THAT SPLITS EVERY TRACE, WHILE RESPECTING EVERY
COEFFICIENT YOU INTRODUCE. The three parts are three different
ways to do this, under three different hypotheses.
(2) The three parts are INDEPENDENT. Route 1 needs real rank zero
of a completion ultrapower. Route 2 does not use central
sequence tricks and does not need a real rank zero assumption;
it starts from strict comparison. Route 3 is nonunital and
works under comparison restricted to finite target ranks.
Coupling between the parts is only the shared machinery:
Section 2 (projection tools, real rank zero) and Section 7
(rank approximation and comparison cutdowns).
(3) THE SHARED OBSTACLE. In every route, moving from "trace close
to half" to "exactly half" requires moving a projection off an
occupied corner. In Route 1, this is done by finding a
partition of the residual projection into finitely many small
pieces, in Section 4. In Route 2, it is done by passing to a
completing limit in D = A^omega along the diagonal, in Sections 8
and 9. In Route 3, it is done by repeated small-splitting
selection in a compact complement, in Sections 13 and 14.
The document does not assume K-theory or UCT. It does assume the
standard equivalence between "uniform property Gamma" in the
completion form and in the all-m form (Section 1.4), which cites
[CCEGSTW].
===========================================================================
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END OF DOCUMENT
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