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CUNTZ COMPARISON AND JIANG--SU ABSORPTION
A complete, readable proof
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HOW TO READ THIS DOCUMENT
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This is a self-contained proof of one implication in the Toms--Winter regularity
program: for a separable simple nuclear non-elementary C*-algebra, strict
comparison of positive elements forces the algebra to absorb the Jiang--Su
algebra Z. We also prove a much more general statement that does not require
the algebra to be simple, and a clean criterion phrased purely in terms of the
Cuntz semigroup.
The mathematics is graduate level. Nothing here can be honestly pitched at a
high school student in the sense of "you can do the exercises." What can be
made accessible is the SHAPE of the argument: what is being asked, why each
large chunk is needed, and how the chunks fit together. Every section below
therefore begins with a short, plain-language paragraph saying what it
accomplishes and why we bother. The technical proof that follows each such
paragraph is complete; no step is left out.
Notation is collected once, in Section 2, and then used without apology. If a
symbol confuses you, the original source file of this proof is the place to
look, and everything critical is restated where it is first used.
Honesty note. This document is a careful rewrite of an existing manuscript. It
improves clarity, motivation, ordering, and internal consistency. It does not
constitute independent verification of a new theorem about C*-algebras; such
verification is the job of referees and experts, not of a rewrite. Where the
original argument had a genuine gap, we have tried to close it or to say so
plainly. Where it was merely hard to read, we have made it easier.
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1. INTRODUCTION
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1.1 The question
Take an algebra A of operators on a Hilbert space, and take two positive
elements a and b in it (or in A tensored with the compact operators). There are
two completely different notions of "a is no bigger than b."
The analytic notion: a is dominated by b if you can shrink b by some operator
(a compression v^*(.)v) and land on a. Concretely, a (=) v^* b v, or
approximated in norm by such things. This is Cuntz comparison, written a <~ b.
The numerical notion: for every way tau of measuring the "size" of a positive
element by a number in [0, infinity] (a trace, in an extended sense), the
measured size of a is at most the measured size of b.
The pivotal question is whether the numerical notion implies the analytic one.
If yes, we say A has strict comparison. Strict comparison is a regularity
property: it says that a single family of numbers captures the whole order
structure of positive elements. For a large class of C*-algebras, strict
comparison is known to be exactly the condition under which the classification
program works, because it forces the algebra to behave like a matrix algebra in
the appropriate limit.
A second, independent regularity property is divisibility. Can a positive
element be split into N pieces of comparable size? Algebraically: for every
class x and every integer N, is there a single class u with
N u <= x <= (N+1) u ?
The two inequalities say u is not too big and not too small, relative to x.
Divisibility of the Cuntz semigroup says positive elements can always be
divided; it is the algebraic shadow of "the algebra is built out of small
matrix pieces."
1.2 The target: Jiang--Su absorption
Jiang and Su built a special simple, infinite-dimensional, projectionless
unital algebra, now written Z, whose Elliott invariant is that of the complex
numbers. Its defining feature is that it is strongly self-absorbing:
Z (=) Z (x) Z,
where (x) is the spatial (minimal) tensor product. An algebra A is Z-stable, or
Z-absorbing, if
A (=) A (x) Z.
Z-stability is the abstract statement that A "has room for arbitrarily small
matrix blocks," and it is the hypothesis needed for classification. The central
theorem of this area, due to Toms--Winter and the preceding decades of work,
connects strict comparison to Z-stability. Our job is to prove the implication
"strict comparison => Z-stability" for separable simple nuclear
non-elementary algebras, together with a nonsimple general form.
1.3 What this document proves
Main Theorem (simple case). Let A be a separable simple nuclear non-elementary
C*-algebra. If A has strict comparison in the extended-functional sense of
Equation (2.4) below, then A is Z-absorbing: A (=) A (x)_min Z.
The statement is proved in two moves.
Move 1 (order theory, Sections 3--6). Strict comparison implies the Cuntz
semigroup of A is almost unperforated and fully almost divisible; both
together are called condition (C). This move needs only exactness (nuclearity
gives it), not simplicity of A once factorization is passed into the Cuntz
semigroup. The hard part is converting comparison into full divisibility with
a relative error bound; we carry that out in Sections 3--6.
Move 2 (central sequences, Sections 7--15). A separable nuclear algebra
satisfying condition (C) admits a unital *-homomorphism Z -> F_omega(A), where
F_omega(A) is the central sequence algebra modulo its annihilator ideal. A
classical absorption step then gives A (=) A (x) Z. This move occupies the
bulk of the document.
We also split the main theorem into its two logically distinct halves. The
following is the genuinely central statement, and it does not mention
comparison at all.
Theorem (absorption from condition (C)). Let A be a nonzero separable nuclear
C*-algebra whose Cuntz semigroup satisfies condition (C). Then for every free
ultrafilter omega, there is a unital *-homomorphism Z -> F_omega(A), and hence
A (=) A (x)_min Z.
Two corollaries restate the main theorem in Cuntz-semigroup language. The
first concerns the canonical map.
Corollary (canonical map). Let A be separable and nuclear, and let
iota : A -> A (x) Z be iota(a) = a (x) 1. If Cu(iota) : Cu(A) -> Cu(A (x) Z) is
an isomorphism, then A (=) A (x) Z.
The second removes the word "canonical" entirely; the isomorphism of Cuntz
semigroups may be any abstract isomorphism whatsoever.
Corollary (abstract Cu criterion). Let A be separable and nuclear. Then
A (=) A (x)_min Z <=> Cu(A) (=) Cu(A (x)_min Z) in the category Cu.
No simplicity, no unitality, and no stable finiteness is assumed. Separability
and nuclearity remain.
1.4 Where the difficulty lives
It is worth saying plainly where the proof is genuinely hard, so a reader can
tell structure from substance.
(i) Turning comparison into divisibility. Comparison gives the existence of a
comparison witness for each fixed pair of elements, but the witness may
depend wildly on the pair, and its norm may blow up. Getting a single
class u that works for a whole sandwich N u <= x <= (N+1) u, with a
uniform bound on the witness, is the delicate content of condition (C).
We do this rung by rung: transfer trace values into small supports,
round by integer spectral cuts, and sum in the Cuntz semigroup with
summable error.
(ii) Central sequences without nuclear dimension bounds. The central sequence
algebra F_omega(A) is defined through the norm ultrapower A_omega. To
build order-zero maps into it we must control, at each coordinate, both
commutators (centrality) and the order-zero identities (orthogonality of
positive elements). Existing machinery that does this (finite nuclear
dimension, uniform property Gamma, tracial boundary) imposes extra
hypotheses we must avoid. The substitute is two independent cone
constructions that we then reconcile: norm-visible cones (which see
irreducible representations) and tracial cones (which see bounded
traces). Neither is central by itself; their join is, after we remove a
relative trace ideal.
(iii) The defect filler. To build a map out of the prime dimension-drop
algebra I(p,p+1) into F_omega(A), we need not just a matrix cone but a
single element filling its defect inside one diagonal corner, with an
approximate-unit-controlled norm. This is the content of Section 13.
1.5 Proof strategy, section by section
The document is organized so that the two moves above are visible as two
arcs. Here is the reading map.
Section 2 Preliminaries: Cuntz comparison, support calculus, trace and rank
conventions, and the equivalence between strict comparison and
almost unperforation.
Sections 3--6 Move 1, the simple route. Trace transfer (Section 3), integer
spectral rounding (Section 3), full almost divisibility via a
summable-error support construction (Section 4), and the resulting
condition (C) (Section 5). Section 6 fixes the support tools used
throughout Move 2.
Sections 7--9 Move 2, part A: the two cone constructions. Two-support
shrinking (Section 7) separates the hereditary supports that
detect representations. It feeds the norm-visible cones of
Section 8. Convex order-zero nuclear approximation and two
pencils (Section 9) produce cones large for bounded traces.
Section 10 Move 2, part B: joining the two kinds of cone. Hereditary
localization and a relative trace ideal convert tracial
centrality into norm centrality.
Section 11 Move 2, part C: reconciling the two errors. Norm visibility gives
finite coordinate multiplicities L_n; the tracial accuracy is
then chosen as eta_n / L_n, so that the reciprocal cancels the
multiplicity. This yields an exact coordinate Cuntz inequality
and bounded comparison witnesses, and hence countably many
orthogonal equivalent central slots.
Section 12 Move 2, part D: packing. Gabe's approximate domination theorem,
in the weighted-inclusion form proved here, packs the individual
witnesses into a single bounded central element filling the
defect of the cone.
Section 13 Move 2, part E: the defect filler becomes a unital map from the
prime dimension-drop algebra I(p,p+1) into F_omega(A).
Section 14 Move 2, part F: from individual block maps to a single unital
map Z -> F_omega(A), using nuclear lifting and bounded
completely positive diagonalization; and the final intertwining
A (=) A (x) Z via connected half-flip unitaries.
Section 15 Assembly: putting the four stated results together.
A useful mental picture: Move 1 builds condition (C) from comparison. Move 2
turns condition (C) into a unital copy of Z inside the central sequence algebra,
and a copy of Z inside the central sequence algebra is exactly what it takes to
absorb Z.
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2. PRELIMINARIES: COMPARISON, SUPPORTS, TRACES, ULTRAPOWERS
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This section collects every background fact used later, so that the main
argument can run without interruption. It is the longest preliminary section
because the rest of the document leans on its notation. A reader on a first
pass can skim the statements, keep Lemma 2.3 (cuts and placement) and the
trace-rank identities (2.5) at hand, and return for proofs as needed.
Throughout: all C*-algebras are over C, and K denotes the compact operators on a
separable infinite-dimensional Hilbert space. Tensor products are spatial
(minimal) unless we explicitly say otherwise. A tilde denotes unitization; we
freely use the adjoined unit in intermediate formulas, but every constructed
element and every map lands in the stated (possibly nonunital) algebra.
For a positive element b of a C*-algebra D we write
her_D(b) = closure of b D b (the hereditary subalgebra of b),
Ideal_D(b) = closure of span(D b D) (the ideal generated by b),
p_b = indicator of (0, infinity)(b) in D** (support projection).
The ambient algebra is dropped from subscripts when it is clear. Matrix
amplifications carry the same definitions entrywise.
2.1 Cuntz comparison and the Cuntz semigroup
A completely positive contractive map is abbreviated c.p.c. It is ORDER ZERO
if it carries orthogonal positive elements to orthogonal positive elements. For
a c.p.c. order-zero map theta : E -> D with unital domain, the order-zero
structure theorem gives
theta(z) = h rho(z) = rho(z) h, h = theta(1), rho(1) = p_h,
where rho : E -> D** is a supporting *-homomorphism whose values act as
multipliers on her(h). If f : [0,1] -> [0,1] is continuous with f(0) = 0, then
f(h) rho(z) lies in D and defines another c.p.c. order-zero map, written f(theta).
A MATRIX CONE is a c.p.c. order-zero map from M_k for some k. Its WEIGHT is the
image of 1, i.e. the element h above.
Fix D. For a, b in (D (x) K)_+, we say a is Cuntz below b, written a <~ b, if
there are v_j in D (x) K with v_j^* b v_j -> a in norm. Mutual comparison is an
equivalence relation; the equivalence classes are the Cuntz classes [a]. The set
of classes, with addition given by orthogonal sum, is the Cuntz semigroup Cu(D).
It carries both an order and a sequential-supremum structure. We use the
following standard facts.
(2.1) [ (a - e)_+ ] is way-below [a] for e > 0; [a] = sup_{e>0} [(a-e)_+].
Here x is way-below y (written x << y) if every increasing sequence with
supremum at least y has some term at least x. We also use
[a + c] <= [a] + [c], (subadditivity)
[ z^* z ] = [ z z^* ], (invariance)
c in her(b)_+ => [c] <= [b], (heredity)
[c] = sup_{n>=1} [ (c - 1/n)_+ ]. (cut approximation)
The class of an orthogonal sum of positives is the sum of the classes: the row
of their square roots compares the sum with its diagonal matrix. Subequivalence
to a fixed positive element is closed under norm limits in the left variable,
by choosing a good enough witness for each approximant. In particular the class
of a finite matrix is handled inside the stable algebra: W(D (x) K), defined via
finite matrices over D (x) K, is naturally isomorphic to Cu(D).
2.2 Cuts, placement, and witness bounds
The next lemma is the workhorse for placing positive elements into prescribed
hereditary corners and for keeping comparison witnesses bounded. Part (1) makes
a cut of the source sit below the target; part (2) bounds an ideal relation by a
finite multiplicity; part (3) keeps witnesses inside the relevant corner; part
(4) converts a strictly better target into a bounded witness with an explicit
norm cap.
Lemma 2.3 (Cuts, placement, witness bounds). Let D be a C*-algebra.
(1) If a, b are in D_+ and ||a - r^* b r|| < e, then there is v in D with
v^* v = (a - e)_+ and v v^* in her(b). Consequently, if a <~ b, the
hereditary algebra of every positive cut of a embeds, by a polar isometry,
into her(b).
(2) If a lies in Ideal(b)_+ then for every e > 0 there is a finite integer L
with [ (a - e)_+ ] <= L [b]. If [a] <= L [b], then every positive cut of a
still compares to L [ (b - d)_+ ] for a sufficiently small d > 0.
(3) Comparisons between elements of a hereditary subalgebra can be witnessed
inside that subalgebra. Comparisons between specified matrix corners can
be witnessed by rectangular matrices supported in those corners.
(4) If u <~ (y - beta)_+ with 0 <= alpha < beta, then for every eta > 0 there
is r in D with
|| u - r^* (y - alpha)_+ r || < eta, ||r||^2 <= (||u|| + eta)/(beta - alpha).
Proof. (1) Choose d with ||a - r^* b r|| < d < e, and a continuous g >= 0 that
vanishes on [0, d] and equals 1 on [e, ||a||]. Put z = b^(1/2) r g(a). Then
(d - ||a - r^* b r||) g(a)^2 <= z^* z <= (||a|| + ||a - r^* b r||) g(a)^2,
so z^* z and g(a)^2 generate the same hereditary algebra, which contains
(a - e)_+. Let w be the polar isometry of z. Conjugation by w maps
her(z^* z) into her(b), and w x lies in D for x in her(z^* z) because w |z| = z
and |z| generates her(z^* z). Taking v = w (a - e)_+^(1/2) gives the claim; the
matrix version gives the hereditary placement between corners.
(2) Approximate a^(1/2) by a finite sum q = sum_i x_i b y_i with
|| a - q^* q || < e/2. Write q = X Y with X = (x_i b^(1/2))_i a row and
Y = (b^(1/2) y_i)_i a column. Then q^* q <= ||X||^2 Y^* Y, while Y Y^* lies in
her(diag(b,...,b)) = her(H). Positive domination together with equality of the
classes of the two squared moduli gives q^* q <~ Y^* Y ~ Y Y^* <~ H. Choose a
comparison witness with error below e/2 and apply (1) to get a cut comparison.
Replacing b by (b - delta)_+ changes any fixed compression by at most
delta ||r||^2, which produces the extra-target-cut statement.
(3) Let C be hereditary in D with approximate unit (e_i). Given a, b in C_+ and
a witness r, pick i so that e_i b e_i and e_i a e_i are close to b and a; then
e_i r e_i lies in C and still compresses a to within the desired error. The
matrix version gives rectangular witnesses. More generally, an element whose
two squared moduli lie in the specified hereditary corners lies in the
rectangular corner, by multiplying by approximate units on both sides.
(4) Choose x with || u - x^* (y - beta)_+ x || < eta, and put
f(t) = ( (t - beta)_+ / (t - alpha)_+ )^(1/2), r = f(y) x,
with the quotient read as 0 where the denominator vanishes. This is continuous
and vanishes near 0, and
f(y) (y - alpha)_+ f(y) = (y - beta)_+, f(y)^2 <= (beta - alpha)^(-1) (y - beta)_+.
The compression and the norm bound follow. If the original comparison lives in
the stabilization, compression to the given corner first lands x in D. QED
2.4 Extended traces and rank
We allow ALL lower semicontinuous extended traces tau : A_+ -> [0, infinity]:
additive and positively homogeneous on positives, lower semicontinuous in norm,
vanishing at 0, and satisfying tau(x^* x) = tau(x x^*) for all x. There is no
dense-finiteness assumption. For b >= 0, the RANK of b at tau is
d_tau(b) = lim_{j -> infinity} tau(b^(1/j)),
with b first rescaled to a contraction if needed. Rank is invariant under
positive scalar rescaling. Traces extend to matrices and to the stabilization
using the unnormalized matrix trace: on a positive infinite matrix, tau is the
sum of the traces of the diagonal entries; additivity and lower semicontinuity
justify the sum, and traciality follows by summing the squared moduli of the
matrix entries.
For a positive CONTRACTION b and s > 0 we use throughout
tau(b) <= d_tau(b), d_tau( (b - s)_+ ) <= s^(-1) tau(b). (2.5)
Both follow from functional calculus. Rank respects Cuntz comparison. For a
bounded trace we use its normal extension to the bidual, so d_tau(b) = tau(p_b).
Every extended trace gives a functional on Cu(A) by [b] |-> d_tau(b). The
converse is what makes the DUAL language useful: if A is exact, every functional
phi : Cu(A) -> [0, infinity] that preserves 0, addition, order, and increasing
suprema is represented by a lower semicontinuous extended trace on the
stabilization,
phi([b]) = d_tau(b), b in (A (x) K)_+. (2.6)
This is the Elliott--Robert--Santiago correspondence together with Haagerup's
theorem in its extended form for exact algebras; neither boundedness nor dense
finiteness is imposed. We record one consequence used repeatedly: every nuclear
algebra is exact. (For an ideal J of D, the quotient norm on A odot (D/J)
induced by (A (x) D)/(A (x) J) is a C*-cross-norm dominating the spatial one,
by lifting elementary tensors; uniqueness of the tensor norm for nuclear A
identifies this quotient with A (x) (D/J).)
2.5 The comparison convention
The precise extended-functional version of strict comparison, referred to in the
introduction, is the following. For all x, y in Cu(A):
if phi(x) <= phi(y) for every functional phi, and
phi(x) < phi(y) whenever 0 < phi(y) < infinity,
then x <= y. (2.7)
The second line only restricts the values at strictly positive finite targets;
arithmetic at 0 and at infinity is the extended nonnegative one. The next lemma
says (2.7) is exactly almost unperforation, which is the usual algebraic form
of comparison. Recall almost unperforation:
(l + 1) x <= l y ==> x <= y for all x, y, l >= 1. (2.8)
Lemma 2.9 (comparison = almost unperforation). For every C*-algebra A, strict
comparison in the sense of (2.7) is equivalent to almost unperforation (2.8).
Proof. Assume (2.7) and (l + 1) x <= l y. Applying a functional gives
phi(x) <= phi(y), with strict inequality on finite positive targets, exactly as
in the proof that (l+1)x <= ly gives x <= y under the functional version. If
phi(y) = 0 then phi(x) = 0, and if phi(y) = infinity the non-strict inequality
is automatic, so (2.7) gives x <= y.
Conversely, assume (2.8) and let x = [a], y = [b] satisfy the two lines of
(2.7). In B = A (x) K put I = Ideal_B(b). The map that is 0 on I_+ and
infinity off I_+ is a lower semicontinuous extended trace (a sum of positives
lies in an ideal exactly when both summands do; z^*z lies in it exactly when
zz^* does; the ideal is norm closed). Its rank functional vanishes at y, so the
first line of (2.7) forces a in I. Lemma 2.3(2) then gives x <= sup_n n y. For
every functional normalized by phi(y) = 1, the finite-target half of (2.7)
gives phi(x) < 1. The comparison criterion for functionals now yields x <= y.
Combined with almost unperforation this gives x <= y, which is (2.7). QED
2.6 Ultrapowers and bounded diagonalization
Fix a free ultrafilter omega on N. For a C*-algebra D put
D_omega = l^infinity(N, D) / { (x_n) : lim_{n -> omega} ||x_n|| = 0 }.
We identify D with its constant copy. For separable A,
Ann(A) = { x in A_omega cap A' : x a = a x = 0 for all a in A },
and the annihilator quotient is
F_omega(A) = (A_omega cap A') / Ann(A).
A sequential approximate unit of a separable nonzero A represents the unit of
F_omega(A). We use completely positive lifting for separable nuclear domains:
a c.p.c. map from a matrix algebra (more generally from a separable nuclear
algebra) into a quotient lifts to a c.p.c. map into the domain. In particular
maps from M_k into F_omega(A) lift to c.p.c. coordinate maps into A.
The following diagonal principle is used many times to turn a family of
"approximately satisfiable on every finite subfamily" conditions into an exact
solution in the ultrapower. It is the engine behind every construction of a
central element below.
Lemma 2.10 (bounded diagonalization). Suppose a countable family of
norm-continuous algebraic and functional-calculus conditions, with coefficients
in a norm ultrapower, is approximately satisfiable on every finite subfamily,
and that each unknown has a fixed individual norm bound in these finite
solutions. Then the conditions can be satisfied exactly in the ultrapower.
Countably many additional commutator conditions, and conditions after
multiplication by a fixed countable subset of the algebra, may be imposed in the
same way.
Proof. For each j choose an ultrapower solution of the first j conditions to
accuracy 1/j with bounded representatives. Enlarge each bound once and truncate
so the bounds are independent of j. There is an omega-large set on which the
first j coordinate conditions hold to accuracy 2/j. Replace these sets by
decreasing finite intersections and intersect the j-th set with {n >= j}. At
coordinate n use the solution of the largest applicable level j <= n, with an
arbitrary bounded choice when no level applies. The chosen level tends to
infinity along omega, so each fixed condition has error tending to zero. The
bounds ensure each unknown represents an element of the norm ultrapower.
Positivity or complete positivity is preserved by choosing positive or
completely positive lifts at each level. QED
A reindexing remark used together with Lemma 2.10: if a central sequence is to
commute with finitely many already-chosen coordinate values, then at each output
coordinate those values are ordinary elements of A; centrality supplies the
required good set of old indices, which may be intersected with finitely many
other good sets before choosing the new index. Any uniform coordinate predicate
retained in this selection keeps holding. In a nonunital quotient, algebraic and
unit defects are tested after multiplication by a dense subset of A, and
boundedness then extends the relations to all of A.
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3. THE SIMPLE ROUTE, I: SMALL SUPPORTS, TRICHOTOMY, TRANSFER, ROUNDING
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This section and the next two carry out Move 1: they convert strict comparison
into condition (C) for a simple exact non-elementary algebra. The present
section builds the analytic toolkit. Three facts about simple algebras drive
everything. First, a simple non-elementary algebra has arbitrarily small
nonzero positive elements, so we can always find a support of tiny rank
(Lemma 3.1). Second, functionals on the Cuntz semigroup come in exactly three
types (zero, everywhere infinite, and "finite-positive-somewhere"), which lets
us measure every rounding error against one fixed positive cut (Lemma 3.2). Third,
we can TRANSPORT a trace value into a small support without changing it
(Lemma 3.3), and then ROUND that value at integer spectral levels with an error
of one support rank (Lemma 3.4). Those last two are where the "error is at most
one support rank" promise of the introduction is made precise.
Throughout Sections 3--5, unless stated otherwise, A is a SIMPLE EXACT
non-elementary C*-algebra, and B denotes A (x) K. We identify Cu(B) with Cu(A).
Matrix algebras over B sit inside stable corners; a placement by the multiplier
isometry V preserves Cuntz classes because b and V b V^* are the two squared
moduli of V b^(1/2).
3.1 Small nonzero classes
The first fact is that simplicity plus "non-elementary" gives us room to make
positive elements as small as we like in the Cuntz order.
Lemma 3.1 (small nonzero classes). If A is simple and non-elementary, then
B = A (x) K is simple with no minimal projections. For every nonzero e in B_+
and integer M >= 1, there is a nonzero h in B_+ with M[h] <= [e].
Proof. Matrix-unit multipliers extract a nonzero matrix entry from any nonzero
ideal of B and put it into every matrix position; with simplicity of A this
makes B simple. If p in B were a minimal projection, then Bp would be a Hilbert
space with inner product x^* y = <x,y> p. Left multiplication is faithful by
simplicity, and the elements x y^* act as rank-one operators. Since p generates
B as an ideal, these span a dense subspace of B, giving B = K. The multiplier
corner 1 (x) e_11 would then identify A with the compacts, contradicting
non-elementarity.
Inside her(e) there are M nonzero mutually orthogonal positives e_1,...,e_M.
Indeed, split a nonzero positive element at two distinct nonzero spectral
points. If none exist, the element is a scalar multiple of a projection whose
corner is not one-dimensional; a shifted nonscalar selfadjoint in that corner
has two distinct positive spectral points and can be split instead. Recursion
gives the family while preserving orthogonality.
For nonzero f, g in B_+, simplicity gives fBg nonzero. If 0 =/= v in fBg, then
[v^* v] = [v v^*] is a nonzero class below both [f] and [g]. Iterating this over
the orthogonal family gives h =/= 0 with [h] <= [e_i] for every i, hence
M[h] <= sum_i [e_i] <= [e]. QED
3.2 The functional trichotomy and one controlling cut
The second fact classifies functionals on Cu(A). This is what lets us handle
unbounded traces and infinite values: although there may be no normalized trace
on a nonunital algebra, one fixed positive cut of any chosen element has finite
positive value at every functional that is finite somewhere.
Lemma 3.2 (functional trichotomy). Let A be simple and nonzero. Every
functional on Cu(A) is one of: zero; infinite on every nonzero class; or
positive on every nonzero class and finite on every class [(c - delta)_+] with c
in (A (x) K)_+ and delta > 0. The third type is exactly the set of functionals
that are finite and positive at some class.
Proof. Every nonzero b in B_+ is full. Lemma 2.3(2) therefore bounds each
positive cut of any c in B_+ by finitely many copies of [b]. If phi([b]) = 0,
preservation of increasing suprema gives phi([c]) = 0 for every c. If
0 < phi([b]) < infinity, the same cut comparison gives finiteness on every cut,
and the first observation gives positivity on every nonzero class. These
alternatives exhaust all functionals. QED
Write Fin(A) for the third type in Lemma 3.2. The notation does not claim
finiteness on all classes. If ||a|| = 1 and e = (a - 1/2)_+, then
0 < phi([e]) < infinity, phi([e]) <= phi([a]) for phi in Fin(A). (3.1)
This fixed cut e is the "controller": every rounding error in the divisibility
argument of Section 5 is measured against phi([e]), which is finite and positive
exactly for the functionals where an error bound matters. The two other types
(zero, and everywhere infinite) are handled separately and trivially.
3.3 Trace transfer into a small support
We can move a positive function of an element, vanishing near zero, into a
prescribed small hereditary support WITHOUT changing any trace value. The point
is that trace values alone cannot see WHERE a positive element sits, only how
big it is, so we are free to relocate the mass into a support we control.
Lemma 3.3 (trace transfer). Let B be a simple stable C*-algebra, let a in B_+ be
a contraction, and let h =/= 0 in B_+. Suppose r : [0,1] -> [0, infinity) is
continuous and vanishes on [0, delta] for some 0 < delta < 1. Then there is c in
her_B(h)_+ such that tau(c) = tau(r(a)) for every lower semicontinuous extended
trace tau on B.
Proof. By fullness of h and Lemma 2.3, [(a - delta/4)_+] <= m[h] for some
finite m >= 1. Apply Lemma 2.3 in M_m(B) with cut parameter e = delta/4 and
comparison error below delta/4, to the source diag((a - delta/4)_+, 0, ..., 0)
and the target H = diag(h, ..., h). The resulting placement matrix has only its
first column nonzero, because all other diagonal entries of its initial squared
modulus are zero. Hence there is a column v = (v_i)_{i=1}^m with
v^* v = (a - delta/2)_+, v v^* in M_m(her_B(h)).
Define g(t) = ( r(t) / (t - delta/2)_+ )^(1/2), read as 0 where the denominator
vanishes. It is continuous and vanishes near zero because r vanishes on
[0, delta]. For z_i = v_i g(a),
sum_i z_i^* z_i = r(a), c := sum_i z_i z_i^* lies in her_B(h),
where the containment uses z_i z_i^* <= ||g||^2 v_i v_i^* together with heredity.
Additivity and traciality give tau(c) = tau(r(a)) for every extended trace, with
the same c. QED
3.4 Integer spectral rounding
The second approximate step rounds a transferred value at integer spectral
levels. The cost is one copy of the transferred element's own rank; crucially
the number of levels does not enter the error.
Lemma 3.4 (integer spectral rounding). Let B be a stable C*-algebra, c in B_+,
and L an integer greater than ||c||. Put w = sum_{i=1}^L [(c - i)_+] in Cu(B).
For every lower semicontinuous extended trace tau, with phi = d_tau,
phi(w) <= tau(c) <= phi(w) + d_tau(c). (3.2)
Proof. With q(t) = min(t,1) on [0, infinity), scalar functional calculus gives
the identity c = sum_{i=1}^L q((c - (i-1))_+). Set R = max(1, ||c||). For every
integer m >= 1,
( (c - i)_+ / R )^(1/m) <= q((c - (i-1))_+),
because below the threshold i the left side is zero and above it the right side
is one. Consequently
d_tau((c - i)_+) <= tau(q((c - (i-1))_+)) <= d_tau((c - (i-1))_+),
the last inequality by (2.5) and heredity. Summing and using (c - L)_+ = 0 gives
(3.2). Only the rank d_tau(c) survives as error; the number L of levels does not
appear. QED
We now combine the two lemmas. If M[h] <= [e] and c comes from trace transfer
into her(h), then M[c] <= [e]. For a functional with phi([e]) < infinity,
rounding thus approximates tau(r(a)) to within phi([e])/M: the error is one
support rank, divided by M. A single cut-supported function does not recover the
full rank of an arbitrary element. The next section writes that rank as a sum of
such trace values and chooses the support ranks so the errors are summable.
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4. THE SIMPLE ROUTE, II: FULL ALMOST DIVISIBILITY
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This is the payoff of Move 1. We show that a simple exact non-elementary
algebra with strict comparison can divide any positive element into N nearly
equal Cuntz pieces, with an explicit error budget, entirely independent of any
trace normalization. The strategy, in one sentence: write the rank as a
telescoping sum of trace values that vanish near zero, transfer each value into
a support of tiny prescribed rank, round each at integer levels, and add the
rounded pieces inside the Cuntz semigroup so the errors form a summable series.
The error budget is fixed first. For a target N, choose a real number s strictly
between 1/(N+1) and 1/N, and integers M_j (j = 1, 2, ...) so large that the
total reciprocals stay below the gap:
1/(N+1) < s < 1/N, E := sum_{j>=1} 1/M_j < s - 1/(N+1). (4.1)
Here s is the "target fraction" (we aim to place mass s times x into u) and E
is the total error we are willing to spend. Every functional will be measured
against ONE fixed cut, so E times that cut's value bounds all the rounding
errors at once.
Theorem 4.2 (full almost divisibility). Let A be a simple exact non-elementary
C*-algebra with strict comparison as in (2.7). For every x in Cu(A) and every
integer N >= 1 there exists u in Cu(A) with
N u <= x <= (N+1) u. (4.3)
Proof. The assertion is immediate for x = 0, so write x = [a] with a in B_+ of
norm one, and put e = (a - 1/2)_+. If Fin(A) is empty then every functional is
zero or everywhere infinite, so (2.7) gives N x <= x while x <= (N+1) x is
automatic; take u = x. Hence assume Fin(A) is nonempty.
STEP 1 (supports of summable prescribed rank). Fix s and M_j as in (4.1).
Lemma 3.1 supplies nonzero h_j in B_+ with M_j [h_j] <= [e].
STEP 2 (recovering the rank from cut-supported functions). Define increasing
cutoffs converging to one on (0,1] and their increments:
F_0 = 0, F_j(t) = min{1, max{0, 2^j t - 1}}, r_j = s (F_j - F_{j-1}) (j >= 1).
The F_j increase and r_j >= 0 vanishes on [0, 2^(-j)]. Put p_j = r_j(a).
Lemma 3.3 gives c_j in her(h_j)_+ with
tau(c_j) = tau(p_j) for all extended traces tau, M_j [c_j] <= [e]. (4.4)
STEP 3 (round and sum inside Cu(A)). Choose integers L_j > ||c_j|| and put
w_j = sum_{i=1}^{L_j} [(c_j - i)_+], u = sup_{k>=1} sum_{j=1}^k w_j. (4.5)
This supremum is represented by an honest positive element of B: choose
representatives of each w_j, rescale them to norm at most 2^(-j), and place them
in mutually orthogonal stable blocks; their sum converges in norm. Each finite
partial sum is Cuntz below the limit, and comparison of all partial sums with
one target passes to the norm limit, so the limit is the least upper bound.
Therefore, for every functional phi in F(Cu(A)),
phi(u) = sum_{j>=1} phi(w_j). (4.6)
STEP 4 (per-piece error). Fix phi in Fin(A) and choose its representing
extended trace via (2.6). Lemma 3.4 and (4.4) give
phi(w_j) <= tau(p_j) <= phi(w_j) + phi([e]) / M_j. (4.7)
The only trace-dependent constant is the single finite number phi([e]).
STEP 5 (rank recovery telescopes). Telescoping,
sum_{j=1}^k tau(p_j) = s tau(F_k(a)),
and we claim
tau(F_k(a)) increases to d_tau(a) = phi(x). (4.8)
The values increase and are at most d_tau(a) by (2.5) and heredity. Given
0 < e < 1, for k large F_k(t) = 1 on [e, 1], so for every integer m >= 1,
((a - e)_+)^(1/m) <= F_k(a). Taking traces and letting m -> infinity gives
d_tau((a - e)_+) <= tau(F_k(a)). The cut-supremum formula (2.1) then gives
(4.8).
STEP 6 (sum the pieces). Sum (4.7), then use (4.6) and (4.8). In extended
nonnegative arithmetic, for every phi in Fin(A),
phi(u) <= s phi(x) <= phi(u) + E phi([e]). (4.9)
If phi(x) < infinity, (3.1) gives (s - E) phi(x) <= phi(u) <= s phi(x); since
phi(x) > 0 and s - E > 1/(N+1) by (4.1), we get the strict interval
phi(x)/(N+1) < phi(u) < phi(x)/N. (4.10)
If instead phi(x) = infinity, then E phi([e]) < infinity and (4.9) forces
phi(u) = infinity; no subtraction of infinite values is used. Since Fin(A) is
nonempty, these conclusions also show u =/= 0.
STEP 7 (compare with the two targets). Zero functionals vanish at x and u;
everywhere-infinite functionals are infinite at both. For phi in Fin(A), (4.10)
gives
phi(N u) <= phi(x) and phi(x) <= phi((N+1) u),
the first strict whenever 0 < phi(x) < infinity. For the second, if
0 < phi((N+1)u) < infinity then phi lies in Fin(A) and the infinite-value case
excludes phi(x) = infinity; (4.10) makes that inequality strict as well.
Applying (2.7) twice proves N u <= x <= (N+1) u. QED
Theorem 4.2 is the whole order-theoretic content of the simple route. Its two
conclusions are exactly the two halves of condition (C), once we recall Lemma
2.9 which supplies the unperforation half from the same comparison hypothesis.
Corollary 4.11 (condition (C) from comparison). Let A be a simple exact
non-elementary C*-algebra with strict comparison as in (2.7). Then Cu(A)
satisfies condition (C): it is almost unperforated (2.8) and fully almost
divisible (4.3).
Proof. Theorem 4.2 gives full almost divisibility; Lemma 2.9 gives almost
unperforation. QED
Remark 4.12. The same sequence (M_j) works for every functional. The numbers
phi([e]) need not be uniformly bounded over any trace family: pointwise
finiteness handles phi(x) = infinity, and the bound phi([e]) <= phi(x) handles
finite values. That is precisely why no trace normalization, and no
tracial-boundary hypothesis, is needed here.
4.3 A unital uniform-trace illustration (not used below)
The main simple route is complete. For completeness we record one separate,
elementary consequence of transfer and rounding when traces CAN be normalized.
It is not used in the absorption construction; it simply shows the mechanism in
the friendliest case. It needs no exactness, nuclearity, or comparison.
Proposition 4.13 (uniform approximation from below). Let A be a unital simple
infinite-dimensional C*-algebra with nonempty tracial state space T(A). For
every p in A_+ and eta > 0 there are a finite integer L and w in M_L(A)_+ with
0 <= tau(p) - d_tau(w) <= eta for all tau in T(A), (4.14)
matrix traces unnormalized.
Proof. Choose N with 1/N < eta. There are N mutually orthogonal nonzero
positive elements in A: repeatedly split a positive element at two distinct
positive spectral values; if a nonzero hereditary corner had no such element it
would contain a minimal projection, and simplicity would identify A with the
compacts, forcing finite-dimensionality by unitality. For any two nonzero
positives e, f, simplicity gives 0 =/= v in eAf, and [v^* v] = [v v^*] is a
nonzero class below both [e] and [f]. Iterating over the orthogonal family gives
0 =/= h in A_+ with d_tau(h) <= 1/N for all tau in T(A).
Fullness gives finitely many s_i with sum_i s_i^* h s_i = 1. To see the exact
equality, choose an invertible finite sum S = sum_i r_i h t_i approximating 1;
for the row R = (r_i h^(1/2))_i and column D = (h^(1/2) t_i)_i, the identity
S = R D gives S^* S <= ||R||^2 D^* D, so D^* D is invertible; set
s_i = t_i (D^* D)^(-1/2). The element c = sum_i h^(1/2) s_i p s_i^* h^(1/2)
lies in her_A(h)_+ and satisfies tau(c) = tau(p) and d_tau(c) <= 1/N for every
tau in T(A). Choose an integer L > ||c|| and let w = direct sum_{j=1}^L (c - j)_+.
Lemma 3.4, applied in the stabilization, gives
d_tau(w) <= tau(c) <= d_tau(w) + d_tau(c) <= d_tau(w) + 1/N,
which is (4.14). QED