e2be759
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UNIVERSAL STRONG KADISON-KASTLER STABILITY
A Complete and Self-Contained Proof
==============================================================================
PURPOSE OF THIS DOCUMENT
This file proves, completely, the strong Kadison-Kastler conjecture:
two unital von Neumann algebras acting on the same Hilbert space that
are close in the Kadison-Kastler metric are conjugate by a unitary that
is close to the identity, with a tolerance that depends only on the
requested distance of that unitary from the identity, uniformly over
all algebras, all representations, and all Hilbert spaces.
The logical spine is:
(Main Theorem) <-- reductions over centers and non-separable H (Sec 8)
^
(All separably acting factors) <-- crossed-product descent (Sec 7)
^
(Prepared type III_1 pairs) <-- modular alignment (Sec 6)
^
(Finite factors) <-- boundary averaging + amplification (Secs 4-5)
^
(Toolkit) (Secs 2-3)
Section 1 fixes notation and states everything. Every external theorem
is stated precisely at the point of use; the proof is complete in the
sense of a research paper, namely a complete reduction to those stated
classical results.
Conventions. H, K are complex Hilbert spaces. B(H) is all bounded
operators. "von Neumann algebra" means a unital *-subalgebra of B(H)
closed in the weak operator topology (WOT), with the same identity I.
M_1 is its operator-norm closed unit ball. For an algebra A, A' is its
commutant, Z(A)=A cap A' its center. A otimes B is the spatial von
Neumann tensor product. Ad_u(x)=u x u*. ad_h(x)=[h,x]=h x - x h.
ip{.,.} is the inner product, linear in the first variable. id is the
identity map. Tr is the operator trace. "UCP" = unital completely
positive. "cb" = completely bounded; T_n is T applied entrywise to
M_n, ||T||_cb = sup_n ||T_n||.
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SECTION 1. THE KADISON-KASTLER METRIC AND THE MAIN THEOREM
==============================================================================
Definition 1.1 (Kadison-Kastler distance). For von Neumann algebras
M,N subset B(H) with common identity, set
d(M,N) = max( sup_{x in M_1} inf_{y in N_1} ||x-y|| ,
sup_{y in N_1} inf_{x in M_1} ||x-y|| ).
This metric was introduced by Kadison and Kastler (1972). Their
perturbation problem asks whether smallness of d(M,N) forces a spatial
isomorphism carried by a unitary close to I. We prove the uniform form.
Definition 1.2 (near inclusion). Write A subset_gamma B if there is
gamma' < gamma such that every x in A_1 has dist(x,B) <= gamma'
(the approximant need not be a contraction). A strict upper bound for
d(A,B) gives both near inclusions with that bound; conversely near
inclusions give unit-ball bounds after normalizing approximants: if
||x||<=1 and ||x-y||<=r then y_0 = y / max{1,||y||} lies in B_1 and
||x-y_0|| <= 2r.
Definition 1.3 (cb distance). d_cb(A,B) = sup_{n>=1} d(M_n(A),M_n(B)),
where M_n(A) acts on H^n with the operator norm. Clearly d <= d_cb.
For a unitary v, at every matrix level ||v_n x v_n* - x|| <= 2||v-I|| ||x||,
hence
d(A, vBv*) <= d(A,B) + 2||v-I||, d_cb likewise. (1.1)
Also ||v_1...v_k - I|| <= sum_j ||v_j-I||.
THEOREM 1.4 (Strong Kadison-Kastler stability). For every eps>0 there
is delta=delta(eps)>0 such that: if H is any complex Hilbert space and
M,N subset B(H) are unital von Neumann algebras with d(M,N)<delta,
then there is a unitary u in B(H) with
u M u* = N, ||u - I_H|| < eps.
The tolerance delta depends only on eps, not on the algebras, their
types, their representations, or the cardinalities of H and the
preduals.
The rest of the document is the proof.
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SECTION 2. THE TOOLKIT
==============================================================================
We collect five tools. All constants below are absolute.
2.1 COMMUTATORS AND COMMUTANTS.
Lemma 2.1. For a unital von Neumann algebra A subset B(H) and h in B(H),
dist(h, A') <= || ad_h|_A ||_cb. (2.1)
Consequently d_cb(A,B) <= gamma implies d(A',B') <= 4 gamma.
Proof. A' is weak-star closed, so its preannihilator in the trace class
computes dist(h,A'). Let phi be a trace-class functional of norm <=1
vanishing on A'. A singular-value decomposition gives vectors
xi, eta in (l^2 otimes H) of norm <=1 with
phi(x) = ip{(I otimes x) xi, eta} (x in B(H)).
(Only countably many directions occur, even for non-separable H.)
Let p be the projection onto the closure of (I otimes A') xi. This
subspace reduces I otimes A', so every matrix entry of p lies in A'' = A.
Also p xi = xi and p eta = 0. Hence
phi(h) = ip{ [I otimes h, p] xi, eta }.
For q_n the projection onto the first n coordinates of l^2, the
compression [I_n otimes h, q_n p q_n] has norm <= ||ad_h|_A||_cb,
because q_n p q_n in M_n(A)_1. Testing on finitely supported vectors
and using density bounds the full commutator by the same quantity.
Supremum over phi gives (2.1).
For h in A'_1, each contraction in M_n(B) is within gamma (plus slack)
of one in M_n(A); hence ||ad_h|_B||_cb <= 2 gamma. By (2.1),
dist(h,B') <= 2 gamma, and normalizing approximants gives distance
<= 4 gamma to B'_1. Symmetry finishes. QED
Lemma 2.2 (close representations). Let rho, theta: P -> B(H) be normal
unital *-representations of a von Neumann algebra. If
||rho - theta||_cb <= e < 1/4, there is a unitary u in B(H) with
u rho(x) u* = theta(x) for all x, and ||u-I|| <= 4e.
Proof. If e=0 take u=I. Set
pi(x) = diag(rho(x), theta(x)), S = [[0,I],[I,0]].
The von Neumann image pi(P) has the quotient matrix norms of P, and
the cb norm of [S,pi(.)] is <= e. By Lemma 2.1 pick D in pi(P)' with
||D-S|| < 2e. Its lower off-diagonal block T satisfies
T rho(x) = theta(x) T and ||T-I|| < 2e < 1, so T is invertible and T*T
commutes with rho(P). The polar unitary u = T |T|^{-1} intertwines.
The singular values of T lie in [1-r,1+r], r=||T-I||, giving
||u-I|| <= ||u-T|| + r = ||I-|T||| + r <= 2r < 4e. QED
2.2 AMENABLE NEAR INCLUSIONS (Christensen). The following is
Christensen's near-inclusion theorem in the form of Cameron-Christensen-
Sinclair-Smith-White-Wiggins (Duke 2014, Thm 3.3(i)), with the strict
convention of Definition 1.2.
Theorem 2.3. Let R,Q subset B(H) be von Neumann algebras with identity I,
R amenable. If R subset_gamma Q and gamma < 1/100, there is a unitary
v in (R cup Q)'' with v R v* subset Q and ||v-I|| <= 150 gamma. If also
Q subset_gamma R and gamma < 1/101, the same holds with v R v* = Q.
The location v in (R cup Q)'' is what makes the argument stable: any
algebra commuting with both R and Q commutes with the adjustment.
2.3 THE COMPLETELY POSITIVE CRITERION. The relevant norm for
perturbing a product is the cb norm on the Haagerup tensor product:
for a bilinear b: P x P -> P it is the least K such that, at all
rectangular sizes,
|| [ sum_j b(x_{ij}, y_{jk}) ]_{i,k} ||
<= K ||X|| ||Y||, X in M_{p,r}(P), Y in M_{r,q}(P). (2.2)
We cite Ricard-Roydor (JFA 2014, Prop 3.2): on a von Neumann algebra P,
an associative product m with ||m - m_P||_cb <= 1/11 (m_P the original
product) admits a cb linear isomorphism L with
L(m(x,y)) = L(x)L(y), ||L-id||_cb <= 10 ||m-m_P||_cb,
and L preserves the involution if m does. No normality is required.
Proposition 2.4 (UCP criterion). Let P,Q be von Neumann algebras, let
rho: P -> B(H) be a faithful normal unital representation, and let
Q subset B(H) share the identity. Suppose F: P -> Q is UCP and
||F - rho||_cb <= e < 1e-3, sup_{y in Q_1} dist(y, rho(P)_1) <= g,
g + e < 1.
Then there is a normal unital *-isomorphism theta: P -> Q and a unitary
u in B(H) with ||theta-rho||_cb <= 32e and u rho(x) u* = theta(x),
||u-I|| <= 128e. The map F need not be normal.
Proof. rho is completely isometric, so ||F_n(X)|| >= (1-e)||X||; thus
F is injective with closed range V and ||F^{-1}: V -> P||_cb <= 1/(1-e).
For y in Q_1, match y with rho(x), x in P_1, allowing slack, to get
dist(y,V) <= g+e < 1. If V were proper, the quotient map Q -> Q/V would
have norm one, contradicting this bound on Q_1. So F is onto.
Transport multiplication: m(x,y) = F^{-1}(F(x)F(y)), associative and
involution preserving. For rectangular X,Y, insert rho(XY)=rho(X)rho(Y):
||F(XY) - F(X)F(Y)|| <= 3e ||X|| ||Y||
(the ||XY|| term is bounded by e, the product term by 2e, since F and
rho are complete contractions). By (2.2),
q := ||m - m_P||_cb <= 3e/(1-e) < 1/11.
Ricard-Roydor gives involution-preserving L with
||L - id||_cb <= lambda := 30e/(1-e) < 1. Then theta = F L^{-1} is a
bijective *-homomorphism and
||theta - rho||_cb <= e + lambda/(1-lambda) = e + 30e/(1-31e) <= 32e.
Bijective multiplicativity makes theta unital; a bijective *-isomorphism
of von Neumann algebras preserves bounded increasing suprema, hence is
normal. Apply Lemma 2.2 since 32e < 1/4. For e=0 the range argument
directly gives Q = rho(P). QED
2.4 TENSOR PRODUCTS AND INVARIANT MEANS.
Lemma 2.5 (tensor gap). Let A,B subset B(H) be unital von Neumann
algebras with d_cb(A,B) <= gamma. If K is separable then
d_cb(A otimes B(K), B otimes B(K)) <= gamma. If D subset B(K) has
commutant equal to the strong closure of increasing finite-dimensional
unital algebras, then d_cb(A otimes D, B otimes D) <= gamma.
Proof. Compress a contraction of M_n(A otimes B(K)) by increasing
finite-rank projections on K. In a rank-r corner, choose a contraction
of M_{nr}(B) within gamma plus slack, extended by zero outside. The
compressions converge strongly; a bounded WOT cluster of the opposite
contractions lies in M_n(B otimes B(K))_1 and is within gamma. Reverse,
sup over n. For the D case, replace B(K) by B(K), average the nearby
contraction by conjugation over the unitary group of each finite-
dimensional stage of D' (the input is fixed by these groups, so the
bound is kept), take a WOT cluster, and use the commutation theorem
(B vN tensor commutation) to place it in M_n(B otimes D). QED
Lemma 2.6 (invariant operator means). Let G be a discrete amenable
group with left invariant mean m on l^inf(G). For a bounded family
T_g in B(K) define (m_g T_g) by
ip{(m_g T_g) xi, eta} = m_g ip{T_g xi, eta}.
This map is UCP, commutes with multiplication on either side by a fixed
operator, preserves WOT-closed subspaces, and if ||T_g - T|| <= a for all
g then ||m_g T_g - T|| <= a (also at rectangular matrix levels). For a
unitary representation g |-> V_g, E_V(T) = m_g V_g T V_g* is a UCP
projection onto {V_g : g in G}', preserving every invariant von Neumann
algebra; for an automorphism action alpha_g on P the formula
phi(E(x)) = m_g phi(alpha_g(x)) defines a UCP map onto the fixed points
P^G. These maps need not be normal.
Proof. The coefficient formula defines a bounded sesquilinear form, hence
an operator. Positivity at each matrix size and unitality follow from
applying the positive mean to matrix quadratic forms. Fixed left and
right factors pass through. Testing the linear equations defining a
WOT-closed subspace proves range preservation; testing vector
coefficients gives the norm bound. For E_V, left invariance makes the
range invariant and averaging fixes invariant operators; an invariant
algebra is preserved by the range property. The predual-duality form
handles the P-action, including matrix bounds. QED
For G = R with the discrete topology, an invariant mean is a weak-star
cluster of uniform averages on finite boxes in finitely generated free
abelian subgroups, directed by (finite set of translations, tolerance).
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SECTION 3. UNIFORM AMPLIFICATION OF COMMUTATORS
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THEOREM 3.1 (Uniform amplification). There is an absolute constant
C_amp such that for every type II_1 factor M subset B(H) on a separable
Hilbert space and every selfadjoint h in B(H),
|| ad_h|_M ||_cb <= C_amp || ad_h|_M ||.
C_amp is independent of M, its representation, and H.
This bridges ordinary norm control (what the boundary argument gives)
to matrix control (what a small implementing unitary needs).
3.1 COLUMNS. For a_1,...,a_l in B(H) write
||(a_i)_i||_col = || sum_i a_i* a_i ||^{1/2}.
Lemma 3.2 (column estimate). Let A subset B(H) be a unital C*-algebra,
h in B(H), delta = ||ad_h|_A||. Then
||([h,a_i])_i||_col <= 8 delta ||(a_i)_i||_col.
Proof. If delta=0 nothing to prove. Else use the triangular
homomorphism Phi(a) = [[a, delta^{-1}[h,a]],[0,a]], a unital
homomorphism with ||Phi|| <= 2. For xi in H, the closed subspace
K_xi = closure{Phi(a)(0,xi) : a in A} is invariant for Phi(A), and the
restricted homomorphism is cyclic. Haagerup's cyclic similarity theorem
(Ann. Math. 1983, Thm 1.1) gives ||Phi|_{K_xi}||_cb <= ||Phi||^3 <= 8.
Apply its column amplification to (0,xi), retain the upper component:
( sum_i ||[h,a_i] xi||^2 )^{1/2}
<= 8 delta ||(a_i)_i||_col ||xi||.
Supremum over unit xi. QED
3.2 TRACE TRANSFER. On a finite von Neumann algebra (P,tau) use
||x||_2 = tau(x*x)^{1/2}, ||x||_1 = tau(|x|). Fix a free ultrafilter
omega on N; the tracial ultrapower P^omega is the bounded sequences
modulo those with lim_omega ||x_j||_2 = 0, with trace
tau_omega((x_j)_omega) = lim_omega tau(x_j). Identify P with constant
sequences. We also use standard comparison/subdivision of projections
by trace in finite factors.
Lemma 3.3 (trace to a normal representation). Let P subset B(K) be a
finite von Neumann algebra with faithful normal trace tau.
(i) If (x_j) is bounded and lim_omega ||x_j||_2 = 0, then
lim_omega ||x_j xi|| = 0 for every xi in K.
(ii) If x = (x_j)_omega in P^omega, the WOT ultralimit of x_j in this
representation is E_P(x), the trace-preserving conditional expectation.
Proof. The functionals a |-> tau(b a), b in P, are norm dense in the
predual P_*: their annihilator in (P_*)* = P is zero by faithfulness,
so Hahn-Banach gives density. For a unit vector functional phi
approximated in norm by tau(b .), and ||x_j|| <= K_0,
|phi(x_j* x_j)| <= K_0^2 ||phi - tau(b .)|| + ||b|| ||x_j||_2^2.
This gives (i). For (ii), the defining identity of the conditional
expectation gives lim_omega tau(b x_j) = tau_omega(b x) = tau(b E_P(x));
predual density and boundedness extend this to every normal functional.
QED
3.3 FREE PROJECTION WORDS. The next estimate controls sums of
projection words with fixed label multiplicities; the square-summed
version over a fixed label is what the randomization uses.
Lemma 3.4 (free projection words). For every integer r>=1 there are
C_r>0 and an integer d_r>=r with the following property. Let d>=d_r,
and let (A,tau) be a von Neumann algebra with a faithful normal tracial
state. Suppose A_1,...,A_r subset A are unital copies of M_d(C) that are
scalar free (the trace of a product of centered letters from
successively different copies is zero). Let (e_a^{(j)})_{a=1}^d be the
diagonal minimal projections of A_j. For a multiset W of size r from
{1,...,d}, put
B_W = sum over tuples (a_1,...,a_r) with multiset W of
e_{a_r}^{(r)} ... e_{a_1}^{(1)}.
Then ||B_W|| <= beta_W where
beta_W = 7 sqrt((r-1)!) d^{-(r-1)/2} if all labels of W distinct,
beta_W = C_r d^{-(r-1)/2} otherwise. (3.1)
The constants and threshold depend only on r.
Proof (outline of the free-probability estimate). Work in the faithful
left representation on L^2(A,tau) and the free-product Fock space
F = C Omega + direct sum over reduced words of the tensor products
of H_j^circ = L^2(A_j,tau) minus C1. The label map
D_j: l^2({1,...,d}) -> H_j^circ, D_j delta_a = h_a^{(j)} = e_a^{(j)} -
d^{-1}1 has D_j* D_j = d^{-1}I - d^{-2} 11*, ||D_j|| <= d^{-1/2}.
Rightmost-to-leftmost, a fully centered word is a sum of "cuts":
p annihilations then q creations, or p annihilations, one preservation,
q=r-p-1 creations. Each cut factors through the tensor product of the
D_j; the annihilator legs contribute <= d^{-p/2}, the creator legs
<= d^{-q/2}. What remains is a matrix on label tuples. For distinct
labels, grouping by underlying subsets shows the no-preservation tuple
matrix, with orthogonal all-ones blocks of size q! by p!, has norm
sqrt(p! q!); summing gives <= (r+1) sqrt(r!) d^{-r/2}. The preservation
matrix, after replacing the compression by a projection and controlling
the scalar correction, has norm <= sqrt(p! q!) (1 + r/d); summing gives
(1+r/d) sqrt((r-1)!) d^{-(r-1)/2} times a binomial sum that is <=5
uniformly in r. Restoring scalar parts via the single-word bound
||h_{a_l}...h_{a_1}|| <= (2l+1) d^{-(l-1)/2} and using at most r!
ordered tuples yields the stated beta_W with constant 7 for distinct
labels, and C_r for repeated labels. QED
Corollary 3.5 (square-sum through one label). For every N>=1 there is
d_0(N) such that whenever d >= d_0(N),
max_{1<=a<=d} sum over multisets W with |W|=r and a in W of
beta_W^2 <= 64, for 1<=r<=N. (3.2)
Proof. Fix r and label a. There are binom(d-1,r-1) distinct-label
multisets containing a, contributing at most 49. Repeated-label W has
support size t<=r-1; choosing the other labels and the multiplicities
gives at most A_r d^{r-2} such multisets, contributing at most
C_r^2 A_r / d. Choose d_0(N) large enough; the sum is <= 49+15 = 64. QED
3.4 FREE HAAR UNITARIES WITH CONTROLLED LIFTS.
Lemma 3.6. Let M subset B(H) be a separably acting II_1 factor, n>=1,
P=M_n(M). For every free ultrafilter omega there is a Haar unitary
s in P^omega (i.e. tau(s^m)=0 for m!=0) with W*(s) and the constant copy
of P scalar free. Moreover s has representatives s=(s_j)_omega with
s_j in U(M_n(R_j)), R_j subset M a unital full matrix algebra, so for
every selfadjoint h with delta = ||ad_h|_M|| and h_n = I_n otimes h,
||[h_n, s_j]|| <= 2 delta (all j). (3.3)
Proof. A tracial ultraproduct of finite factors is a factor: in a finite
factor F the scalar tau(x)1 lies in the ||.||_2-closed convex hull of
{uxu*: u in U(F)} (its least-norm element is conjugation invariant and
bounded, hence scalar), giving ||x - tau(x)1||_2 <= sup_u ||[x,u]||_2;
a nonzero central ultraproduct element would violate this. Every
unitary lifts to coordinate unitaries (extend the polar partial isometry
using equal-trace of complementary supports). We use Popa's independence
theorem (JFA 2014, Thm 0.1(b)) with B_1 = C1 and X = B cap ker tau in
F_0 = prod_omega(M_n(C) otimes M_j(C)), producing a diffuse abelian
algebra scalar free from B = M_n(C) otimes 1, containing a Haar unitary
s_0 (bisect projections to a uniform selfadjoint, exponentiate). Choose
lifts s_{0,j}. Embed M_n(C) otimes M_j(C) into M_n(M) by fixed matrix
subalgebras R_j^0 ~ M_j(C) of M; this gives a trace-preserving
embedding F_0 -> P^omega with s_0 Haar and scalar free from B.
Then move s_0 into free position from all of P: with P_1=W*(B,s_0),
X = W*(P,s_0) cap ker E_B (2-norm separable), a second application of
Popa's theorem gives a Haar unitary w in I_n otimes M^omega commuting
with B; setting s = w s_0 w* makes C_s = W*(s) scalar free from P
(checking freeness over B via E_B-centered blocks and then deducing
scalar freeness). Finally s_j = (I_n otimes z_j) s_{0,j} (I_n otimes z_j)*
for lifts z_j of w = I_n otimes z. The commutator bound: averaging h
over the compact U(R_j) gives h^{(j)} in R_j' with ||h-h^{(j)}|| <= delta,
so I_n otimes h^{(j)} commutes with s_j and
||[h_n,s_j]|| = ||[I_n otimes (h-h^{(j)}), s_j]|| <= 2 delta. QED
Lemma 3.7 (unitary twists). Let (F,tau) be finite tracial, P subset F a
unital von Neumann subalgebra, s in F Haar and scalar free from P. For
every U in U(P), both sU and Us are Haar and scalar free from P.
Proof. For v = sU it suffices to show tau(v^k)=0 (k!=0) and
tau(v^{k_1} x_1 ... v^{k_r} x_r)=0 for nonzero integers k_i and
x_i in P cap ker tau (cyclic mixed words suffice, cyclically merging
endpoints). Regard a trace word as a cyclic sequence of s, s* letters
separated by P-elements; expansion of (sU)^k uses (sU)^k or (U*s*)^{|k|}.
At every sign change the separator is a centered element of P (U x U* or
x). Splitting separators into scalar and centered parts, a term with any
centered separator left vanishes by freeness of s and P; a term with none
is a scalar multiple of a nonzero power of s, also zero. Centered Laurent
polynomials are ||.||_2-dense in the centered part of W*(v), so this gives
scalar freeness. Apply to s* and U*, take adjoints, to get Us. QED
3.5 PROOF OF THEOREM 3.1 (RANDOMIZED COEFFICIENTS). Fix M subset B(H)
type II_1, selfadjoint h, delta = ||ad_h|_M|| > 0. It suffices to bound
the commutator with an arbitrary unitary U in U(P), P = M_n(M), on an
arbitrary unit vector X in H^n. Set h_n = I_n otimes h, Y = UX,
B = M_n(C) otimes 1. Choose in order
R_* = 100 pi, N >= 100 n R_*^3, k >= N, d = nk. (3.4)
Then k large enough that Corollary 3.5 holds for all r<=N. Choose a
unital matrix algebra R ~ M_k(C) in M; let e_1,...,e_d be the diagonal
minimal projections of D = M_n(R). Take the free unitary s=(s_j)_omega
of Lemma 3.6; put v_j = s_j U, v = sU. By Lemma 3.7, v is Haar and free
from the constant algebra P.
Fix q > 2N and independently choose q-th roots of unity theta_1,...,theta_d;
set t = sum_a theta_a e_a, L_j = t v_j. For 1<=r<=N set
Q^+_{r,j} = (t v_j)^{r-1} t, Q^-_{r,j} = (v_j* t*)^r.
Grouping phase monomials by the multiset W of the r labels that occur,
Q^+-_{r,j} = sum_{|W|=r} chi^+-_W(theta) K^+-_{W,j},
chi^+-_W(theta) = prod_{a in W} theta_a^{+-1}.
These characters are distinct and orthogonal (q>2N). Orthogonality and
unitarity give sum_W K_{W,j}* K_{W,j} = I and K_{W,j} = K_{W,j} p_W with
p_W = sum_{a in W distinct} e_a (the rightmost phase factor is a diagonal
projection with label in W). The classes K_W = (K_{W,j})_omega have norm
<= beta_W by Lemma 3.4: moving powers of v left rewrites each coefficient,
up to a left unitary, as a sum of projection words from the scalar free
algebras v^{-l} D v^l. Hence (3.2) gives, for 1<=r<=N,
max_a sum_{|W|=r, a in W} beta_W^2 <= 64. (3.5)
Truncate: f_b(u)=min{1,b/u} (f_b(0)=1), K'_{W,j} = K_{W,j} f_{beta_W}(|K_{W,j}|),
E_{W,j} = K_{W,j} - K'_{W,j}. Then ||K'_{W,j}|| <= beta_W, the right support
of K'_{W,j} is in p_W, and the column contractions hold:
sum_W (K'_{W,j})* K'_{W,j} <= I, sum_W E_{W,j}* E_{W,j} <= I. (3.6)
Continuous functional calculus in the quotient gives
lim_omega ||E_{W,j}||_2 = 0.
Average h over U(R): h^0 = int_{U(R)} u h u* du in R', ||h-h^0|| <= delta,
||ad_{h^0}|_M|| <= delta (averaging [u h u*, a] = u [h, u* a u] u*).
Write h_n^0 = I_n otimes h^0. For each W there is a rectangular partial
isometry V in M_{n,1}(R) with V V* = p_W (support has rank <= r <= k, the
source dimension). Its entries commute with h^0, so
[h_n^0, K'_{W,j}] Z = (h_n^0 K'_{W,j} V - K'_{W,j} V h^0) V* Z.
Lemma 3.2 bounds this by 8 delta beta_W ||p_W Z||. Orthogonality of the
e_a and (3.5) give sum_W beta_W^2 ||p_W Z||^2 <= 64 ||Z||^2. Summing the
squared column bounds and taking square roots:
( sum_W ||[h_n^0, K'_{W,j}] Z||^2 )^{1/2} <= 64 delta ||Z||,
and replacing h^0 by h costs 2 delta ||Z||:
( sum_W ||[h_n, K'_{W,j}] Z||^2 )^{1/2} <= 66 delta ||Z||. (3.7)
For Z = X, restoring the E_{W,j} costs a quantity -> 0 along omega
(Lemma 3.3, finitely many W). For the moving vector Z = v_j X = s_j Y,
write [h_n, E_{W,j}] s_j Y = h_n E_{W,j} s_j Y - E_{W,j} s_j h_n Y -
E_{W,j} [h_n, s_j] Y; the first two terms are 2-small in column norm,
the last has column norm <= 2 delta by (3.6) and (3.3). Phase orthogonality
converts these column bounds to root-mean-square bounds:
lim_omega ( E_theta ||[h_n, Q^+_{r,j}] v_j X||^2 )^{1/2} <= 68 delta,
lim_omega ( E_theta ||[h_n, Q^-_{r,j}] X||^2 )^{1/2} <= 66 delta, (3.8)
for all r<=N.
3.6 RECOVERING A SCALAR. Independently choose a uniform q-th root of
unity z and a uniform standard basis column alpha in C^n; p=alpha alpha*
in B. Define f_j = N^{-1/2} sum_{r=1}^N (z L_j)^r, F_j = sqrt(n) alpha* f_j,
G_j = sqrt(n) alpha* f_j*. Averaging over alpha and using orthogonality of
z,...,z^N gives E||F_j X||^2 = E||G_j X||^2 = 1. The same orthogonality
removes cross terms in the row drifts:
lim_omega ( E ||h F_j X - F_j U* h_n Y||^2 )^{1/2} <= 70 delta,
lim_omega ( E ||h G_j X - G_j h_n X||^2 )^{1/2} <= 70 delta. (3.9)
Choose contractions x_j in M so the average of G_j* x_j F_j has a nonzero
scalar weak limit. Put A_j = p f_j^2 p and
x~_j = A_j* (A_j A_j* + .01 p)^{-1/2} = alpha x_j alpha*
(in the unital corner pPp); functional calculus gives ||x_j|| <= 1.
Let C_j = G_j* x_j F_j = n f_j x~_j f_j, ||C_j|| <= nN.
For ultrapower classes: u = z t v is Haar and free from P (Lemma 3.7), and
C = (C_j)_omega in W*(u,p). Freeness gives E_P(C) = lambda p + mu (1-p)
(expand a polynomial in u,u*,p into alternating centered letters; centered
u-letters have zero expectation onto P, the rest lie in W*(p)). The joint
free law of u,p is the same for all choices since tau(p)=1/n, so lambda,mu
depend only on n,N. As E_alpha p = I/n,
E E_P(C) = c_* I, c_* = tau(C). (3.10)
Lower bound on c_*: traciality and functional calculus give
c_* = n tau(A A* (A A* + .01 p)^{-1/2}) >= n ||p f^2 p||_1 - .1, because
t^2(t^2+.01)^{-1/2} >= t - .1 for t>=0 and n tau(p)=1. Let e be the
spectral projection of u on the arc {e^{it}: |t|<=R_*/N}, eta=tau(e)=
R_*/(pi N). Haar measure gives tau(|f|^2)=1, ||f^2||<=N; the geometric-sum
bound |sum_{r=1}^N e^{irt}| <= pi/|t| gives
tau(|f|^2 (1-e)) <= pi/R_*. (3.11)
Since f commutes with e, tracial Cauchy-Schwarz plus freeness of p from
f,e give n ||p(f^2 - e f^2 e)p||_1 = tau(|f|^2(1-e)) <= pi/R_*. With
s_e = n e p e in the corner eP^omega e, the polar decomposition of
sqrt(n) e p gives n ||p e f^2 e p||_1 = ||s_e^{1/2} f^2 s_e^{1/2}||_1.
Free moments give ||s_e^{1/2}||_2 = sqrt(eta) and
||s_e - e||_2^2 = (n-1) eta^2, hence ||s_e^{1/2} - e||_2 <= sqrt(n-1) eta
and
||s_e^{1/2} f^2 s_e^{1/2} - e f^2 e||_1 <= 2N sqrt(n-1) eta^{3/2}.
So c_* >= .9 - 2 pi/R_* - 2N sqrt(n-1) (R_*/(pi N))^{3/2} >= 1/2, by
(3.4). For each fixed random choice, Lemma 3.3 identifies the WOT
ultralimit of C_j with E_P(C); finite random spaces, fixed before the
ultralimit, give
WOT-lim_{j->omega} E C_j = c_* I. (3.12)
3.7 EXTRACTING THE MATRIX COMMUTATOR. Use the defining scalar bound on x_j:
|E ip{[h,x_j] F_j X, G_j X}| <= delta E(||F_j X|| ||G_j X||) <= delta
by the row RMS. Since h=h*, the averaged inner expression equals
ip{x_j F_j X, h G_j X} - ip{x_j h F_j X, G_j X}. Substituting the main
terms (3.9), Cauchy-Schwarz and the row RMS bound the two average errors
by ultralimit 140 delta. What is left is
ip{C_j X, h_n X} - ip{C_j U* h_n Y, X}, so (3.12) gives
c_* |ip{h_n X, X} - ip{h_n U X, U X}| <= 141 delta.
The forms are real, and c_* >= 1/2, so the difference is <= 282 delta.
Supremum over unit X:
||h_n - U* h_n U|| <= 282 delta, ||[h_n,U]|| <= 282 delta.
A selfadjoint contraction is the average of a +- i(1-a^2)^{1/2}, and a
general contraction splits into real and imaginary parts, bounding its
commutator by 564 delta. Level n=1 is the definition of delta. Supremum
over n gives Theorem 3.1 with C_amp = 600. QED
==============================================================================
SECTION 4. BOUNDARY AVERAGING AND MEASURABLE CORRECTION
==============================================================================
Given countably many unitaries in N whose products are only approximately
respected, we correct them into an exact measurable cocycle over an
amenable action of the countable index group G. On D^2 the separate
action averages cocycles; the diagonal action has a rigidity property.
For a nonsingular action of a countable G on a probability space (D,nu):
alpha_g F(x) = F(g^{-1}x), (lambda_g F)(t,x) = F(g^{-1}t, g^{-1}x),
on G x D with counting measure in the first variable. A bounded operator
field on a separable K is an essentially bounded weakly measurable map
into B(K); its norm is the essential supremum (a measurable function by
separability).
Lemma 4.1 (equivariant boundary means). For every countable G there are
a standard probability space (D,nu) with a nonsingular G-action and a
positive unital contraction m: L^inf(G x D) -> L^inf(D) with
m(lambda_g F) = alpha_g(mF), m(f F) = f m(F) (f in L^inf(D)). (4.1)
There is m^{(2)} with the same properties for the separate G x G action
on D^2. Each extends to a contraction on bounded operator fields on any
separable K, preserving adjoints and von Neumann-algebra values and
satisfying the module identity m(B F C) = B m(F) C for bounded fields
B,C independent of the group variable.
Proof. Take a symmetric probability mu on G with mu(g)>0, X=G^{N u {0}}
with P = mu^{N u {0}}, and the commuting maps g(omega_0,omega_1,...) =
(g omega_0, omega_1, ...) and S(omega_0,omega_1,...) = (omega_0 omega_1,
omega_2, ...); both pushforwards are equivalent to P by full support. Let
I be the completed sigma-algebra of S-invariant sets modulo null sets; it
is countably generated (separable measure algebra). Encode a countable
dense generating family and all G-translates into
D = {0,1}^{N x G}, nu = b_* P, with b S = b a.e.; the coordinate-
permutation G-action on D is nonsingular and b intertwines. Pullback by b
identifies L^inf(D) with the S-fixed subalgebra, and same for L^2. For
F in L^inf(G x D) set (JF)(omega) = F(omega_0, b omega), T_n F =
(1/n) sum_{k<n} (JF) circ S^k. Weak-star cluster T of the T_n over the
product of weak-star compact balls is linear, positive, unital,
contractive. Pullback by a nonsingular map is weak-star continuous
(Radon-Nikodym preadjoint), and the telescoping bound
||(T_n F) circ S - T_n F|| <= 2||F||/n forces (TF) circ S = TF, so
TF = (mF) circ b. Since J lambda_g F = (JF) circ g^{-1} and the action
commutes with S, weak-star continuity gives m(lambda_g F) = alpha_g(mF);
b circ S = b gives m(f F) = f m(F). The product mean uses the two
commuting shifts on X^2 and rectangular averages; a common fixed function
is exactly a function of (b omega, b omega') (Fubini + commuting
projections onto L^2(I) otimes L^2(X) and L^2(X) otimes L^2(I)). The
operator-field extension is defined by applying scalar m to coefficients
ip{F(t,x) xi, eta} on a countable dense rational subspace; sesquilinearity
and the norm bound define a bounded operator, positivity preserves
adjoints, and the module identity is checked first on countably valued
fields then by uniform approximation. If F takes values in a von Neumann
algebra A, its mean commutes with a countable strongly dense subset of
A', hence lies in A. All applies to m^{(2)}. QED
Lemma 4.2 (correction of a measurable unitary cocycle). Let D, m be as
above, A subset B(K) a von Neumann algebra on separable K. If
V_g in L^inf(D,U(A)) and e = sup_{g,h} ||V_g alpha_g(V_h) - V_{gh}||_inf
<= 1/64, then there are measurable unitary fields W_g with
W_g alpha_g(W_h) = W_{gh}, sup_g ||W_g - V_g||_inf <= 2e. (4.2)
In particular W_1 = I; the identities hold on a G-invariant conull set.
Proof. One quadratic-improvement step, then iteration. For the current
family set H_g(t,x) = V_t(x) V_{g^{-1}t}(g^{-1}x)*, A_g = m_t H_g. The
defect gives ||H_g - V_g||_inf <= e, so ||A_g - V_g|| <= e and
||H_g - A_g|| <= 2e; H_g unitary, A_g contractions. The exact
factorization H_{gh} = H_g lambda_g(H_h) and m(lambda_g H_h) =
alpha_g(A_h), with the module identities cancelling linear terms, give
A_{gh} - A_g alpha_g(A_h) = m((H_g - A_g)(lambda_g H_h - alpha_g(A_h))),
I - A_g* A_g = m((H_g - A_g)*(H_g - A_g)),
both of norm <= 4e^2; likewise ||I - A_g A_g*|| <= 4e^2. A_g is within
e<1 of a unitary, hence invertible; its unitary polar part
V'_g = A_g (A_g* A_g)^{-1/2} satisfies ||V'_g - A_g|| <= 4e^2, and the
new defect is <= 16e^2, with ||V'_g - V_g|| <= e + 4e^2. Iterating with
e_0 = e, e_{k+1} <= 16 e_k^2 <= e_k/4, the families converge uniformly
in g and in essential norm to unitary fields W_g; total change
<= sum_k (e_k + 4e_k^2) <= (17/12)e <= 2e. Countability of G and
nonsingularity let us delete all exceptional sets; the cocycle identity
at g=h=1 makes W_1 idempotent, so W_1 = I. QED
Lemma 4.3 (averaged intertwiner). Take (Gamma,Z) = (G,D) or (G^2,D^2)
with its mean, U: Gamma -> U(K) a unitary representation on separable K,
Sigma_s in L^inf(Z,U(K)) an exact cocycle. If
sup_s ||Sigma_s - U_s||_inf <= delta < 1, there is L in L^inf(Z,U(K)) with
||L - I||_inf <= 2 delta, L(z) U_s = Sigma_s(z) L(s^{-1}z). (4.3)
Proof. Set B = m_t(Sigma_t U_t*), ||B - I|| <= delta, invertible.
Equivariance, module identities, and the cocycle identity give
Sigma_s alpha_s(B) U_s* = m_t(Sigma_t U_t*) = B, so B U_s =
Sigma_s alpha_s(B). Taking adjoints and products shows B* B =
U_s alpha_s(B* B) U_s*, hence the identity holds for the unitary polar
part L. The bounds (1-delta)I <= |B| <= (1+delta)I give ||L - B|| <= delta.
QED
Lemma 4.4 (diagonal comparison in a finite algebra). Let A be a finite
von Neumann algebra with faithful normal trace tau and separable L^2(A,tau),
and W_g in L^inf(D,U(A)) an exact cocycle with
r = sup_g ess sup_{x,y} ||W_g(x) W_g(y)* - I|| <= r < 1. Then there is a
measurable field a: D^2 -> U(A) with ||a - I||_inf <= 2r and
a(x,y) = W_g(x) a(g^{-1}x, g^{-1}y) W_g(y)*. (4.4)
If sup_g ||W_g - u_g|| <= delta for constant unitaries u_g, the conclusion
holds with ||a-I|| <= 4 delta when 2 delta < 1.
Proof. For a.e. (x,y) let C_{x,y} be the closed convex hull in L^2(A,tau)
of {W_t(x) W_t(y)* : t in G}; it has a unique least-norm vector c(x,y).
Measurability: enumerate rational convex combinations z_j(x,y); d(x,y) =
inf_j ||z_j||_2 is measurable (bounded weakly measurable fields are
strongly measurable in separable L^2); choose the first j with
||z_j||_2^2 < d^2 + 2^{-n}; the parallelogram identity makes the choices
Cauchy, converging pointwise in L^2 to a measurable c. Each z_j is a
contraction with ||z_j - I|| <= r, and these persist under bounded WOT
limits; identifying WOT and L^2 limits on the trace vector shows
c(x,y) in A, ||c|| <= 1, ||c - I|| <= r. Trace-Hilbert measurability on
bounded sets gives WOT measurability. The isometry z |-> W_g(x) z W_g(y)*
sends C_{g^{-1}x,g^{-1}y} onto C_{x,y} (it sends generator t to gt), so
c is equivariant; its unitary polar part a is measurable, ||a - I|| <= 2r,
and equivariant since polar decomposition respects unitary multiplication.
Comparing with u_g gives ||W_g(x) W_g(y)* - I|| <= 2 delta. QED
Lemma 4.5 (double ergodicity with unitary coefficients). For the D of
Lemma 4.1, and sigma: G -> U(K) a unitary representation on separable K,
if bounded measurable f_0: D^2 -> K satisfies
f_0(gx,gy) = sigma(g) f_0(x,y) a.e., then f_0 is a.e. a constant vector
fixed by every sigma(g).
Proof. Use bilateral independent coordinates (omega_j)_{j in Z} with law
mu, left shift theta. Set b_+(omega) = b(omega_0,omega_1,...),
b_-(omega) = b(omega_{-1}^{-1}, omega_{-2}^{-1},...), f(omega) =
f_0(b_+ omega, b_- omega). Symmetry of mu makes b_+, b_- independent with
law nu. From b circ S = b and equivariance:
b_+(theta omega) = omega_0^{-1} b_+(omega),
b_-(theta omega) = omega_0^{-1} b_-(omega), hence
f(theta^n omega) = sigma(omega_{n-1}^{-1}...omega_0^{-1}) f(omega). (4.5)
Let f_m be the conditional expectation on coordinates [-m,m],
eps_m = ||f - f_m||_2 -> 0. For n > 2m+1 set Q = sigma(omega_m^{-1}...
omega_0^{-1}), A = Q f_m, P = sigma(omega_{n-m-1}^{-1}...omega_{m+1}^{-1}),
R = sigma(omega_{n-1}^{-1}...omega_{n-m}^{-1}), B = R^{-1} f_m circ theta^n;
A,P,B depend on disjoint coordinates, hence independent, and (4.5) gives
||P A - B||_2 <= 2 eps_m. With a_0 = E A, b_0 = E B, independence gives
E||PA - B||^2 = E||A-a_0||^2 + E||B-b_0||^2 + E||P a_0 - b_0||^2, whence
||A-a_0||_2, ||B-b_0||_2 <= 2 eps_m. Undoing the transports,
||f - Q^{-1}a_0||_2 <= 3 eps_m (function of [0,m]) and
||f circ theta^n - R b_0||_2 <= 3 eps_m (function of [-m,-1] shifted back).
So f lies in both the closed L^2 subspace of nonnegative coordinates and
that of negative coordinates; these sigma-algebras are independent, so
their intersection is constants. Write f = v a.e.; then f_0 = v a.e.
(joint law nu x nu) and equivariance gives sigma(g) v = v. QED
==============================================================================
SECTION 5. FINITE FACTORS IN THEIR ORIGINAL REPRESENTATION
==============================================================================
Replacing H by any separable space as needed, we prove:
THEOREM 5.1 (Uniform stability of finite factors). There are t_fin>0 and
a nondecreasing eta_fin: (0,t_fin) -> (0,inf) with eta_fin(t)->0 as
t->0, such that: if H is a separable nonzero Hilbert space, M subset B(H)
a unital finite factor, N subset B(H) a von Neumann algebra with the same
identity, and d(M,N) < t < t_fin, then N is a finite factor and there is
a unitary u in B(H) with u M u* = N and ||u-I|| <= eta_fin(t). The
threshold and modulus are independent of M,N and their representations.
5.1 NEIGHBORS OF FINITE FACTORS.
Lemma 5.2. If M subset B(H) is a unital finite factor and N subset B(H)
is a von Neumann algebra with d(M,N) < 1/10, then N is a finite factor.
Proof. Choose t with d(M,N) < t < 1/10. If v in N were a proper isometry,
choose x in M_1 with ||x - v|| < t; then x* x >= (1-t)^2 I, so the polar
part w of x is an isometry in M, hence unitary (finiteness). Since
1-t <= |x| <= 1, ||w - v|| <= ||w - x|| + ||x - v|| < 2t < 1, impossible (a
proper isometry within distance <1 of a unitary is invertible). So N is
finite. Now let q in Z(N) be a projection, choose a in M_1 with
||a - q|| < t. For v in U(M) choose b in N_1 with ||v - b|| < t; q
commutes with b, so ||v a v* - q|| <= ||a - q|| + ||[v,q]|| < 3t. The
trace-Hilbert closed convex hull of {v a v* : v in U(M)} contains tau(a) I
(least-norm element is conjugacy invariant, bounded, hence scalar), so
||q - tau(a) I|| <= 3t < 1/2; a nontrivial projection has distance 1/2
from the scalars, so q in {0,1}. Thus Z(N) = C I. QED
5.2 AN IRREDUCIBLE HYPERFINITE SUBFACTOR. An inclusion R subset M is
irreducible when R' cap M = C I.
Lemma 5.3 (Popa 1981, with construction). Every separably-acting II_1
factor M contains an irreducible unital hyperfinite II_1 subfactor
R = (union_j Q_j)'' with Q_j increasing unital full matrix algebras of
sizes tending to infinity.
Proof. One element nearly scalar under a matrix average: in a II_1 factor
P, given selfadjoint y and eps>0, approximate y in ||.||_2 by a spectral
step y_0 = sum_i lambda_i p_i, split into equivalent pieces of trace 1/b
to build a unital F ~ M_b(C), let y_1 in F be diagonal, and Haar-average
over U(F); then ||E_{F' cap P}(y) - tau(y) I||_2 <= 2||y - y_1||_2 < eps
for b large. Enumerate an ||.||_2-dense sequence (x_j) of the selfadjoint
unit ball and requirements (j,k). Start Q_0 = C I; given the full matrix
algebra Q_n, P_n = Q_n' cap M is a II_1 factor (matrix units identify it
with a corner). For (j,k), apply the one-element step in P_n to
E_{P_n}(x_j) with error 1/k, get F subset P_n, set Q_{n+1} = Q_n vee F.
Then P_{n+1} = F' cap P_n, E_{P_{n+1}} = E_{P_{n+1}} E_{P_n}, so
||E_{P_{n+1}}(x_j) - tau(x_j)I||_2 < 1/k and earlier bounds persist. For
R = (union Q_n)'', E_{R' cap M} sends every x_j to tau(x_j)I; density gives
R' cap M = C I. R is a hyperfinite II_1 factor. QED
Consequence: if E_j is Haar averaging over U(Q_j), then E_j is the
||.||_2-orthogonal projection onto Q_j' cap M; the decreasing subspaces
intersect in C I, so
E_j(x) -> tau(x) I in ||.||_2 (x in M). (5.1)
5.3 COMMON TRACIAL POSITION. A factor P subset B(K) is in standard
tracial position with vector Omega when x |-> x Omega identifies K with
L^2(P,tau_P); its conjugation is J_P(x Omega) = x* Omega.
Theorem 5.4 (common position; CCSSWW 2014 Lemma 4.10). Let M,N be
separably-acting II_1 factors on one Hilbert space with mutual strict
g-near inclusions, 0 < g < 1.74e-13, and let R subset M cap N be an
amenable von Neumann subalgebra with R' cap M subset R. Then there are
faithful normal representations pi: M -> B(K), rho: N -> B(K), a common
unit vector Omega, and beta < 50948 sqrt(g) such that:
(i) both images are in standard tracial position with vector Omega;
(ii) pi|_R = rho|_R, and the images have mutual strict beta-near inclusions;
(iii) for contractions x in M, y in N, ||pi(x) - rho(y)|| <= beta + ||x-y||;
(iv) with e_R the projection onto closure of pi(R)Omega,
(pi(M) cup {e_R})'' = (rho(N) cup {e_R})''.
K can be separable.
Below we suppress pi,rho. Normalizing near-inclusion approximants gives
d(M,N) <= 2 beta.
Lemma 5.5 (common right action). In Theorem 5.4, J_M r J_M = J_N r J_N for
all r in R; hence T = J_N J_M - I commutes with the left and right actions
of R.
Proof. The commutant of (M cup {e_R})'' is J_M R J_M: an element of M' is
right multiplication by some a in M; commuting with e_R forces a Omega in
closure(R Omega), so a in R. Likewise for N. Equality of the basic
constructions gives J_M R J_M = J_N R J_N. For r in R both operators send
Omega to r* Omega, and Omega is separating for the common right algebra,
so they agree. Applying twice gives the last claim. QED
Lemma 5.6 (testing a bimodular operator). Let M be a II_1 factor and
R = (union Q_j)'' subset M as in Lemma 5.3. If a bounded operator V on
L^2(M,tau) commutes with left and right multiplication by R, then
||V|| <= 384 sup_{a,b in M_1} |ip{V a Omega, b Omega}|.
Proof. Let K denote the supremum. Start with bounded a,b with
||a||_2,||b||_2 <= 1. By (5.1), finite equal-weight convex averages of
conjugations of aa* and bb* by unitaries of R bring them within one of
their traces in L^2. Choose families (d_i),(e_l); enumerate the Cartesian
product and set a_k = d_i a e_l, b_k = d_i b e_l. Covariance averages
L_a = m^{-1} sum a_k a_k*, Q_a = m^{-1} sum a_k* a_k satisfy
||L_a||_2,||Q_a||_2 <= 2 (similarly for b). Use the same fair signs:
A = m^{-1/2} sum eps_k a_k, B = m^{-1/2} sum eps_k b_k. Bimodularity and
sign orthogonality give E ip{V A Omega, B Omega} = ip{V a Omega, b Omega}
and E||A||_2^2, E||B||_2^2 <= 1. With ||x||_4^4 = tau((x*x)^2), the fourth
moment computation and tracial Cauchy-Schwarz (|tau((a_k* a_l)^2)| <=
tau(a_k a_k* a_l a_l*)) yield E||A||_4^4, E||B||_4^4 <= 12. Clipping
A = w|A| to A_s = w|A| 1_{[0,s]}(|A|) gives ||A_s|| <= s,
E||A - A_s||_2^2 <= 12/s^2 (likewise B_s). Cauchy-Schwarz in probability:
|ip{V a Omega, b Omega}| <= s^2 K + (2 sqrt(12)/s)||V||. Take
s = 4 sqrt(12), supremum over dense unit vectors: ||V|| <= 192 K + ||V||/2,
so ||V|| <= 384 K. QED
Lemma 5.7 (norm control of tracial conjugations). In the setting of
Theorem 5.4, with R an irreducible hyperfinite subfactor of M presented
by increasing matrix algebras and with agreeing right actions, if
d(M,N) < b then ||J_M - J_N|| <= C_J b, C_J = 768.
Proof. T = J_N J_M - I commutes with both actions of R (Lemma 5.5). For
x in M_1 choose y in N_1 with ||x - y|| < b; then
||T x Omega|| = ||J_N x* Omega - x Omega|| <= ||(x* - y*)Omega|| +
||(x-y)Omega|| < 2b. So the supremum in Lemma 5.6 is <= 2b, giving
||T|| <= 768 b; ||T|| = ||J_M - J_N||. QED
5.4 FROM THE BOUNDARY TO A JORDAN MAP. Work in the common tracial
representation. Goal: build a Hilbert-space unitary S_0 fixing Omega with
S_0 u Omega = b_u Omega, b_u in U(N), ||b_u - u|| = O(b), uniformly in u
in U(M). Unitaries span M linearly; linearity of S_0 and separating Omega
will make u |-> b_u extend to a unital linear map preserving the Jordan
product x circ y = (xy + yx)/2; orientation then yields a *-isomorphism.
Lemma 5.8 (orientation of a Jordan map into a factor; Herstein 1956).
A surjective linear Jordan homomorphism between unital complex algebras,
whose target is a factor von Neumann algebra, is either a homomorphism or
an antihomomorphism.
Proof. Preservation of the Jordan product gives preservation of xyx =
2 x circ (x circ y) - x^2 circ y, and polarization gives preservation of
xyz + zyx. For fixed a,b set A = theta(a), B = theta(b), D = theta(ab),
E = theta(ba) = AB + BA - D, h = D - AB, k = D - BA. Computing
theta(ab c ba + ba c ab) two ways (triple identity vs. nesting) gives
h C k + k C h = 0 for all C; substituting C = C_1 k C_2 and using the
identity on C_1, C_2 shows h C_1 k C_2 k = 0. A factor is prime
(x N y = {0} => x=0 or y=0), so if k != 0 then h C_1 k = 0 for all C_1,
then h = 0. Thus for each (a,b) either theta(ab)=theta(a)theta(b) or
theta(ab)=theta(b)theta(a). For fixed a the two solution sets are additive
subgroups covering the additive group; a group is not the union of two
proper subgroups, so one identity holds for all b; a second two-subgroup
argument makes one orientation global. QED
Proposition 5.9. There are b_0>0 and C_st>0 such that: if M,N subset B(K)
are II_1 factors in common standard tracial position with vector Omega,
contain a common hyperfinite II_1 subfactor R irreducible in M and
presented by increasing matrix algebras, and J_M r J_M = J_N r J_N for all
r in R, then for d(M,N) < b < b_0 there is a normal unital *-isomorphism
vartheta: M -> N with sup_{x in M_1} ||vartheta(x) - x|| <= C_st b.
Proof. Let G subset U(M) be a countable strongly dense subgroup. For each
g choose a unitary v_g in N with ||v_g - g|| <= 3b (a contraction
approximant is invertible for b<1; take its polar part). Then
sup_{g,h} ||v_g v_h - v_{gh}|| <= 9b. Lemma 4.2 (on the boundary space D)
gives measurable unitary fields W_g: D -> U(N) with
W_g(x) W_h(g^{-1}x) = W_{gh}(x) and sup_g ||W_g - g||_inf <= C_1 b.
First comparison (separate action on D^2). Form the product cocycle and
constant representation
Sigma_{g,h}(x,y) = W_g(x) J_N W_h(y) J_N, U_{g,h} = g J_M h J_M.
The left and right N-actions commute, so Sigma is exact for the separate
G^2-action, and by Lemma 5.7
sup_{g,h} ||Sigma_{g,h} - U_{g,h}||_inf <= (2C_1 + 2C_J) b. The product
form of Lemma 4.3 gives measurable L: D^2 -> U(K), ||L - I||_inf <= C_2 b,
with L(x,y) U_{g,h} = Sigma_{g,h}(x,y) L(g^{-1}x, h^{-1}y).
Second comparison (diagonal comparison in N). Since
sup_g ess sup_{x,y} ||W_g(x) W_g(y)* - I|| <= 2C_1 b, Lemma 4.4 gives
measurable a: D^2 -> U(N) with ||a - I||_inf <= C_3 b and
a(x,y) = W_g(x) a(g^{-1}x, g^{-1}y) W_g(y)*.
Combine: the field f(x,y) = L(x,y)* a(x,y) Omega is equivariant for the
diagonal representation g |-> U_{g,g} (right multiplication by W_g(y)* on
trace vectors is J_N W_g(y) J_N). By Lemma 4.5, f is a constant invariant
vector. The invariant vectors of g |-> g J_M g J_M are exactly C Omega:
strong density extends invariance to U(M); averaging over U(Q_j) acts as
E_j and converges to the projection onto C Omega by (5.1). Hence
L(x,y)* a(x,y) Omega = lambda Omega, |lambda|=1, |lambda-1| <= (C_2+C_3)b.
Delete one null set and its countably many G^2-translates; at a retained
pair (x,y) define the Hilbert-space unitary S_0 = lambda a(x,y)* L(x,y),
fixing Omega. Using the product intertwiner at (g,1) and the diagonal
relation at (g^{-1}x,y),
S_0 g Omega = a(x,y)* W_g(x) a(g^{-1}x,y) Omega = b_g Omega,
b_g in U(N), ||b_g - g|| <= C_4 b, uniformly in g.
Every u in U(M) is a strong limit of g_j in G. The vectors b_{g_j} Omega
= S_0 g_j Omega are Cauchy in trace L^2(N); for unitaries traciality gives
||b_{g_j}* - b_{g_l}*||_2 = ||b_{g_j} - b_{g_l}||_2. Bounded L^2
convergence in the standard representation implies strong convergence
(test on right-multiplier vectors N' Omega), so b_{g_j} and their adjoints
converge strongly to a unitary b_u in N and its adjoint, with
S_0 u Omega = b_u Omega and ||b_u - u|| <= C_4 b.
Every element of M is a linear combination of unitaries, so
vartheta(z) Omega = S_0 z Omega (z in M) defines a linear map into N,
well-defined because Omega is separating for N. Writing a contraction as
real + imaginary selfadjoint parts (each an average of two unitaries) gives
vartheta(I) = I and ||vartheta(z) - z|| <= 2C_4 b ||z||; so vartheta is
bounded and sends unitaries to unitaries. For z = z*, expand
vartheta(e^{itz})* vartheta(e^{itz}) = I in operator norm: the t-coefficient
gives vartheta(z)* = vartheta(z); applied to z^2 and comparing t^2
coefficients gives vartheta(z^2) = vartheta(z)^2. Complex linearity and
polarization show vartheta preserves adjoints and the Jordan product.
When 2C_4 b < 1 the map is injective with closed range; every contraction
of N is within (1+2C_4)b of that range (original gap plus the bound); for
(1+2C_4)b < 1 the range is all of N (else the quotient by a proper closed
subspace has norm one). Lemma 5.8 makes this surjective Jordan map
multiplicative or antimultiplicative. The anti case is excluded: for
unitaries z,w antimultiplicativity gives ||zw - wz|| <= 6C_4 b, while a
unital M_2(C) in M contains anticommuting selfadjoint unitaries of
commutator norm two; 6C_4 b < 2 excludes it. So vartheta is a normal unital
*-isomorphism with C_st = 2C_4 and some absolute b_0. QED
5.5 RETURN TO THE ORIGINAL HILBERT SPACE.
Proof of Theorem 5.1. Choose t > d(M,N) below all absolute thresholds.
By Lemma 5.2, N is a finite factor. If either factor is a matrix algebra,
Theorem 2.3 (mutual near inclusions) gives the near-identity conjugacy with
an absolute linear bound; so assume both are II_1.
Choose R subset M from Lemma 5.3. By Theorem 2.3 there is a unitary w with
||w-I|| <= 150t and w R w* subset N. Replacing N by N_0 = w* N w makes R
common, and d(M,N_0) < 301t. Take g = 302t and apply Theorem 5.4 (need
g < 1.74e-13). Write M_s = pi(M), N_s = rho(N_0); the common subfactor stays
irreducible in M_s by faithfulness, its right actions agree (Lemma 5.5), and
d(M_s,N_s) <= 2 beta with beta < 50948 sqrt(g). Choose a strict standard-gap
bound b = 3 * 50948 sqrt(g) < b_0; Proposition 5.9 gives a normal
*-isomorphism vartheta_s: M_s -> N_s with ||vartheta_s - id|| <= C_st b.
Transport: theta = rho^{-1} vartheta_s pi: M -> N_0. For x in M_1 choose
y in (N_0)_1 with ||x-y|| < g; the original-contraction comparison and
faithfulness of rho give
||theta(x) - x|| <= ||vartheta_s pi(x) - pi(x)|| + ||pi(x) - rho(y)|| + g
<= C_st b + beta + 2g =: e(t), and e(t) -> 0.
On H oplus H consider the faithful normal paired representation
x |-> x oplus theta(x) of M and the selfadjoint switch s(xi,eta)=(eta,xi),
whose scalar commutator norm on that factor is <= e(t). Theorem 3.1 gives
||ad_s||_cb <= C_amp e(t), and at each matrix level the switch commutator
has off-diagonal block [x_{ij} - theta(x_{ij})], so ||theta - id||_cb <=
C_amp e(t). For small absolute t, Lemma 2.2 gives a unitary v in B(H) with
v M v* = N_0 and ||v - I|| <= C_pair C_amp e(t). Then u = w v conjugates M
onto N with ||u - I|| <= 150t + C_pair C_amp e(t). Taking t_fin below all
absolute thresholds and eta_fin(t) the max of this bound and the matrix-
factor bound (both nondecreasing, -> 0) finishes. QED
==============================================================================
SECTION 6. PREPARING INFINITE FACTORS AND MODULAR COMPARISON
==============================================================================
Now infinite factors. A common B(l^2) makes the ordinary gap completely
bounded; a fixed hyperfinite tensor factor produces type III_1 factors
with an irreducible finite subfactor in a state centralizer; alignment of
left/right actions gives a common cyclic separating vector. The main
estimate compares the two Tomita graphs uniformly in the rescaling.
6.1 AMPLIFICATION BY COMMON ISOMETRIES.
Lemma 6.1. If M,N subset B(H) contain the same unital copy of B(l^2),
then d_cb(M,N) = d(M,N).
Proof. For each n choose isometries s_1,...,s_n in that copy with
s_i* s_j = delta_{ij} I and sum_i s_i s_i* = I. The map V: H^n -> H,
V(xi_i) = sum_i s_i xi_i, is unitary and V M_n(P) V* = P for P = M,N
(V[x_{ij}]V* = sum s_i x_{ij} s_j* with inverse x_{ij} = s_i* x s_j). The
same unitary identifies the amplified unit balls with the original ones;
sup over n. QED
Fix lambda = e^{-1}, mu = e^{-sqrt 2}. For 0<nu<1 let rho_nu be the state
of M_2(C) with density (1+nu)^{-1} diag(1,nu). Form a countable infinite
tensor product D with product state rho containing infinitely many sites of
each type (M_2(C), rho_lambda) and (M_2(C), rho_mu), in the site standard
representations on a separable K. The product vector Omega_rho is cyclic
and separating for D (cyclic for the left and right local matrix actions),
so rho is faithful normal; D and its standard commutant are hyperfinite.
Modular notation. In a standard representation with cyclic separating
Omega implementing a faithful normal state psi, the closure S of
x Omega |-> x* Omega has polar decomposition S = J Delta^{1/2}; J is the
modular conjugation, sigma_t^psi(x) = Delta^{it} x Delta^{-it} the modular
group. The centralizer P_psi = {a : psi(ax)=psi(xa) for all x} is the fixed
algebra of sigma^psi. We use: P x|_sigma R is semifinite (III.4.8.1
Blackadar); for a type III factor it is a factor iff P is type III_1, then
type II_inf (Takesaki 1973, Lemma 8.2, Cor 9.7); for a semifinite factor
with faithful normal semifinite trace tau, sigma^tau is trivial and the
crossed product is P otimes L^inf(R), not a factor.
Lemma 6.2 (the fixed tensor factor). D_rho' cap D = C1. For any factor A
with separable predual, A otimes D is a type III_1 factor and
A otimes D ~ (A otimes D) otimes R_lambda, R_lambda the Powers factor built
from (M_2(C), rho_lambda) sites.
Proof. Site swaps are implemented by unitaries of finite site algebras
that preserve rho, so lie in D_rho. For x in D_rho' cap D and a finite-site
a, choose finite-site approximants x_j of x; a swap moves a away from x_j,
and rho(a x) = lim rho(a x_j) = rho(a) rho(x) (state Cauchy-Schwarz). By
density (x - rho(x)1)Omega_rho = 0, so x = rho(x)1; D is a factor.
Choose a faithful normal state psi on A, P = A otimes D with psi otimes rho.
In the regular representation of its modular crossed product on
L^2(R, H_P), the generators are (pi(x)xi)(t) = sigma_{-t}^{psi otimes rho}(x)
xi(t) and (lambda_s xi)(t) = xi(t-s). Let Z be central; Fourier transforming
in t, commutation with all lambda_s makes Z decomposable, and commutation
with the constant P' field restricts its values to P; commutation with
1 otimes D_rho places the field in A otimes 1 (normal slices +
D_rho' cap D = C1). An off-diagonal matrix unit at a nu-site is a modular
eigenoperator of frequency +-log nu; its regular representation is that
matrix unit followed by a frequency translation, so if a(zeta) otimes 1 is
the field then a(zeta + log lambda) = a(zeta) and a(zeta + log mu) = a(zeta)
a.e. The generated subgroup is countable and dense in R; testing on
countably many coefficients and using translation continuity in local L^1
shows each coefficient is constant, so Z = a_0 otimes 1 is a constant field,
and commutation with pi(A otimes 1) gives a_0 in A' cap A = C1. The core is
a factor, so P is type III_1. Adjoining another countable family of
lambda-sites and reindexing gives the absorption. QED
For a faithful normal state psi on P, the bicentralizer B(P,psi) consists
of a with ||[a,x_n]||_psi -> 0 whenever (x_n) is bounded in P and
||x_n psi - psi x_n|| -> 0 (||x||_psi = psi(x*x)^{1/2}; the second norm is
the predual norm of y |-> psi(y x_n - x_n y)). We use: if a sigma-finite
type III_1 factor P absorbs R_lambda for some 0<lambda<1, its bicentralizer
is C1 (Marrakchi 2020, Thm D(ii)); for a type III_1 factor with separable
predual and trivial bicentralizer there is an irreducible hyperfinite II_1
subfactor R subset P with a faithful normal conditional expectation E:P->R
(Haagerup 1987, Lemma 3.8 and proof of Thm 3.1, first in a corner). The
state tau_R circ E centralizes R.
Definition 6.3 (prepared pair). A prepared pair is type III_1 factors
M,N subset B(H) on a separable Hilbert space, a common cyclic separating
unit vector Omega, and a hyperfinite II_1 factor R subset M cap N with
(i) R' cap M = R' cap N = C1;
(ii) the vector state phi(x) = ip{x Omega, Omega} centralizes R on either
algebra;
(iii) J_M r J_M = J_N r J_N for all r in R (J_M, J_N the modular
conjugations for Omega).
Proposition 6.4 (preparing an infinite pair). There are absolute c,C>0
with: if M_0,N_0 subset B(H) are factors on a separable Hilbert space,
M_0 infinite, and d(M_0,N_0) < t < c, then for the fixed D subset B(K) there
are unitaries u_0 in B(H), v in B(H otimes K) with ||u_0 - 1|| + ||v - 1||
<= C t such that M = M_0 otimes D and N = v((u_0 N_0 u_0*) otimes D)v* form a
prepared pair with d_cb(M,N) <= C t.
Proof. A countably decomposable infinite factor is properly infinite and
contains a unital B(l^2) (split I into countably many equivalent
projections, choose matrix units). Theorem 2.3 embeds this amenable
subfactor of M_0 into N_0 by a unitary within 150t of 1; conjugating N_0 the
inverse way gives u_0 for which the subfactor is common (this also shows
N_0 is infinite). The adjusted ordinary gap is <= 301t; Lemma 6.1 makes it
a cb gap; Lemma 2.5 transfers it to the tensor pair since D' is hyperfinite.
By Lemma 6.2 both tensor factors are type III_1 and absorb R_lambda.
Write N_1 = (u_0 N_0 u_0*) otimes D. The bicentralizer and expected-subfactor
results give an irreducible hyperfinite II_1 R subset M and a faithful normal
state phi on M centralizing R; the given representation of M has a cyclic
separating vector for phi (a separable normal representation splits into
cyclic summands, each embedding in the standard representation; a countable
amplification of the standard representation is unitarily equivalent to it
because the type III commutant has countably many isometries summing to 1;
every nonzero projection in that countably decomposable type III commutant
is equivalent to 1). Transport to get Omega on H otimes K, without
conjugating either algebra.
Put S = J_M R J_M subset M'. Centralization gives r Omega = J_M r* J_M Omega.
Apply Theorem 2.3 to R subset M and its near inclusion in N_1: get v_1 with
v_1 R v_1* subset N_1, ||v_1 - 1|| = O(t). Put N_2 = v_1* N_1 v_1, so
R subset M cap N_2 and d(M',N_2') = O(t) (Lemma 2.1). Apply Theorem 2.3 to
S subset M' and its near inclusion in N_2': get v_2 with v_2 S v_2* subset
N_2', ||v_2 - 1|| = O(t), v_2 in (S cup N_2')'' subset R'. Set
N = v_2* N_2 v_2; it still contains R (v_2 commutes with R) and S subset N'.
The unitary in the statement is v = v_2* v_1* with the stated bound; the cb
gap is O(t).
Irreducibility and the vector: choose increasing matrix subfactors R_j
strongly generating R; Haar averaging over U(R_j) converges weakly on M to
x |-> phi(x)1 (bounded clusters commute with all R_j, hence scalar, and
averages preserve phi). If a nontrivial projection p in R' cap N existed, a
close approximant x in M would give dist(p,C1) <= ||p - x|| = O(t), against
1/2. So R' cap N = C1; in the commutants with S and the vector state on M',
S' cap N' = C1. Let p be the projection onto closure(N' Omega) in N. By
r Omega = J_M r* J_M Omega, that subspace reduces R (r x' Omega = x' r Omega),
so p in R' cap N, so p = 1; likewise closure(N Omega) projection lies in
S' cap N' and is 1. So Omega is cyclic and separating for N. The common
right action shows the state on N is U(R)-invariant, hence centralizes R.
J_M r J_M and J_N r J_N (both in N') agree on Omega, and Omega is separating
for N', so they agree. All of Definition 6.3 holds. QED
6.2 FLATTENING RESCALED TOMITA GRAPHS. Fix a prepared pair on H; choose
increasing matrix subfactors R_j strongly generating R. For P = M,N let
S_P be the closure of x Omega |-> x* Omega, S_P = J_P Delta_P^{1/2}. For
c>0 let E_c^P be the projection in H oplus conj(H) onto the closure of
g_c(x) = (x Omega, c conj(x* Omega)), x in P. (6.1)
Proposition 6.5 (uniform graph comparison). There is an absolute C with,
for every prepared pair with d_cb(M,N) <= gamma,
sup_{c>0} ||E_c^M - E_c^N|| <= C gamma.
Proof. For s in R put R_s = J_M s* J_M = J_N s* J_N. On either graph,
g_c(x) |-> g_c(r x s), r,s in U(R), is the restriction of the common
unitary (r R_s) oplus conj(s* R_{r*}) on H oplus conj(H); both graphs
reduce these unitaries, so T = (1-E_c^N) E_c^M commutes with them.
Bounded element retaining the signal. Take c = sqrt(k), k in N. If T != 0
choose x in M with ||g_c(x)|| = 1 and ||T g_c(x)|| >= (1/2)||T|| (dense in
the graph). Write alpha = phi(x*x), beta = phi(x x*), alpha + k beta = 1.
Choose independent Haar u_i,v_i in U(R_j) and signs eps_i, and set
x_i = u_i x v_i, y = m^{-1/2} sum eps_i x_i. The sign second moment and the
Hilbert fourth-moment bound give E_eps ||T g_c(y)||^2 = ||T g_c(x)||^2 and
E_eps ||T g_c(y)||^4 <= 3 ||T g_c(x)||^4, so with sign probability >= 1/12,
||T g_c(y)|| >= ||T||/(2 sqrt2). For simultaneous fourth moments use the
Haar averages E_j(a) = int u a u* du -> phi(a)1 weakly; centralizer cycling
and Haar invariance give the expected limits of the three sign-pairing
terms in phi((y*y)^2) as alpha^2, alpha beta, and a term bounded by
alpha beta (state Cauchy-Schwarz), with the all-equal contribution
m^{-1} phi((x*x)^2); similarly for y y*. Choose m so the weighted diagonal
is <= 1 and then j so the weighted off-diagonal is <= 7; Markov bounds the
bracket E[||g_c(y)||^2 + phi((y*y)^2) + k phi((y y*)^2)] <= 10 failure by
1/24. With the sign bound this yields a realization. Clip y at K_1:
z = y f_{K_1}(|y|), f_b(t) = min(1,b/t), f_b(0)=1; functional calculus gives
||g_c(y - z)||^2 <= K_1^{-2}[phi((y*y)^2) + k phi((y y*)^2)]. Choose K_1 so
sqrt(240)/K_1 <= 1/(4 sqrt2); with a_0 = 1/(4 sqrt2), the signal loss is
||T g_c(z)|| >= a_0 ||T||, so a fixed K_1 suffices independent of ||T||, and
||z|| <= K_1, phi(z*z) <= 240, phi(z z*) <= 240/k.
Bounded row retaining the normalized signal. Replicate z into the row
Y = [r_1 z ... r_k z], A = Y Y*, with independent Haar r_i in U(R_j).
Use the normalized row norm ||(eta_i)_i||_av^2 = k^{-1} sum ||eta_i||^2;
equivariance gives normalized signal >= a_0 ||T|| for every choice of r_i.
With alpha_z = phi(z*z), beta_z = phi(z z*), Haar averaging and cycling give
E phi(A^2) -> k phi((z z*)^2) + k(k-1) beta_z^2 and
E[k^{-1} sum_i phi(Y_i* A Y_i)] -> phi((z*z)^2) + (k-1) alpha_z beta_z,
absolutely bounded (use phi((z z*)^2) <= K_1^2 beta_z etc.). Choose a stage
and realization with phi(A^2) + k^{-1} sum_i phi(Y_i* A Y_i) <= C_1 (absolute).
Clip the row on the left, X = f_{K_2}(A^{1/2}) Y, ||X|| <= K_2; if Q = 1 -
f_{K_2}(A^{1/2}) then Q^2 <= A/K_2^2, Y - X = QY, and the normalized squared
graph error is <= C_1/K_2^2. Choose absolute K_2 so sqrt(C_1)/K_2 <= a_0/2,
giving normalized signal ||(T g_c(X_i))_i||_av >= (a_0/2)||T||.
Comparison at one matrix level and all rescalings. Cb closeness gives a row
Z in M_{1,k}(N) with ||X - Z|| <= K_2 gamma. For any row [e_i] of norm <= eta,
k^{-1} sum ||g_c(e_i)||^2 = k^{-1} sum phi(e_i* e_i) + phi(sum e_i e_i*)
<= 2 eta^2 (the graph-vector formula makes sense for all operators on H).
Because g_c(X_i) in ran E_c^M and g_c(Z_i) in ran E_c^N,
||(T g_c(X_i))_i||_av <= sqrt2 K_2 gamma, which with the signal bounds
||(1-E_c^N)E_c^M|| by an absolute multiple of gamma. Interchanging M,N gives
the reverse. The antiunitary flip (xi, conj eta) |-> (eta, conj xi) sends
g_c(x) to c g_{1/c}(x*), so the bound holds at c = 1/sqrt(k); for c>=1 take
k = floor(c^2) so c/sqrt(k) in [1,sqrt2], and scaling the second leg has
condition number <= sqrt2, multiplying either one-sided gap by <= sqrt2;
reciprocal scaling handles 0<c<1. Finally ||E - F|| = max(||(1-F)E||,
||(1-E)F||) for projections. QED
Corollary 6.6 (logarithmic modular cuts). For a prepared pair with
d_cb(M,N) <= gamma, put L_P = log Delta_P. There is absolute C with, for
every a in R,
||1_{(-inf,a-1/2]}(L_N) 1_{[a+1/2,inf)}(L_M)|| <= C gamma, (6.2)
and the same with M,N interchanged.
Proof. The upper-left block of the projection onto a closed operator graph
is (1 + A* A)^{-1}; applied to (6.1) this is (1 + c^2 Delta_P)^{-1}. With
g(s) = (1 + e^s)^{-1}, Proposition 6.5 gives ||g(L_M - a) - g(L_N - a)|| <=
C gamma. Let p = 1_{(-inf,a-1/2]}(L_N), q = 1_{[a+1/2,inf)}(L_M), regard pq
as an operator qH -> pH, and let A,B be the restrictions of g(L_N - a),
g(L_M - a) to those subspaces; then A >= g(-1/2)1, B <= g(1/2)1, and
||A(pq) - (pq)B|| <= C gamma. For D = A(pq) - (pq)B, integration of the
derivative of e^{-tA}(pq)e^{tB} gives pq = int_0^inf e^{-tA} D e^{tB} dt,
so ||pq|| <= ||D||/(g(-1/2) - g(1/2)); the denominator is positive. Reverse. QED
==============================================================================
SECTION 7. ALIGNING THE MODULAR ACTION
==============================================================================
PROPOSITION 7.1 (common modular action). There are absolute c_mod,C_mod>0
with: if (M,N,Omega,R) is a prepared pair on separable H and
d_cb(M,N) < gamma < c_mod, there is a unitary q in B(H) with
||q - I|| <= C_mod gamma, d_cb(q M q*, N) <= C_mod gamma,
Delta_N^{it}(q M q*) Delta_N^{-it} = q M q* (all t). (7.1)
The algebra N and its modular group are unchanged.
For a concrete factor P subset B(H) let
G_P = U(P) U(P'), N_P = {v in U(B(H)) : v P v* = P}. G_P subset N_P.
7.1 DISTANCE TO INFINITESIMAL NORMALIZERS.
Lemma 7.2. For a separably acting properly infinite von Neumann algebra
P subset B(H) and selfadjoint h,
dist(h, P') <= 4 ||ad_h|_P||.
Proof. Let d = ||ad_h|_P||. Choose a unital B(l^2) subset P with matrix
units e_{ij}; let S_n = span{e_{ij}: i,j<=n} + C(I - sum_{i<=n} e_{ii}),
which increase to S. Average h over U(S_n), take a WOT cluster h_0; each
average is selfadjoint, within d of h, has scalar commutator bound <= d on
P, and these pass to the cluster, so h_0 in S'. For any k choose isometries
s_1,...,s_k in S with s_i* s_j = delta_{ij} and sum s_i s_i* = I; the
unitary V: H^k -> H identifies M_k(P) with P and h_0^{(k)} with h_0, so
||ad_{h_0}|_P||_cb <= d. Lemma 2.1 gives dist(h_0,P') <= d, so
dist(h,P') <= 2d. QED
Lemma 7.3 (uniform infinitesimal estimate). There is absolute K<inf with,
for every separably acting type III factor P subset B(H) and selfadjoint h,
dist(h, P + P') <= K sup_{x in P_1} dist([h,x], P). (7.2)
(P + P' is a linear subspace; no closedness is asserted.)
Proof. Suppose no common K. Scaling and subtracting a selfadjoint sum from
P+P' (its commutator with P lies in P, so neither distance changes) gives
P_j subset B(H_j) and selfadjoint h_j with dist(h_j,P_j+P_j')=1,
||h_j||<=2, eps_j = sup_{x in (P_j)_1} dist([h_j,x],P_j) -> 0. Take a norm
ultraproduct A = prod_omega P_j subset E = prod_omega B(H_j). A is simple:
a nonzero positive a has representatives a_j >= 0 and t>0 with e_j =
1_{[t,inf)}(a_j) nonzero on an omega-large set; in a separably acting type
III factor each nonzero projection is equivalent to I, so pick isometries
v_j with v_j v_j* = e_j there, giving v_j* a_j v_j >= t I and v* a v
invertible in the ideal (a). So every nonzero closed ideal of A contains I.
[H_j] in E defines a bounded derivation D: A -> A. Its range is in A:
for bounded x_j in P_j choose y_j in P_j with ||[h_j,x_j] - y_j|| <=
(eps_j + 1/j) sup||x_i||; the y_j are uniformly bounded and define the same
class (independence of choices, linearity, Leibniz follow in E). Sakai's
theorem (a derivation of a simple unital C*-algebra is inner) gives a in A
with D(x) = [a,x]; replacing a by (a + a*)/2 keeps this (h_j selfadjoint).
Choose bounded selfadjoint representatives a_j in P_j. Equality of the
derivations forces lim_omega ||ad_{h_j - a_j}|_{P_j}|| = 0 (else contractions
witnessing a fixed positive commutator norm would contradict equality on
their class). Lemma 7.2 then gives 1 = dist(h_j,P_j+P_j') <=
dist(h_j - a_j, P_j') <= 4||ad_{h_j - a_j}|_{P_j}|| ->_omega 0, absurd. QED
Lemma 7.4 (path of approximate normalizers). There are absolute e_0,C_path>0
with: if P subset B(H) is a separably acting type III factor, I subset R an
interval containing 0, v: I -> U(B(H)) norm continuous with v(0)=I,
0<=e<e_0, and sup_{t in I} sup_{x in P_1} dist(v(t) x v(t)*, P) <= e, then
sup_{t in I} dist(v(t), G_P) <= C_path e. The constants are independent of
the interval's length.
Proof. b(t) = dist(v(t), G_P) is continuous (1-Lipschitz). Suppose b(t) < a
fixed radius and pick p in G_P with ||p* v(t) - I|| < b + eta. For small
b + eta, p* v(t) = e^{ih}, h selfadjoint, ||h|| <= 2(b + eta); p normalizes
P, so the conjugation has the same bound e. The twice-integrated
conjugation formula ||e^{ih} x e^{-ih} - x - i[h,x]|| <= 2||h||^2||x|| gives
sup_{x in P_1} dist([h,x], P) <= e + 2||h||^2. Lemma 7.3 and approximation of
h by a selfadjoint sum a + b' with a in P, b' in P' (so e^{i(a+b')} =
e^{ia} e^{ib'} in G_P, using ||e^{ih} - e^{ik}|| <= ||h - k||) give, letting
slacks vanish, b(t) <= K e + 8K b(t)^2. Choose r in the radius with 8Kr <=
1/4. If Ke < r/2, the continuous b starting at 0 cannot first reach r (else
r < 3r/4). So b(t) < r on the interval from 0 to any t; absorbing the
quadratic term gives b(t) <= 2Ke. QED
7.2 ANALYTIC APPROXIMATION OF THE MODULAR COMPARISON. On K = L^2(R,H) put
D = -i d/dt with Fourier convention hat f(xi) = (2pi)^{-1/2} int e^{-it xi}
f(t) dt, and P_a = 1_{(a,inf)}(D).
Lemma 7.5 (analytic extension of a frequency-preserving multiplier). If
multiplication by a bounded field B(t) on L^2(R,H) preserves P_0 K, it is
the boundary field of a bounded operator-norm holomorphic function on the
upper half-plane, given by the Poisson integral
B(t + ir) = int_R r/(pi((t-s)^2 + r^2)) B(s) ds, r>0, (7.3)
of norm at most ess sup ||B(s)||. For two such fields and constant x, the
extension of the boundary product B(s) x S(s) is B(z) x S(z).
Proof. With w = (t-i)/(t+i) and normalized circle measure, the map
(Jf)(t) = (1/sqrt(pi)(t+i)) f((t-i)/(t+i)) is unitary L^2(T,H) -> K
(change of variables d theta/(2pi) = dt/(pi(1+t^2))). For n>=0, J(w^n) is a
combination of (t+i)^{-k}, 1<=k<=n+1, whose Fourier transforms are
supported in [0,inf); for n<0 of (t-i)^{-k}, supported in (-inf,0]. The
images of circle modes form an orthonormal basis, so J identifies the
nonnegative-mode subspace with P_0 K. Conjugating the multiplier gives a
circle multiplier B~(w); applying the invariant-subspace property to every
constant vector kills all negative Fourier coefficients, so
F(w) = sum_{n>=0} b_n w^n converges in operator norm locally uniformly,
is holomorphic, and equals the circle Poisson integral of B~; the positive
Poisson kernel of mass one gives ||F(w)|| <= ||B~||_inf, and changing
variables gives (7.3). The product identity is checked without pointwise
boundary limits: radial Poisson integrals converge in L^2(T,H) on each
fixed vector for B~, B~*, S~, S~*; a subsequence of radii realizes these
for a dense family, uniform boundedness extends to all vectors, the
products converge strongly-* a.e. to B~ x S~, each Fourier coefficient of
the bounded analytic product F x G converges to the coefficient of that
boundary product, so its negative coefficients vanish and its nonnegative
ones are the Taylor coefficients of F x G. QED
Lemma 7.6 (from modular cuts to near normalizers). Let M,N subset B(H) be a
prepared pair on separable H with M type III and d_cb(M,N) <= gamma. Put
U_P(t) = e^{it L_P}, L_P = log Delta_P, W(t) = U_N(t) U_M(t)*. Assume the
cut estimate (6.2) and Lemma 7.4. There are absolute gamma_0, C with, for
0<gamma<gamma_0, a unitary n_t with n_t M n_t* = M and
||n_t - W(t)|| <= C gamma for every t (no regularity in t asserted).
Proof. Initial approximate normalizers: U_M(t), U_N(t) normalize their
algebras, so moving U_M(t)* x U_M(t) into M_k(N)_1, conjugating by U_N(t),
and moving back gives dist(W(t)^{(k)} x W(t)^{(k)*}, M_k(M)_1) <= 2 gamma for
x in M_k(M)_1, and likewise for W(t)*; hence d_cb(W(t) M W(t)*, M) <= 2 gamma.
Frequency nest and signs. On K let C_s be multiplication by e^{ist}, U_P~
multiplication by U_P(t). Spectral calculus gives U_P~* D U_P~ = D + L_P and
C_s* D C_s = D + s. With W~ = U_N~ U_M~*, moving spectral projections gives
(1-P_a) C_1 W~ P_a = C_1 U_N~ 1_{(-inf,a-1]}(D+L_N) 1_{(a,inf)}(D+L_M) U_M~*,
whose Fourier-transformed fiber is 1_{(-inf,a-1-xi]}(L_N)
1_{(a-xi,inf)}(L_M); its thresholds have separation one, midpoint
a - 1/2 - xi, so (6.2) bounds it by C gamma uniformly in a, xi. The same
with W~*. Arveson's nest distance formula (distance to the nest algebra =
sup_a ||(1-P_a) T P_a||) gives nest-preserving T_B, T_S with
||T_B - C_1 W~||, ||T_S - C_1 W~*|| <= delta = C gamma.
Character averaging. The UCP map E(T) = m_s C_s T C_s* (invariant mean on
the discrete R) has range in the commutant of the scalar characters, hence
(by L^1 annihilator argument) in the decomposable operators; conjugation by
C_s preserves the frequency nest algebra, and averaging preserves it. Both
C_1 W~, C_1 W~* commute with every C_s, so E(T_B), E(T_S) are multipliers by
fields B(s), S(s) with ess sup ||B(s) - e^{is} W(s)||, ||S(s) - e^{is} W(s)*||
<= delta, preserving P_0 K, so Lemma 7.5 applies (extensions B(z), S(z)).
Analytic products. The boundary products satisfy
||B(s) S(s) - e^{2is} I||, ||S(s) B(s) - e^{2is} I|| <= 2 delta + delta^2
a.e., and by the product identity the same in the upper half-plane; and for
every k and x in M_k(M)_1, dist(B(z)^{(k)} x S(z)^{(k)}, M_k(M)) <= beta and
likewise with S(z)^{(k)} x B(z)^{(k)}, beta = 2 gamma + 2 delta + delta^2
(proved by testing against normal functionals annihilating M_k(M), whose
norm distance is the supremum over such; Hahn-Banach). For z = t + ir,
0<r<=gamma, set A_r(t) = e^{-iz} B(z), C_r(t) = e^{-iz} S(z). Because
|e^{-iz}| = e^r, we get ||A_r(t)||, ||C_r(t)|| <= 1 + C gamma,
||A_r C_r - I||, ||C_r A_r - I|| <= C gamma, and the two algebra-distance
bounds, uniformly in r,t,k.
Polar smoothing. For small gamma the products are invertible, providing
matching inverses with ||A_r(t)^{-1}|| <= 1 + C gamma; with the norm bound,
spectrum(|A_r(t)|) is in [1-C gamma, 1+C gamma]. The polar unitary
V_r(t) = A_r(t)(A_r(t)* A_r(t))^{-1/2} satisfies ||V_r(t) - A_r(t)|| <= C gamma
and ||V_r(t)* - C_r(t)|| <= C gamma, hence V_r(t) approximately normalizes M
in both directions (all matrix levels). For fixed r>0, A_r(t) is norm
continuous by holomorphy with uniformly bounded inverse, so V_r(t) is norm
continuous; E_r(t) = V_r(t) V_r(0)* is a norm-continuous unitary path with
E_r(0) = I and sup_{x in M_1} dist(E_r(t) x E_r(t)*, M) <= C gamma (compose
via an approximant m of V_r(0)* x V_r(0)). Lemma 7.4 gives a_r(t) in U(M),
b_r'(t) in U(M') with ||E_r(t) - a_r(t) b_r'(t)|| <= C gamma.
Boundary comparison at every time. Set F_0(s) = e^{is} W(s) (bounded
strongly-* continuous) and D_r(t) = e^{-i(t+ir)} int r F_0(s)/(pi((t-s)^2 +
r^2)) ds. The Poisson formula and the boundary errors give
||A_r(t) - D_r(t)|| <= e^r delta, so ||V_r(t) - D_r(t)|| <= C gamma,
||D_r(t)|| <= e^r. At each fixed t the Poisson approximate identity gives
D_r(t) -> W(t) strongly-* as r->0 (split |s-t|<h and its complement; strong
continuity controls the first, vanishing Poisson mass the second; same for
the adjoint). So Q_r(t) = D_r(t) D_r(0)* -> W(t) strongly-*, and
||E_r(t) - Q_r(t)|| <= C gamma.
UCP limit recovers an actual normalizer. Fix t. For a sequence r->0 define
UCP Phi_r: M -> M, Phi_r(x) = a_r(t) x a_r(t)* (a restriction of
Ad(a_r(t) b_r'(t)) since b_r'(t) commutes with M). Then at every matrix
level ||Phi_r^{(k)}(x) - Q_r(t)^{(k)} x Q_r(t)^{(k)*}|| <= C gamma ||x|| (the
cb norm of the difference of x |-> T x T* and x |-> R x R* is at most
(||T||+||R||)||T-R||). Compactness of the product of WOT compact balls gives
a subnet with Phi_r(x) -> Phi(x) weakly for all x; the limit is UCP (linear,
unital, positivity at each matrix level pass to the limit; normality not
needed). Bounded strong-* continuity gives Q_r(t) x Q_r(t)* -> W(t) x W(t)*
strongly-*, and WOT lower semicontinuity of norm gives
||Phi - rho_t||_cb <= C gamma with rho_t(x) = W(t) x W(t)*. Here rho_t is
faithful and normal with range at two-sided cb gap <= 2 gamma from M
(the W-range gap), so Proposition 2.4 gives a unitary q_t with
||q_t - I|| <= C gamma and q_t rho_t(M) q_t* = M. Then n_t = q_t W(t) is a
unitary normalizer of M with ||n_t - W(t)|| = ||q_t - I|| <= C gamma. QED
7.3 CORRECTION WITHIN THE NORMALIZER GROUP.
Lemma 7.7 (small exact normalizers). There are absolute r_0,C_0>0 with: if P
is a separably acting type III factor, v in N_P, ||v - I|| < r_0, then v =
a b' with a in U(P), b' in U(P'), ||a - I|| + ||b' - I|| <= C_0 ||v - I||.
Proof. v = e^{ih} (principal log), h selfadjoint, ||h|| <= 2||v - I||. For
r_0 small, log(Ad_v) = i ad_h as norm-convergent series; each term preserves
P (v P v* = P), so i ad_h restricts to a bounded *-derivation of P, a simple
unital C*-algebra (same projection argument as Lemma 7.3). Sakai gives a
selfadjoint a_0 in P with h - a_0 in P'; shift the scalar to the midpoint of
the spectrum so ||a_0|| <= ||h|| (a selfadjoint a in P has spectral diameter
<= ||ad_a|_P||, since a partial isometry between projections near the two
spectral endpoints compresses [a,w] to the endpoint gap; apply to a_0 with
||ad_{a_0}|_P|| = ||ad_h|_P|| <= 2||h||). Then b_0' = h - a_0 has norm
<= 2||h||, they commute, v = e^{ia_0} e^{ib_0'}. QED
Lemma 7.8 (correction of normalizer representations). There are absolute
e_1, C_1>0 with: if P subset B(H) is a separably acting type III factor and
v_t in N_P, t in R, satisfy e = sup_{t,s} ||v_t v_s - v_{t+s}|| < e_1, there
is an exact unitary representation z: R -> N_P of the discrete group with
sup_t ||z_t - v_t|| <= C_1 e.
Proof. With the invariant operator mean (discrete R), set
F_t(s) = v_{t+s} v_s*, A_t = m_s F_t(s). Each F_t(s) is a unitary normalizer
within e of v_t, so ||A_t - v_t|| <= e. Translation invariance and the exact
factorization F_{t+r}(s) = F_t(r+s) F_r(s) give A_{t+r} - A_t A_r =
m_s[(F_t(r+s) - A_t)(F_r(s) - A_r)], bounded by 4e^2; and
I - A_t* A_t = m_s(F_t(s) - A_t)*(F_t(s) - A_t), also <= 4e^2. Taking polar
parts in B(H) alone would not keep membership in N_P; use the local product
structure. By Lemma 7.7, for every s,t choose F_t(s) v_t* = a_{t,s} b_{t,s}'
with a_{t,s} in U(P), b_{t,s}' in U(P'), displacement <= C_0 e (arbitrary
choices). Their averages a_t = m_s a_{t,s} in P, b_t' = m_s b_{t,s}' in P'
satisfy ||m_s(a_{t,s} b_{t,s}') - a_t b_t'|| <= 4C_0^2 e^2, and
||I - a_t* a_t||, ||I - (b_t')* b_t'|| <= 4C_0^2 e^2. For small e the
averages are invertible (within C_0 e of I); their polar unitaries a_t^, b_t'^
in P, P' differ from them by <= C e^2. Thus w_t = a_t^ b_t'^ v_t in N_P,
||w_t - A_t|| <= C_2 e^2, giving ||w_t - v_t|| <= e + C_2 e^2 and
||w_t w_r - w_{t+r}|| <= (4 + 3C_2) e^2 (compare products using ||A_t|| <= 1,
||w_t|| = 1). Iterate with fixed absolute constants: choose e_1 so each new
defect is at most half the previous one and each displacement at most twice
the previous defect; total displacement <= 4e. Uniform-in-t limit z_t in
operator norm satisfies z_t z_r = z_{t+r}; N_P is norm closed; the law gives
z_0 = I, z_{-t} = z_t*. QED
Proof of Proposition 7.1. Put U_t = Delta_N^{it}, W(t) = Delta_N^{it}
Delta_M^{-it}. Lemma 7.6 gives n_t in N_M with sup_t ||n_t - W(t)|| <= C gamma.
Set v_t = n_t Delta_M^{it} in N_M, sup_t ||v_t - U_t|| <= C gamma; since U is
exact, sup_{t,s} ||v_t v_s - v_{t+s}|| <= 3C gamma. For small gamma, Lemma
7.8 gives an exact representation z_t in N_M with sup_t ||z_t - U_t|| <=
C' gamma. Take T = m_t z_t U_t*, within C' gamma of I, hence invertible; for
every s translation invariance and the two exact laws give z_s T U_s* = T.
Its polar unitary w = T|T|^{-1} then also satisfies z_s w = w U_s, and
||w - I|| <= 2||T - I|| <= 2C' gamma. So z_s = w U_s w*, giving strong
continuity of z. Set q = w*. Since z_s normalizes M, U_s(q M q*) U_s* =
q M q* for all s. Conjugation by q^{(k)} moves contractions by <= 2||q - I||,
so d_cb(q M q*, N) <= gamma + 2||q - I|| <= C_mod gamma. QED
==============================================================================
SECTION 8. COMMON CROSSED PRODUCTS AND DESCENT
==============================================================================
Let P subset B(H) be a unital von Neumann algebra on separable H and
(U_s)_{s in R} a strongly continuous unitary group with U_s P U_s* = P. On
K = L^2(R,H) define
(lambda_s xi)(t) = xi(t-s), (rho_s xi)(t) = xi(t+s),
(pi(x) xi)(t) = U_t* x U_t xi(t).
pi is a faithful normal representation of all of B(H), a complete isometry
(its matrix norm is the essential sup of unitary conjugates). Write
C(P,U) = (pi(P) cup {lambda_s : s in R})'', the concrete crossed product
for Ad(U_s)|_P.
Lemma 8.1 (common-action fixed points). C(P,U) = (P otimes B(L^2(R)))^alpha
with alpha_s = Ad(U_s otimes rho_s). For (Q_a xi)(t) = e^{iat} xi(t),
beta_a = Ad(Q_a) preserves C(P,U) and C(P,U)^beta = pi(P).
Proof. pi(x), lambda_r lie in P otimes B(L^2(R)) and are alpha-fixed. For the
reverse: a fixed X in the tensor product, f,g in C_c(R),
lambda(f) = int f lambda, Y = lambda(f) X lambda(g)* stays fixed. Define
R_{f,t} xi = int f(t-s) xi(s) ds (norm <= ||f||_2, norm continuous in t).
On compactly supported vector functions Fubini gives the kernel
k(t,r) = R_{f,t} X R_{g,r}*, (Y xi)(t) = int k(t,r) xi(r) dr, bounded and
jointly norm continuous, with values in P (evaluation operators intertwine
the constant P'-action). Fixedness gives k(t+s,r+s) = U_s* k(t,r) U_s,
first a.e. then everywhere by norm continuity, so
k(t,r) = U_t* k(0,r-t) U_t. To rebuild Y inside C(P,U) use Fejer-type masks
chi_j(v) = (1 - |v|/j)_+ (a contraction multiplier, the weak integral of
Ad(Q_a)); then Y_j = int chi_j(v) pi(k(0,v)) rho_v dv in C(P,U), and Y_j -> Y
weakly then in norm (dominated convergence; strong continuity of
v |-> k(0,v)). A strong approximate identity lambda(f_n) -> I gives
lambda(f_n) X lambda(f_n)* -> X strongly, so C(P,U) = the fixed algebra.
For beta: beta_a(pi(x)) = pi(x), beta_a(lambda_s) = e^{ias} lambda_s, so
beta preserves C(P,U). A beta-fixed element commutes with scalar
multiplication on L^2(R); if also in P otimes B(L^2(R)) it is multiplication
by a bounded measurable P-valued field F(t) (a.e. constant via countable
coefficient tests and smoothing). The remaining alpha-fixedness reads
F(t+s) = U_s* F(t) U_s a.e.; then G(t) = U_t F(t) U_t* is translation
invariant, hence a.e. constant (countable dense coefficient tests,
translation-invariant L^inf classes -> constants after smoothing), and since
U_t normalizes P the constant x lies in P, so F(t) = U_t* x U_t a.e. QED
For a unitary action V_s, E_V(T) = m_s V_s T V_s* is UCP, fixes invariant
operators, and preserves any invariant von Neumann algebra.
Lemma 8.2 (cb gap of common crossed products). If P,Q subset B(H) are
normalized by the same U and d_cb(P,Q) < gamma, then
d_cb(C(P,U), C(Q,U)) <= gamma.
Proof. Lemma 2.5 (tensor with B(L^2(R))) keeps the upper bound. For a
contraction X of a matrix level of C(P,U), choose a contraction Y of the
same level of Q otimes B(L^2(R)) within the bound. E_V (V_s = U_s otimes rho_s)
fixes X, takes Y into C(Q,U) (Lemma 8.1), and preserves the norm bound and
the error; reverse. QED
Proposition 8.3 (conjugacy through a common crossed product). There are
absolute gamma_0>0 and nondecreasing zeta: (0,gamma_0) -> (0,inf),
zeta(gamma) -> 0, with: if P,Q subset B(H) are unital von Neumann algebras on
separable H, d_cb(P,Q) < gamma < gamma_0, a strongly continuous unitary group
U normalizes both, and C(Q,U) is a type II_inf factor, then a unitary a in
B(H) has a P a* = Q and ||a - I|| <= zeta(gamma). No factoriality or modular
interpretation of C(P,U) is assumed.
Proof. Write A = C(P,U), B = C(Q,U); Lemma 8.2 gives d_cb(A,B) < gamma. The
semifinite factor B contains matrix units (e_{ij}) with sum e_{ii} = I
strongly and e = e_{11} finite nonzero: a faithful normal semifinite trace
and a separable representation give increasing finite-trace projections with
sup I; split successive differences by trace continuity and group into
countably many projections of one fixed positive finite trace, giving
exhausting matrix units; S = (e_{ij})'' ~ B(l^2) is amenable.
Theorem 2.3 on S subset_gamma A gives a unitary w with w S w* subset A,
||w - I|| <= 150 gamma. Set A_1 = w* A w, so S subset A_1 cap B and
d_cb(A_1, B) < 301 gamma =: kappa. Compress by e: A_0 = e A_1 e, B_0 = e B e
on eK, d(A_0,B_0) < kappa. B_0 is a II_1 factor; for kappa < 1/10,
Lemma 5.2 makes A_0 a finite factor, so Theorem 5.1 applies on the separable
eK, giving v_0 on eK with v_0 A_0 v_0* = B_0 and ||v_0 - e|| <=
eta_fin(kappa). Inflate: V = sum_{i} e_{i1} v_0 e_{1i} converges strongly to
a unitary, a direct sum of copies of v_0 on the e_{ii}K; ||V - I|| =
||v_0 - e||, V e_{ij} = e_{ij} V. For x in A_1 the transported entries
e_{1i} x e_{j1} in A_0 give entries of V x V* in B_0; finite matrix
compressions lie in B and converge strongly, so V A_1 V* = B. Thus
Z = V w*, Z A Z* = B, ||Z - I|| <= 150 gamma + eta_fin(301 gamma).
Descend to H. Average the character action beta on all of B(K) (Lemma 2.6),
giving UCP E_beta (V_a = Q_a). By Lemma 8.1 it maps B into pi(Q) and fixes
pi(x) for x in B(H). So F(x) = pi^{-1}(E_beta(Z pi(x) Z*)) is a UCP map
P -> Q, and since pi is completely isometric on B(H),
||F - id_P||_cb <= 2||Z - I|| <= 300 gamma + 2 eta_fin(301 gamma). The
ordinary gap d(P,Q) < gamma. Proposition 2.4 then gives the conjugating
unitary with displacement zeta(gamma) (a constant multiple of
gamma + eta_fin(301 gamma), nondecreasing). QED
THEOREM 8.4 (Uniform stability of separably acting factors). There are
absolute t_fac>0 and nondecreasing eta_fac: (0,t_fac) -> (0,inf),
eta_fac(t) -> 0, such that for every separable complex Hilbert space H and
every pair of unital factors M_0,N_0 subset B(H) with d(M_0,N_0) < t <
t_fac there is a unitary u in B(H) with u M_0 u* = N_0 and
||u - I|| <= eta_fac(t), independent of the factors, their types, and
representations.
Proof. If either factor is finite, Theorem 5.1 applies (with the
finite-neighbor conclusion, small gap makes both finite). So both infinite.
Proposition 6.4 (keeping its representation changes) gives, on H otimes K,
the fixed hyperfinite D subset B(K) and unitaries u_0 in B(H), v in
B(H otimes K) with M = M_0 otimes D, N = v((u_0 N_0 u_0*) otimes D) v* a
prepared type III_1 pair, ||u_0 - I|| + ||v - I|| <= C_1 t,
d_cb(M,N) < C_2 t. Let U_s = Delta_N^{is} be the modular group of the common
vector. Proposition 7.1 gives a unitary q with ||q - I|| <= C_3 t,
P = q M q* normalized by every U_s, d_cb(P,N) < C_4 t. N and its modular
group are unchanged, so C(N,U) is its type II_inf modular core (Cor 9.7
Takesaki 1973); apply Proposition 8.3 with gamma = C_4 t to get a P a* = N,
||a - I|| <= zeta(C_4 t).
On the actual tensor product, inverse adjustments give
b = (u_0* otimes I_K) v* a q with b(M_0 otimes D) b* = N_0 otimes D and
||b - I|| <= (C_1 + C_3)t + zeta(C_4 t) =: r(t), which -> 0 uniformly.
Choose a unit vector xi in K and its normal vector state omega_xi on B(K).
Slicing defines F_0(x) = (id otimes omega_xi)(b(x otimes I_K) b*), a UCP map
M_0 -> N_0 with, at every matrix level,
||F_0 - id_{M_0}||_cb <= 2||b - I|| <= 2r(t) (slice is defined on all of the
tensor product, compare with x otimes I_K). The reverse ordinary gap is < t.
Proposition 2.4 gives the conjugating unitary on H with displacement an
absolute constant times r(t). Choose t_fac below all thresholds; the maximum
of eta_fin(t) and a large constant times r(t) (made positive, nondecreasing)
is eta_fac. QED
==============================================================================
SECTION 9. CENTERS AND ARBITRARY HILBERT SPACES
==============================================================================
9.1 ALIGNING THE CENTERS.
Lemma 9.1. For unital von Neumann algebras M,N subset B(H),
d(Z(M), Z(N)) <= 3 d(M,N). Hence if d(M,N) < g < 1/303 there is a unitary w
in B(H) with w Z(M) w* = Z(N) and ||w - I|| <= 450 g; for N_0 = w* N w one
has Z(N_0) = Z(M) and d(M,N_0) < 901 g.
Proof. Fix z in Z(M)_1 and choose y in N_1 with ||z - y|| < g. For v in
U(N) choose a in M_1 with ||v - a|| < g; z commutes with a, so ||[z,v]|| < 2g
and ||v y v* - z|| <= ||y - z|| + ||v z v* - z|| < 3g. Dixmier's averaging
theorem says the norm-closed convex hull of {v y v* : v in U(N)} meets Z(N);
an element of the intersection is a contraction within 3g of z. Interchange
and let g -> d(M,N). The centers are amenable; the mutual conclusion of
Theorem 2.3 with bound 3g gives w; conjugation changes a gap by <= 2||w-I||. QED
9.2 COMMON-CENTER DECOMPOSITION. If M acts on separable H with center
Z = Z(M), there is a standard measure space (X,mu) and a measurable field of
separable Hilbert spaces with
H = int_X^oplus H_t d mu(t), Z = L^inf(X,mu), M = int_X^oplus M_t d mu(t),
M_t a factor a.e. (finite, completed mu; a bounded measurable field with
values in M_t a.e. is an element of M, norm = ess sup fiber norm; central
case of Blackadar III.1.6.2-4). For two algebras with the same center, use
the same decomposition and the same diagonal algebra (countably generated
C*-subalgebras can be represented simultaneously on fibers, III.5.1.10-14).
Lemma 9.2 (fiber gap). If M,N subset B(H) act on separable H with the same
center, then d(M_t,N_t) <= d(M,N) for a.e. t.
Proof. Choose separable unital C*-subalgebras A subset M, B subset N with
strong closures M,N (separable H makes bounded ball strong topology
metrizable separable); choose countable norm-dense {a_j} subset A_1,
{b_j} subset B_1. Off a null set, M_t = A(t)'', N_t = B(t)''. Density: the
quotient *-homomorphism A -> A(t) has image of A_1 norm dense in A(t)_1
(lift contractions with norm <= 1 + s, rescale), so {a_j(t)} is norm dense in
A(t)_1 and strongly dense in (M_t)_1 (Kaplansky); same for B. With
d_0 = d(M,N), for each j,n choose y_{j,n} in N_1 with ||a_j - y_{j,n}|| <
d_0 + 1/n; the essential-sup norm and countability give a common conull set
on which y_{j,n}(t) in (N_t)_1 and ||a_j(t) - y_{j,n}(t)|| <= d_0 + 1/n. Fix
such t, x in (M_t)_1, n; choose a net a_j(t) -> x strongly, a subnet
y_{j,n}(t) -> y weakly, y in N_t; norm balls are WOT closed, so
||x - y|| <= d_0 + 1/n. Let n grow; same the other direction on a conull set.
QED
Lemma 9.3 (measurable implementers). In the setting of Lemma 9.2, if for
some a>0 and a.e. t there is a unitary v_t in B(H_t) with v_t M_t v_t* = N_t
and ||v_t - I|| < a, then there is a unitary v in B(H) with v M v* = N and
||v - I|| <= a.
Proof. On the standard Borel model of X keep the countable contraction
fields a_j(t), b_j(t); take Borel versions of the dimension function,
trivializations on its level sets, and the contraction fields; discard a
Borel null set of exceptions and the implementer-hypothesis failures. On
each remaining dimension stratum identify fibers with a fixed separable K
(omit zero-dimensional fibers). U(B(K)) in strong-* topology is Polish; fix
a dense (xi_l). On pairs (t,u), impose ||u - I|| <= a (countable vector-norm
tests) and, for all j,m,k, the Borel predicates: exists q with
max_{l<=m} ||(u a_j(t) u* - b_q(t)) xi_l|| < 1/k; and exists q with
max_{l<=m} ||(u* b_j(t) u - a_q(t)) xi_l|| < 1/k. Strong density of the
contraction fields shows these encode u M_t u* subset N_t and the reverse,
i.e. exactly the implementers with the imposed bound; sections are nonempty.
Jankov-von Neumann uniformization gives a selection measurable for the
sigma-algebra generated by analytic sets (the relation is Borel, hence
analytic); analytic sets are universally measurable, so the selector is
mu-measurable. Its direct integral is a unitary v with ||v - I|| <= a, and
for x in M the field v_t x_t v_t* is a bounded measurable field in N_t, so
v x v* in N; reverse likewise. QED
THEOREM 9.4. For every a>0 there is delta_sep(a)>0 such that if M,N are
unital von Neumann algebras on a separable Hilbert space and
d(M,N) < delta_sep(a), there is a unitary u with u M u* = N and
||u - I|| < a, independent of the algebras and the representation.
Proof. Zero H is immediate. By Theorem 8.4 choose tau>0 so any pair of
factors on separable H at distance < tau has an implementer within a/2.
Choose delta_sep(a)>0 with Lemma 9.1 applicable at g = delta_sep(a) and
450 delta_sep(a) < a/2, 901 delta_sep(a) < tau. For a pair in hypothesis,
align centers by w, N_0 = w* N w; the common-center pair M,N_0 has distance
< tau, and by Lemma 9.2 almost every fiber pair has distance < tau. Apply
the factor theorem on each fiber and Lemma 9.3 with bound a/2, giving v with
v M v* = N_0, ||v - I|| <= a/2; then u = w v implements and
||u - I|| <= ||w - I|| + ||v - I|| < a. QED
9.3 SEPARABLE TESTS IN AN ARBITRARY REPRESENTATION.
Lemma 9.5. Let M,N subset B(H) be unital von Neumann algebras and g > d(M,N).
Given finite E subset M, F subset N, V subset H (V containing a nonzero
vector), there are separable unital C*-subalgebras A subset M, B subset N and
a nonzero separable closed K subset H with: E subset A, F subset B, V subset
K; K reduces A and B; d(A,B) < g and d((A|_K)'', (B|_K)'') < g.
Proof. Choose r with d(M,N) < r < g; start A_0 = C*(E,I), B_0 = C*(F,I). Given
separable unital A_j subset M, B_j subset N, adjoin to B_j N_1-approximants
within r of a countable norm-dense subset of (A_j)_1, and to A_j M_1-
approximants within r of a countable dense subset of (B_j)_1; let A_{j+1},
B_{j+1} be the generated unital C*-algebras. A = closure(union A_j),
B = closure(union B_j) are separable, contain the tests, and stage unit balls
are norm dense, so d(A,B) <= r < g. Let K = closure(span C*(A,B) V), separable
(V finite), containing V, reducing A and B. Restriction gap: for
x in (A|_K)_1 and any s>0 there is a quotient lift a in A with a|_K = x,
||a|| <= 1 + s; the contraction a/(1+s) has a B_1-approximant within r plus
slack, whose restriction gives dist(x,(B|_K)_1) <= s/(1+s) + r; s -> 0, reverse.
Finally for a contraction in (A|_K)'' use Kaplansky density (strong net of
contractions from A|_K), approximate in (B|_K)_1 within r plus slack, take a
bounded WOT cluster (in ((B|_K)'')_1 with the same bound); reverse. QED
Proof of Theorem 1.4. Let eps>0. By Proposition 2.4 (uniform; its
implementer bound -> 0 with the cb error), choose a>0 and g_0>0 so it gives an
implementer within eps whenever the reverse ordinary gap is <= g_0 and the cb
error is <= 2a. Choose 0 < delta < min{g_0, delta_sep(a)}. Let M,N subset B(H)
with d(M,N) < delta; for H = {0} take the identity. Else fix a unit vector
xi_0 and g with d(M,N) < g < delta_sep(a). Consider the directed set of
quadruples i = (E_i,F_i,V_i,k_i), E_i subset M, F_i subset N, V_i subset H
finite, xi_0 in V_i, k_i in N positive, ordered by inclusion and the usual
order on k_i. For each i, Lemma 9.5 gives A_i,B_i,K_i; set
P_i = (A_i|_{K_i})'', Q_i = (B_i|_{K_i})'' on nonzero separable K_i,
d(P_i,Q_i) < g. By Theorem 9.4 there is a unitary u_i in B(K_i) with
u_i P_i u_i* = Q_i, ||u_i - I|| < a; extend by I on K_i^perp to U_i in B(H).
For all matrix sizes ||Ad(U_i)|_M - id_M||_cb <= 2||U_i - I|| < 2a.
What the local conjugacy says on H: if x in E_i, then K_i reduces x, and the
restriction of U_i x U_i* lies in Q_i with norm <= ||x||; Kaplansky density
approximates this restriction on the finitely many coefficients from V_i by
an element of B_i|_{K_i} of norm <= ||x|| (arbitrary slack); lift through the
quotient to b_{i,x} in B_i subset N with ||b_{i,x}|| <= ||x|| + 1 and
|ip{(U_i x U_i* - b_{i,x}) xi, eta}| < 1/k_i for xi,eta in V_i. The whole
conjugated algebra need not lie in N; this coefficient bound is what survives.
For each x in M the operators U_i x U_i* lie in the WOT compact ball of radius
||x||; compactness of the product of these balls gives a subnet with
F(x) = WOT-lim_i U_i x U_i* for all x. Linearity and unitality pass to the
limit; positivity at each matrix level (a WOT-closed cone) makes F completely
positive; the WOT-closed norm ball of radius 2a||X|| gives
||F - id_M||_cb <= 2a. Range: fix x, cofinality makes x in E_i eventually, so
pick b_{i,x} as above on the tail; every fixed pair xi,eta in H is eventually
in V_i with k_i -> inf, so b_{i,x} -> F(x) weakly, and N is WOT closed, so
F(x) in N for all x. Thus F: M -> N is UCP.
The original inclusion M -> B(H) is faithful, normal, unital, with reverse
ordinary gap < delta < g_0 from N and cb error <= 2a. Proposition 2.4 yields a
unitary u in B(H) with u M u* = N and ||u - I|| < eps. The choices of a, g_0,
delta depend only on eps; the intervening tests, subspaces, and lifts change
none of the bounds. Hence the same tolerance works for all algebras, all
representations, and all Hilbert spaces. QED
==============================================================================
END OF PROOF
==============================================================================
SUMMARY OF THE DEPENDENCY CHAIN
Theorem 1.4 (Main)
<- Section 9: centers (Lemma 9.1), central decomposition (9.2), measurable
implementers (9.3), separable algebras (9.4), separable tests (9.5),
UCP limit; uses Theorem 8.4 and Proposition 2.4.
Theorem 8.4 (separably acting factors)
<- Theorem 5.1 (finite), Proposition 6.4 (prepare), Proposition 7.1
(modular alignment), Proposition 8.3 (core descent), slicing + 2.4.
Proposition 8.3 (core descent)
<- Theorem 2.3 (amenable embedding), Lemma 5.2, Theorem 5.1 (finite corner),
Lemma 8.1/8.2 (crossed-product fixed points and gap), Lemma 2.6, 2.4.
Proposition 7.1 (modular alignment)
<- Lemma 7.2 (commutant bound), 7.3 (infinitesimal estimate via norm
ultraproduct + Sakai), 7.4 (path barrier), 7.5 (analytic multiplier),
7.6 (cuts -> near normalizers via Arveson), 7.7/7.8 (normalizer
correction via invariant mean).
Theorem 5.1 (finite factors)
<- Lemma 5.2 (neighbor finite), 5.3 (irreducible hyperfinite R),
Theorem 5.4 (common position), 5.5 (common right action), 5.6 (bimodular
test), 5.7 (||J_M - J_N||), 4.1-4.5 (boundary: means, cocycle
correction, averaged intertwiner, diagonal comparison, double
ergodicity), Proposition 5.9 (Jordan -> *-iso), 5.8 (Herstein),
Theorem 3.1 (uniform amplification), Lemma 2.2 (paired representations).
Theorem 3.1 (amplification)
<- Lemma 3.2 (column/Haagerup cyclic similarity), 3.3 (trace transfer),
3.4/3.5 (free words, square-sum), 3.6 (free Haar with lifts, Popa),
3.7 (unitary twist), and the randomized-coefficient computation.
Toolkit (Section 2)
<- Lemma 2.1 (commutant/derivation), 2.2 (paired reps), Theorem 2.3
(Christensen amenable near inclusions), Proposition 2.4 (UCP criterion,
Ricard-Roydor), Lemma 2.5 (tensor gap), 2.6 (invariant means).
All external inputs used are: Christensen's amenable near-inclusion theorem;
Ricard-Roydor's multiplication perturbation theorem; Haagerup's cyclic
similarity theorem; Popa's independence theorem; Haagerup's bicentralizer/
subfactor construction and Marrakchi's absorption criterion; Sakai's
derivation theorem; Arveson's nest distance formula; Herstein's Jordan
orientation theorem; Takesaki's continuous decomposition (modular crossed
product of a type III_1 factor is a type II_inf factor); Dixmier averaging;
Kaplansky density; the CCSSWW common-position theorem; the Jankov-von Neumann
analytic uniformization theorem; and standard comparison/subdivision of
projections and central decomposition. Every constant is absolute, so the
tolerance delta(eps) depends only on eps.