0032560 ded5eb4
THE TRACE CONE CLASSIFIES RAZAK-JACELON STABILIZATIONS
A self-contained rewrite for a first serious reading.
Note to the reader. The technical engine of this paper is operator
algebra: C*-algebras, traces, tensor products, and several published
classification theorems by other authors. Those cannot be learned in a
page. What we do here is state every result cleanly, explain each object
in plain words, and give the complete logical spine of the proof so
that nothing is left to guesswork. Where a step is a citation to a
published theorem, the citation is named and its exact hypothesis is
displayed. Everything else is proved.
Notation used throughout:
C* means C-star: an algebra of operators closed under adjoint.
(x) means the spatial tensor product of two C*-algebras.
K means the compact operators on the separable Hilbert space l^2.
W means the Razak-Jacelon algebra (defined in Section 1.3).
T(C) means the trace cone of C (defined in Section 1.2).
Cu(C) means the Cuntz semigroup of C.
C_+ means the positive elements of C.
[0,infty] means the extended nonnegative reals, with 0 * infty = 0.
a_+ means max(a,0) by functional calculus; (a-eps)_+ = max(a-eps,0).
d_tau means the rank functional of a trace tau (Section 2.1).
1. WHAT IS BEING PROVED
1.1 Background the proof really uses.
A C*-algebra C is a complex algebra of bounded operators on a Hilbert
space, closed under the adjoint * and closed in operator norm. An
element a is positive, written a in C_+, if a = x*x for some x.
Separable: C has a countable dense subset. Nearly every algebra here is
separable, so countable families of tests exist.
Nuclear: C can be approximated, in the completely positive contractive
(cpc) sense, by finite-dimensional matrix algebras. This is the only
regularity hypothesis we place on the input algebras A and B.
A trace on C is a linear functional tau with tau(x*y) = tau(yx); an
extended trace takes values in [0,infty] and need not be finite.
1.2 The invariant T(C).
Let T(C) be the set of all maps tau : C_+ -> [0,infty] satisfying, for
a, b in C_+ and real r > 0,
(i) tau(0) = 0,
(ii) tau(a + b) = tau(a) + tau(b),
(iii) tau(r a) = r tau(a),
(iv) tau(x*x) = tau(x x*) for every x in C,
(v) tau is lower semicontinuous on C_+ along norm-convergent nets.
Arithmetic is extended: sums live in [0,infty] and 0 * infty = 0.
Crucially we do NOT require {a in C_+ : tau(a) < infty} to be dense.
That single decision is what makes the invariant strong.
Consequently every closed two-sided ideal I of C contributes the weight
tau_I(a) = 0 if a in I_+,
tau_I(a) = infty if a in C_+ \ I_+.
These ideal weights encode the whole ideal lattice, so T(C) sees the
ideal structure of C, however complicated.
The canonical topology on T(C) is: tau_i -> tau iff, for every a in C_+
and every eps > 0,
limsup_i tau_i((a - eps)_+) <= tau(a) <= liminf_i tau_i(a)
A cone isomorphism T(A) -> T(B) is a homeomorphism preserving addition,
the zero weight, and multiplication by every positive real scalar.
1.3 The Razak-Jacelon algebra W.
W is a simple, monotracial, stably projectionless, separable nuclear
C*-algebra (Jacelon). Simple means no nontrivial closed ideals;
monotracial means exactly one tracial state; stably projectionless
means no nonzero projections in M_n(W) for any n. Its role is to erase
K-theory: tensoring by W turns the Cuntz semigroup into its
realification and lets traces carry the whole invariant.
As black boxes from the literature, each with its precise hypothesis:
W (x) W is isomorphic to W (Nawata).
W (x) Q is isomorphic to W, Q the universal UHF (Jacelon, Nawata).
KK(W, W) = 0 (Nawata; used in Section 5).
The flip of W (x) W is approximately inner (Jacelon);
equivalently realification behaves functorially (Robert).
1.4 The main theorem.
THEOREM (main). Let A and B be separable nuclear complex C*-algebras.
If T(A) and T(B) are isomorphic as topological cones, then
A (x) W (x) K is isomorphic to B (x) W (x) K,
all tensor products spatial.
The conclusion is an honest C*-algebra isomorphism, not merely an
equality of invariants. The algebras may be nonunital, may contain both
finite and infinite subquotients, and may have an arbitrary primitive
ideal space.
Stronger form. Given a cone isomorphism, transport it to
F : T(B (x) W (x) K) -> T(A (x) W (x) K);
then there is an isomorphism Phi : A (x) W (x) K -> B (x) W (x) K with
tau o Phi = F(tau) for every extended trace tau. Proved last, in
Section 8.
1.5 Why some stabilization is unavoidable.
Tensoring by W (and K) is what makes classification possible at all.
By Theorem 2.1 below, for any C*-algebra C,
Cu(C (x) W) is isomorphic to Cu(C) realified,
so the Cuntz semigroup after tensoring by W is determined by trace data.
Nothing similar is expected without the W factor.
1.6 What the reader should expect from the proof.
Five genuinely different difficulties become the remaining chapters.
(2) STRUCTURE. From the trace cone we recover a common ideal lattice
for P = A (x) W (x) K and Q = B (x) W (x) K, pointwise order on
Cuntz classes via ranks, and an isomorphism G : Cu(P) -> Cu(Q).
We fix matrix conventions (folding, rational averaging).
(3) SEQUENCE MODELS. Errors are organized by local error ideals J(U)
inside D = l^infty(Q)/c_0(Q). Ranks fix Cuntz comparisons up to
scalar slack; a diagonalization lemma keeps controllers bounded.
(4) SMALL STABILIZERS. From countably many small errors we build a
homomorphism whose image lies inside J(U) but is large enough to
absorb every prescribed error. UHF fractions and cone
quasidiagonality enter here.
(5) UNIQUENESS. Two models with the same trace data become equal in
every J(U) after tracial alignment and averaging; a spare model,
obtained by splitting each model into UHF halves, removes the
difference in norm via Gabe's ideal-related machinery.
(6) TRACIAL APPROXIMATION. Convex separation plus von Neumann algebra
realization builds a cpc map with prescribed first and second
trace moments, plus a lower ideal comparison that stops it from
collapsing infinity to a finite value.
(7) CONE CORRECTION. A hereditary-support construction corrects the
multiplicative defects; a second stabilizer preserves the lower
ideal support. A finite-ideal trace-determination lemma proves
the full cone formula, including weights finite only on a proper
ideal.
(8) POINT MODELS AND INTERTWINING. An exhausted line model plus
translation covariance gives a crossed-product model; a full
projection with a finite positive partition identity gives a
point model. Uniqueness makes the coordinate lifts Cauchy modulo
unitaries, so a subsequence converges to an actual homomorphism.
Repeating backwards and closing with approximate intertwining
finishes the proof.
Sections 2 through 8 follow. Every numbered result is either proved in
full or is a named theorem from the literature with its hypothesis
displayed and its use pinpointed.
2. THE INVARIANT, ITS IDEALS, AND MATRIX CONVENTIONS
Throughout, P = A (x) W (x) K and Q = B (x) W (x) K. The given cone
isomorphism T(A) = T(B), extended through K and W and transported,
becomes a fixed cone isomorphism
F : T(Q) -> T(P).
Everything downstream is derived from F, and we never use the specific
algebras A, B again.
2.1 Ranks and the realification theorem.
For a in (C (x) K)_+ with 0 <= a <= 1, extend tau to finite matrices by
the unnormalized matrix trace, and define the rank
d_tau(a) = lim_{k -> infty} tau(a^{1/k}).
Rescaling an element to a contraction does not change d_tau, and d_tau
depends only on the Cuntz class [a] of a. Two identities are used
constantly, in extended arithmetic, for every tau in T(C):
d_tau(a) = sup_{eps>0} d_tau((a - eps)_+),
tau(a) = integral_0^{||a||} d_tau((a - t)_+) dt. (2.1)
THEOREM 2.1 (trace and realification; ERS, Robert). For every C*:
(a) T(C) with the topology of 1.2 is compact Hausdorff, and second
countable when C is separable.
(b) If C is exact, tau -> d_tau identifies T(C) with the topological
cone of additive, positively homogeneous, lower-semicontinuous
functionals on Cu(C) vanishing at 0.
(c) Cu(C (x) W) is canonically isomorphic to the realification of
Cu(C), determined functorially by the functional cone in (b), with
pointwise order, pointwise addition, and positive real scalar
multiplication, and with the same functional cone as Cu(C).
The functional-cone realization used in (c): if T is a functional cone
(a cone of extended functions), the realification L(T) consists of
pointwise suprema f = sup_n f_n, where each f_n : T -> [0,infty] is
additive, positively homogeneous, lower semicontinuous, zero at zero,
and for some 0 < eps_n < 1,
f_n <= (1 - eps_n) f_{n+1},
f_n is continuous at every tau with f_{n+1}(tau) < infty.
This specific approximation condition is essential; not every lower-
semicontinuous homogeneous function is a Cuntz class. A cone
homeomorphism preserves all the displayed properties, so it produces
the isomorphism G below.
Consequently F induces compatible isomorphisms
F : T(Q) -> T(P), G : Cu(P) -> Cu(Q), d_tau(G[x]) = d_{F tau}(x). (2.2)
In particular, for x, y in Cu(Q),
x <= y <=> d_tau(x) <= d_tau(y) for every tau in T(Q). (2.3)
Scalar multiplication by every positive real is an order automorphism
of Cu(Q) preserving increasing suprema and compact containment.
2.2 Finite ideals, zero ideals, and open supports.
For tau in T(C), let Fin(tau) be the closed ideal generated by the
positive elements on which tau is finite, and let Ker(tau) be the zero
ideal. Then tau is densely finite on Fin(tau) and tau(a) = infty for a
in C_+ \ Fin(tau)_+. A useful characterization is
a in Fin(tau) <=> d_tau((a - eps)_+) < infty for every eps > 0. (2.4)
Forward: approximate a by a finite-trace positive b in Fin(tau); if
||a - b|| < eps/2 then d_tau((a - eps)_+) <= (2/eps) tau(b) < infty.
Backward: finite rank of every positive cut gives finite trace of the
cuts, since tau((a - eps)_+) <= ||a|| d_tau((a - eps)_+).
PROPOSITION 2.2 (transport of ideals). F and G identify the complete
ideal lattices of P and Q, in a way compatible with G, and they carry
Fin(tau) and Ker(tau) of a weight tau across to Fin(F tau) and
Ker(F tau). Writing P(U), Q(U) for the ideals of an open set U of the
common primitive space X, we have G(Cu(P(U))) = Cu(Q(U)) and
Fin(tau) = Q(U) <=> Fin(F tau) = P(U), and likewise for Ker.
Proof. The additive idempotents of T(C) are exactly the ideal weights
tau_I: tau + tau = tau forces every value to be 0 or infty and the zero
set is a closed ideal; and I subset J iff tau_I + tau_J = tau_I. Hence
F is an order isomorphism of ideal lattices. If F(tau_I) = tau_J then
[a] in Cu(J) iff d_{F tau_I}(a) = 0 iff d_{tau_I}(G[a]) = 0 iff
G[a] in Cu(I), proving compatibility. The topology recovers the two
other ideals through the ERS limits
lim_{r -> 0} r tau = tau_{Fin(tau)}, lim_{r -> infty} r tau = tau_{Ker(tau)}, (2.5)
which follow directly from (2.4) in the topology of 1.2; continuity and
positive homogeneity of F preserve both limits. An ideal-lattice
isomorphism preserves prime ideals and hull-kernel topology, and in a
separable C* every prime ideal is primitive (Dixmier), giving the
homeomorphism Prim(P) = Prim(Q) = X. Finally the cone distinguishes the zero
algebra: if C = 0 then T(C) = {0}, whereas every nonzero C has the
nonzero ideal weight tau_{0}. So we may assume P, Q nonzero. QED.
For a positive c in Q(U), say c is compactly Cuntz supported in Q(U) if
[c] <= [(b - delta)_+] for some b in Q(U)_+, delta > 0. (2.6)
This is about the class of c, not compactness of a subset of the
possibly non-Hausdorff space X.
LEMMA 2.3 (countable supports and common controllers). The open-set
lattice of X has a countable basis closed under finite unions, and
every open U is a union of basis opens V compactly contained in U. For
V compactly contained in U there is a positive b_{V,U} in Q(U),
compactly Cuntz supported, with Q(V) inside the ideal generated by
b_{V,U}; and every compactly Cuntz supported c in Q(V)_+ then satisfies
[c] <= N [b_{V,U}] for some finite integer N. (2.7)
The element b_{V,U} is independent of c; the integer N need not be.
Proof. A norm-dense sequence (a_j) of positive contractions in Q gives
supports of (a_j - r)_+, r rational and positive, forming a countable
basis, and the support of (a_j - r)_+ is compactly contained in that
of (a_j - r/2)_+ because a directed union of ideals containing the
latter puts the more deeply cut element in one member. Choosing two
rational cut levels gives basis open compactly contained in the
prescribed open. Let h > 0 in Q(U); the (h - 1/k)_+ increase to a
dense union of Q(U), so some Q(V) sits inside Ideal_Q((h - 1/k)_+);
take that cut as b_{V,U}. If c satisfies (2.6) inside Q(V) then
[c] << [z] for some z in Q(V)_+; the ideal generated by [b_{V,U}] is
everything below sup_N N[b_{V,U}], so [z] <= sup_N N[b_{V,U}], and
compact containment gives (2.7). QED.
For the domains used below set S_Y = C_0(Y) (x) P and
S_Y(U) = C_0(Y) (x) P(U); the primitive space is Y x X. For s in S_Y,
let U_s be the projection to X of the open support of the ideal
generated by s. Projection is open, and
s in S_Y(U) <=> U_s subset of U. (2.8)
So every element has a smallest coarse label; this lower-semicontinuity
is exactly what the ideal-related machinery in Section 5 needs.
2.3 Folding and rational averaging.
Two matrix operations recur. Stabilization adds internal matrix
corners, whose Cuntz ranks add with no normalization. Rational
averaging uses a separate UHF factor, whose normalized trace supplies
convex coefficients. The next lemma lets both end up in Q while
respecting the trace convention.
LEMMA 2.4 (trace-preserving folding). Finite matrix amplifications,
including sizes varying with the coordinate, may be folded into Q so as
to preserve every trace and rank computation with unnormalized internal
matrix traces. The folding of a fixed first corner can be made
arbitrarily close to the identity on any prescribed finite subset of Q
by a multiplier-unitary adjustment.
If h_1,...,h_l are cpc maps into finite matrix amplifications of Q and
lambda_i > 0 are rational with sum 1, the normalized diagonal average
h(s) = direct-sum over i of lambda_i copies of h_i(s), folded back into
Q, is cpc and satisfies, for s >= 0,
tau(h(s)) = sum_i lambda_i tau(h_i(s)),
tau(h(s)^2) = sum_i lambda_i tau(h_i(s)^2),
d_tau((h(s)-eps)_+) = sum_i lambda_i d_tau((h_i(s)-eps)_+),
all in extended arithmetic, with the traces on the right including the
unnormalized internal matrices.
Proof. By W-absorption Q = C (x) UHF (x) K. A Hilbert-space unitary
U_k : C^k (x) l^2 -> l^2 folds a k-fold matrix amplification by
id (x) Ad(U_k), preserving the unnormalized matrix extension of every
trace. For a finite set E in Q pick a finite-rank projection e in the
K factor with ||a - eae|| small on E; a unitary of B(l^2) sends the
folded first-corner copy of e l^2 back to e l^2 and fixes the specified
identification there, so conjugating the folding by this unitary fixes
the compressed finite tests. Writing lambda_i = l_i/L and choosing
orthogonal projections e_i in the UHF factor with normalized traces
l_i/L summing to one, h(s) = sum_i h_i(s) (x) e_i has norm the maximum
of the block norms and is blockwise functional-calculus compatible,
giving complete positivity, contractivity, and the three identities
before the final UHF identification. That identification is chosen by
interleaving tensor factors of a fixed infinite tensor-product model of
the UHF algebra; finite tensor-permutation unitaries match the
first-factor embedding with the identity to any accuracy on finite
tests, and the trace identity tau~(a (x) e) = tr(e) tau(a) on finite
matrices passes to positive cuts for extended weights. Multiplier
conjugation preserves every extended trace by applying the trace
identity to a^{1/2} u. All choices are made separately per coordinate,
so finite sets, matrix sizes, and accuracies may vary with the
coordinate. QED.
We always keep the internal matrix directions distinct from the
normalized averaging direction: an internal sector sum is not divided by
its number of sectors. All later padding and folding uses Lemma 2.4
with the existing constants and finite tests included among the aligned
elements.
3. SEQUENCE MODELS AND LOCAL ERROR IDEALS
Set D = Q_infinity = l^infty(Q) / c_0(Q), the bounded sequences modulo
the null sequences, and identify Q with the constant sequences in D.
This section has two purposes: turn trace identities into uniform cut
comparisons, and retain a local error bound when approximation stages,
matrix sizes, and controllers vary with the coordinate.
3.1 Regularized limits and their model data.
PROPOSITION 3.1 (regularized limits). Let omega be a free ultrafilter on
the natural numbers and tau_n in T(Q) arbitrary. For a bounded positive
sequence (x_n), the formula
rho([(x_n)]) = sup_{eps>0} lim_{n -> omega} tau_n((x_n - eps)_+) (3.1)
defines a lower-semicontinuous tracial weight rho on D, with rank
functional
d_rho([(x_n)]) = sup_{eps>0} lim_{n -> omega} d_{tau_n}((x_n - eps)_+). (3.2)
If sigma = lim_{n -> omega} tau_n in the topology of 1.2, then
rho restricted to Q equals sigma.
Proof. Pull tau_n back to l^infty(Q) by the n-th coordinate
homomorphism. The compact Hausdorff cone T(l^infty(Q)) has a unique
limit along omega, say rho~. For every positive x in the product the
defining topology gives
lim_omega tau_n((x_n - eps)_+) <= rho~(x),
rho~((x - eps)_+) <= lim_omega tau_n((x_n - eps)_+),
and taking suprema over eps proves (3.1) on the product. Every fixed cut
of an element of c_0(Q)_+ is eventually zero, so rho~ vanishes on that
ideal and descends to rho on the quotient; this also shows independence
of representative. Constant sequences give rho|_Q = sigma.
For the rank formula rescale so 0 <= x_n <= 1 and let L be the right
side of (3.2). For every k and t > 0,
tau_n((x_n^{1/k} - t)_+) <= d_{tau_n}((x_n - t^k)_+), so regularization
and the supremum over k give d_rho(x) <= L. Conversely pick a continuous
g_eps in [0,1] that is zero on [0,eps/2] and one on [eps,1]; then
tau_n((g_eps(x_n)-t)_+) >= (1-t) d_{tau_n}((x_n-eps)_+), and the support
of g_eps(x) is below that of x, so d_rho(x) >= rho(g_eps(x)) >=
lim_omega d_{tau_n}((x_n-eps)_+); let t -> 0 and take the supremum in
eps. QED.
We call the rho in (3.1) a regularized limit. The coordinate weights
need not be bounded, normalized, or densely finite. Passing to a
subsequence is included by taking an ultrafilter concentrated on it.
Let Y be a point, a half-open interval, or the real line, with a
full-support Radon measure m (finite on a cone interval, locally finite
on the line), and S = S_Y = C_0(Y) (x) P. For nu in T(P) the product
trace m (x) nu is the integrated densely finite trace on
C_0(Y) (x) Fin(nu), extended to be infinity on positives outside that
ideal. On elementary positive tensors, with 0 * infty = 0,
d_{m (x) nu}(f (x) a) = m({f > 0}) d_nu(a). (3.3)
Full support of m ensures a nonzero positive continuous function has
support of positive measure.
DEFINITION 3.2 (model). A model for (Y, m, F) is a homomorphism
p : S_Y -> D such that, for every regularized limit rho with
sigma = rho|_Q,
rho o p = m (x) F(sigma). (3.4)
Fix a countable collection of positive elementary tensors to test this:
in each basis ideal P(V) take a countable dense positive set, its
rational positive cuts, and the less deeply cut enlarged tests; in
C_0(Y) take dense positives on a compact exhaustion with rational cuts
and enlarged compact supports; take all finite tensor products,
including the zero tensor, and all positive rational cuts of each
factor. For the line or a varying interval a compact test and its
enlarged support are chosen inside one common open interval before an
exhaustion index is advanced.
LEMMA 3.3 (elementary determination). A trace theta in T(S_Y) equals
m (x) nu if its ranks on the preceding countable test collection
satisfy (3.3). In particular, checking (3.3) for every regularized
limit suffices for (3.4).
Proof. First extend the rank identities to all elementary tensors, then
identify where the weights are densely finite, then compare their
bounded restrictions there. Extension: if positive contractions f_j,
a_i approximate f, a within eta, the cut inequalities give
(f-3eta)_+ (x) (a-3eta)_+ << (f_j-2eta)_+ (x) (a_i-2eta)_+ << f (x) a;
the middle belongs to the test collection for rational eta, and enlarged
versions give the same with any fixed cuts of f, a on the left. Ranks
respect increasing suprema of cuts, so applying to theta and m (x) nu
and letting eta -> 0 proves (3.3) on compactly supported elementary
tensors; exhaustion finishes.
Finite ideal: put I = Fin(nu). The finite ideal of theta is exactly
C_0(Y) (x) I. One inclusion: a compactly supported function and a
compactly Cuntz supported positive of I give a tensor of finite
theta-rank. Other inclusion: let an open rectangle V x U lie in the
support of Fin(theta), take nonzero f in C_c(V)_+ and cut-supported a
in P(U)_+, choose f_0 in C_c(V)_+ equal to one on supp(f) and b in
P(U)_+, delta > 0, with a << (b-delta)_+; then
f (x) a << (f_0 (x) b - delta/2)_+ lies in Fin(theta), so f (x) a is
compactly Cuntz supported there and has finite theta-rank by (2.4);
(3.3) and 0 < m(f>0) < infty give d_nu(a) < infty. Such elements
generate P(U), so P(U) inside I; rectangles exhaust the open support.
Equality on the finite ideal: choose h > 0 in I, e_n = (h - 1/n)_+,
increasing compactly supported f_n with supports exhausting Y, and the
hereditary algebras H_n = Her(f_n (x) e_n) = C_0({f_n>0}) (x) Her(e_n),
increasing with dense union in C_0(Y) (x) I. Both traces are finite
support-rank on H_n, hence bounded and normally extended there, and
two-variable spectral integration with (3.3) gives theta(f (x) a) =
(integral f dm) nu(a) on elementary tensors of H_n; boundedness and
algebraic tensor density give equality on H_n. Take a positive
contractive approximate unit (v_n) of the ideal inside the union of the
H_j; then theta(v_n x v_n) = theta(x^{1/2} v_n^2 x^{1/2}) <= theta(x)
and v_n x v_n -> x, so lower semicontinuity gives theta(x) =
lim theta(v_n x v_n), and likewise for m (x) nu; equality on the
localizations gives equality on the finite ideal. Both are infinity
outside it, hence agree everywhere. QED.
3.2 Uniform comparison and cut sandwiches.
LEMMA 3.4 (bounded witnesses into a fixed cut). Let a, b be positive in
a matrix algebra over any C*, and suppose a << (b - delta)_+ with
delta > 0. For every eta > 0 there is w with
||w* b w - a|| < eta, ||w||^2 <= (||a|| + eta) / delta. (3.5)
Proof. Choose v with ||v*(b-delta)_+ v - a|| < eta, let
g_delta(t) = sqrt((t-delta)_+/t) for t > 0 and g_delta(0) = 0, and put
w = g_delta(b) v. Then w* b w = v*(b-delta)_+ v and
g_delta(b)^2 <= delta^{-1} (b-delta)_+, giving (3.5). QED.
LEMMA 3.5 (strict comparison from regularized limits). Let x, y in D_+.
If for some 0 < alpha < 1,
d_rho(x) <= alpha d_rho(y) for every regularized limit rho, (3.6)
then x << y; the same holds after finite matrix amplification and after
coordinatewise folding.
Proof. Take bounded positive representatives x_n, y_n. If comparison
fails there is eps > 0 with (x-2eps)_+ not << y, so for every fixed
delta > 0 there are arbitrarily late coordinates with
(x_n - eps)_+ not << (y_n - delta)_+ (otherwise Lemma 3.4 supplies
bounded witnesses showing (x-eps)_+ << y). Choose increasing n_k where
this holds for delta = 1/k. By (2.3) each failure gives a trace whose
rank on the source exceeds its rank on (y_{n_k}-1/k)_+; normalize
(dividing by that rank if finite, by one plus the smaller rank if not)
to get tau_k with d_{tau_k}((x_{n_k}-eps)_+) >= 1 and
d_{tau_k}((y_{n_k}-1/k)_+) <= 1. The regularized limit along these
indices satisfies d_rho(x) >= 1, and for fixed t > 0 the cut
(y_{n_k}-t)_+ is eventually below (y_{n_k}-1/k)_+, giving
d_rho(y) <= 1, contradicting (3.6). Matrix folding preserves the
comparisons and ranks by Lemma 2.4. QED.
PROPOSITION 3.6 (sandwich comparisons). Let p : S_Y -> D be a model,
s = f (x) a >= 0, and 0 < m_0 := m(f > 0) < infty. For r > 0 take
c_r in Q_+ representing r G[a]. Then
c_r << p(s) if 0 < r < m_0, p(s) << c_r if r > m_0. (3.7)
Fixed-cut versions: if eta, delta > 0, 0 < r < m(f > eta), and c
represents r G[(a-delta)_+], then for every 0 < kappa < eta delta,
c << (p(f (x) a) - kappa)_+; (3.8)
and if r' > m_0 and eps > 0, a representative b of r' G[a] has a
positive cut (b - kappa)_+ such that
(p(f (x) a) - eps)_+ << (b - kappa)_+. (3.9)
The choices in (3.8)-(3.9) use only the constant classes, not p.
Proof. For a regularized limit rho, sigma = rho|_Q, t = d_{F sigma}(a),
the ranks of c_r and p(s) are r t and m_0 t, with strict scalar slack,
so Lemma 3.5 gives (3.7) including t = 0 and t = infty. Apply the lower
comparison to s' = (f-eta)_+ (x) (a-delta)_+, whose support is in the
region f (x) a > eta delta, so s' << (f (x) a - kappa)_+ when
0 < kappa < eta delta; applying p gives (3.8). For the upper comparison
choose m_0 < r < r' and delta > 0 with ||f|| delta < eps/2; the norm
perturbation gives (p(f (x) a) - eps)_+ << p(f (x) (a-delta)_+) << c
with [c] = r G[(a-delta)_+], and since positive scalars preserve compact
containment, [c] << r G[a] <= r' G[a] = [b], so some kappa > 0 gives
[c] <= [(b-kappa)_+]. Conversely, applying any regularized rank to the
fixed-cut comparisons and letting delta, eta -> 0 and the rational
scalars approach m_0 give (3.3) for the pullback on every compact test;
suprema of cuts handle zero or infinite ranks and compact exhaustion
handles infinite-measure supports, so Lemma 3.3 gives (3.4). QED.
COROLLARY 3.7 (constant ideals and all traces). For a model p and every
s in S,
Ideal_D(p(s)) = Ideal_D(Q(U_s)). (3.10)
Given countably many models p_i : S -> D, there is a separable
subalgebra D_0 of D containing their ranges and the constants Q such
that, for every gamma in T(D_0),
gamma o p_i = m (x) F(gamma|_Q) for every i. (3.11)
It may also contain any prescribed countable subset of D.
Proof. For a nonzero elementary tensor of finite function support the
upper sandwiches put its image inside the ideal generated by the
corresponding constants, and the lower trimmed sandwiches recover the
ideals of all cuts of a, hence the constant ideal Q(U_a). For general
s, the ideal generated by s in S is generated by positive elementary
tensors whose rectangle supports lie in its open support, whose
projected supports exhaust U_s; the elementary lower bounds give
Ideal_D(Q(U_s)) inside Ideal_D(p(s)), and approximating s by finite sums
of such tensors with the upper bounds gives the reverse. For (3.11),
start with the separable algebra generated by the ranges, constants,
and extra elements, and for every countable test, rational cut and
scalar in Proposition 3.6, and every model, adjoin the matrix entries of
comparison witnesses at accuracies tending to zero; the resulting D_0 is
separable and the comparisons hold in Cu(D_0). For gamma in T(D_0) the
rank of a constant representing r G[a] is r d_{F sigma}(a) with
sigma = gamma|_Q, so applying d_gamma to the retained sandwiches gives
the elementary rank formula for each gamma o p_i, and the converse
argument of Proposition 3.6 and Lemma 3.3 applies without gamma being a
regularized limit. QED.
3.3 The ideals of locally small elements.
DEFINITION 3.8 (local error ideal). For open U in X, an element
x = [(x_n)] in D belongs to J(U) if, for every eps > 0, there are a
positive c compactly Cuntz supported in Q(U), a sequence r_n > 0 with
r_n -> 0, and an index N, such that
d_tau((|x_n| - eps)_+) <= r_n d_tau(c) (n >= N, tau in T(Q)). (3.12)
The controller c, the decay r_n, and N may depend on x, U, eps; once
chosen, the bound holds for all traces and all coordinates in the tail.
Because r_n > 0, the right side is unambiguously infinity when
d_tau(c) = infinity. No estimate on the uncut element is required.
Example. Fix a nonzero positive contraction a compactly Cuntz supported
in Q(U). In an extra UHF factor choose projections e_n of positive
rational trace t_n -> 0. After the folding of Lemma 2.4 the elements
x_n = a (x) e_n satisfy ||x_n|| = ||a|| and d_tau((x_n-eps)_+) =
t_n d_tau((a-eps)_+) <= t_n d_tau(a); so [(x_n)] lies in J(U) and is
nonzero. This is the smallness that lets a stabilizer keep its norm
while the coefficient in its local rank bound tends to zero.
PROPOSITION 3.9 (properties of the local error ideals). Definition 3.8
is independent of representatives and makes J(U) a closed two-sided
ideal of D. The assignment is order preserving, and
J(U) subset Ideal_D(Q(U)). (3.13)
If rho is a regularized limit with sigma = rho|_Q densely finite on
Q(U), then rho vanishes on J(U). All assertions hold in finite matrix
algebras with the same conventions.
Proof. We need only the spectral cut estimates, for bounded operators
and every tracial rank:
d_tau((|x+y|-eps)_+) <= d_tau((|x|-eps/4)_+) + d_tau((|y|-eps/4)_+),
d_tau((|bx|-eps)_+), d_tau((|xb|-eps)_+) <= d_tau((|x|-eps/(2M))_+)
for ||b|| <= M, M > 0,
d_tau((|x*|-t)_+) = d_tau((|x|-t)_+),
||x-y|| < eps/4 implies d_tau((|x|-eps)_+) <= d_tau((|y|-eps/2)_+).
Cut reduction: set x_t = x f_t(|x|) with f_t(s) = (s-t)_+/s (s>0) and
f_t(0) = 0; then ||x - x_t|| <= t and |x_t| = (|x|-t)_+. Use the
perturbation comparison ||z-w|| < eps => (|z|-eps)_+ << |w|: for the
sum take w = x_{eps/4} + y_{eps/4} and |w| << |x_{eps/4}| + |y_{eps/4}|;
for a product take w = b x_{eps/(2M)}, whose modulus is Cuntz below
|x_{eps/(2M)}|; for the perturbation take w = y_{eps/2} so
||x-w|| < 3eps/4; adjoints give the other product order. Each follows
from the positive cut perturbation lemma and [v*v] = [vv*], and the
additive rank functional proves the displays for extended weights.
If x, y in J(U), choose controllers at tolerance eps/4; a folded finite
block sum of the controllers is compactly Cuntz supported in Q(U), and
the maximum of the two decay sequences proves the bound for x+y by the
sum estimate. The product estimates give closure under multiplication on
either side by any bounded sequence, the adjoint identity gives self-
adjointness, and the perturbation estimate gives independence of
representative. Norm closure: fix eps, choose one element within eps/8
in quotient norm (representatives eventually within eps/4), and use the
controller at tolerance eps/2. Monotonicity in U is immediate. For
(3.13): fix a cut of |x|, choose its controller c << (b-delta)_+ in
Q(U), wait until r_n <= 1, use pointwise order to get
(|x_n|-eps)_+ << c << (b-delta)_+, and the bounded witnesses of Lemma
3.4 give comparison with the constant b in the quotient. Vanishing: fix
eps and, for each coordinate weight, d_{tau_n}(c) <= (4/delta)
tau_n((b-3delta/4)_+) because on b > delta the element
(b-3delta/4)_+ is at least delta/4. Put h = (b-delta/2)_+; dense
finiteness on Q(U) and (2.4) give sigma(h) < infty, while the cone
convergence inequalities give lim_omega tau_n((b-3delta/4)_+) =
lim_omega tau_n((h-delta/4)_+) <= sigma(h); so the ranks are bounded on
an omega-large set, and multiplying by r_n -> 0 in (3.12) yields
d_rho(|x|) = 0, hence rho vanishes on J(U)_+. QED.
3.4 A diagonalization with fixed controllers.
The diagonalization used below retains two different controls: a fixed
positive controller bounds a local error's rank, and a fixed positive
target cut bounds the norm of a comparison witness. Both are chosen
before the stages advance. We call each stage's sequence of coordinate
approximations a row; its finite matrix size may depend on the row and
the coordinate, and all matrices are folded by Lemma 2.4.
LEMMA 3.10 (local diagonalization). Let A_alg be a countable dense
(Q + iQ) *-subalgebra of S = S_Y containing the localized dense sets to
be tested. For each j let L_j : A_alg -> D be a map with bounded
coordinate representatives L_{j,n}(a) for every a. Assume that for each
fixed input, limsup_j ||L_j(a)|| <= ||a||, and that the quotient errors
in every fixed rational linearity, involution, and multiplicativity
relation tend to zero as j -> infinity (only bounds on each fixed input
are required). Then one can choose j(n) -> infinity so that
L(a) = [(L_{j(n),n}(a))] extends to a contractive homomorphism
L : S -> D.
The same diagonal can retain the following countable data.
(i) For each l, fix basis opens V_l compactly in U_l and row errors
e_l^{(j)} in J(V_l) for all large j, with representatives e_{l,n}^{(j)}
and sup_j ||e_l^{(j)}|| < infty; then e_l = [(e_{l,n}^{(j(n))})] lies in
J(U_l).
(ii) For each prescribed comparison let a^{(j)}, b^{(j)} in D_+ have
uniformly bounded norms and bounded positive representatives, with
a^{(j)} << (b^{(j)}-delta)_+ for a fixed delta > 0; if a, b are the
diagonals, then a << (b-delta')_+ for any preselected
0 < delta' < delta.
For local errors from maps on S it suffices to schedule dense localized
inputs in pairs V compactly in U. The conclusion extends to every open
U and every input in S(U) if the coordinate error maps are uniformly
equicontinuous on bounded sets, or if their diagonals are continuous
expressions in the resulting quotient maps (this includes differences
from fixed equicontinuous maps and continuous multilinear errors with
other variables on countable dense lists). In particular a diagonal of
models is again a model when their data agree on the fixed compact
tests and their enlarged versions; every subsequence of a model keeps
its model data.
Proof. Common controller and coordinate choice. Enumerate all algebraic
relations and positive rational cut tests; reindex the rows so row j
contains the first j requirements with quotient errors tending to zero,
include the bounds on each prescribed error and comparison element, and
set as yet unspecified early-row errors to zero. Let l enumerate pairs
of a prescribed local error and a positive rational tolerance eps_l
(repeating the opens and representatives with each tolerance), and
choose once and for all the controller b_l = b_{V_l,U_l} of Lemma 2.3.
In row j the l-th error after folding has a bound
d_tau((|e^{(j)}_{l,n}| - eps_l)_+) <= r^{(j)}_{l,n} d_tau(c^{(j)}_l)
with r^{(j)}_{l,n} -> 0 uniformly in tau eventually in n, and the
controller c^{(j)}_l is compactly Cuntz supported in Q(V_l), so there is
a finite integer N_{j,l} with [c^{(j)}_l] <= N_{j,l}[b_l], absorbing
matrix sizes and repeated blocks.
Fix eta_j -> 0. After the controllers and integers N_{j,l} for l <= j
are chosen, choose increasing thresholds n_j >= j so large that
N_{j,l} r^{(j)}_{l,n} <= eta_j (n >= n_j, l <= j), (3.14)
and impose the first j norm and algebraic requirements with coordinate
error at most eta_j (possible because a quotient norm is the limsup of
coordinate norms). For n_j <= n < n_{j+1} put j(n) = j and use row j.
The diagonal norm bounds make each value a bounded sequence, and for a
fixed local test l, (3.14) eventually gives
d_tau((|e_{l,n}|-eps_l)_+) <= eta_{j(n)} d_tau(b_l) with j(n) -> infty
and a fixed controller, exactly the required J(U_l) bound; enumerating
all rational cut tolerances proves membership. The norm and algebraic
requirements make L contractive and multiplicative.
Extension of local errors: for arbitrary open U the ideals S(V) with
basis V compactly in U' inside U have dense union in S(U) (finitely many
pairs V_i compactly in U'_i collapse to their finite unions by Lemma
2.3), so the union of the scheduled localized dense sets is dense in
S(U); each retained error on those sets lies in J(U') inside J(U), and
closedness of J(U) with the assumed continuity extends the containment;
for several variables approximate each on its scheduled set. Comparison
witnesses with a fixed target cut: include both source and target norms
in the coordinate requirements; for a chosen source cut and accuracy,
Lemma 3.4 applied with uncut target (b^{(j)}-delta')_+ and cut size
delta-delta' gives squared witness norm at most (source norm + accuracy)
/ (delta - delta'); lift to a bounded coordinate sequence and impose the
finite-test equation at n_j. This retained slack gives bounded witnesses
on the diagonal and, enumerating source cuts and accuracies, the
required Cuntz comparisons, so the order of choices is: fixed tests and
target cuts, row controllers and witnesses, comparison multiplicities,
and finally the thresholds. For rows of models the lower bounds use
target thresholds kappa < eta delta from fixed trimmed inputs, the upper
bounds use cuts of fixed constants from the scalar slack and fixed
source tolerance, and both thresholds are row-independent and hence
retained. The converse of Proposition 3.6 identifies the diagonal model
data; on increasing intervals the same argument uses the common compact
tests and unchanged measure, and restriction to a subsequence keeps all
fixed-cut comparisons. QED.
REMARK 3.11. When the resulting quotient map is a homomorphism with
nuclear separable domain, the completely positive lifting theorem
(Choi-Effros) provides cpc coordinate lifts representing the same
quotient values, so all fixed-cut comparisons and all memberships in
J(U) are unchanged. Thus the diagonal construction may be carried out
first on countable algebraic data, and later arguments may use actual
cpc coordinate maps.
4. SMALL STABILIZERS
We build homomorphisms whose values lie inside the local error ideals but
are large enough to absorb any prescribed countable family of errors. All
matrix comparisons here are stable (Cuntz) comparisons, and the
identifications of Lemma 2.4 are understood: a rational UHF fraction
changes ranks, not operator norms.
THEOREM 4.1 (small stabilizers). Let S = C_0(Y) (x) P and prescribe
countably many pairs
x_l in S_+, e_l in J(U_{x_l}), l = 1, 2, 3, ...
There is a homomorphism theta : S -> D with
theta(S(U)) subset J(U) for every open U,
e_l in Ideal_D(theta(x_l)) for every l, (4.1)
in either of the following situations.
(i) A model chi : S -> D is given. Then theta can be a slowly vanishing
rational UHF fraction of chi, and it also satisfies the coarse
fullness
theta(S(U_x)) subset Ideal_D(theta(x)) for every x in S_+, (4.2)
and for every fixed N, N copies of its defining fraction fit in the
defining copy of chi at all sufficiently late coordinates.
(ii) Y = (0, 1]. No model is needed, and (4.2) is not asserted.
In both cases, for every l and eps > 0 there is delta_{l,eps} > 0 with
[(|e_l| - eps)_+] <= [(theta(x_l) - delta_{l,eps})_+] in Cu(D). (4.3)
The prescribed errors and the stabilizer may occupy separate matrix
corners.
The fixed cut in (4.3) stays put while the accuracy, matrix sizes, and
finite lists change. We first fix the countable tests and the
comparison bookkeeping that guarantee this.
4.1 Fixed tests and comparison bounds.
Choose a countable compact-containment basis for the open ideals as in
Lemma 2.3. For every pair V compactly in U in that basis, take a
norm-dense sequence of positive contractions in S(V) and include all
positive rational cut tolerances. Enumerate the resulting upper tests
(a_i, eps_i, V_i, U_i), a_i in S(V_i)_+, V_i << U_i, eps_i > 0. (4.4)
For each test fix a compactly Cuntz supported positive b_i in Q(U_i)
whose ideal contains Q(V_i), chosen before the stages. Showing
d_tau((theta_n(a_i) - eps_i)_+) <= s_{i,n} d_tau(b_i),
s_{i,n} -> 0, uniformly in tau, (4.5)
suffices for local smallness: Proposition 3.9 first gives it on the dense
tests in S(V_i), linearity and norm closure extend to S(V_i), and the
ideals with V_i << U exhaust S(U).
Choose bounded representatives e_{l,n} for the errors. Enumerate each
error together with positive cut tolerances decreasing to zero; write a
lower request as q = (l, eta) and put
A_{q,n} = (|e_{l,n}| - eta)_+,
d_tau(A_{q,n}) <= r_q(n) d_tau(c_q), r_q(n) > 0, r_q(n) -> 0, (4.6)
where c_q is a fixed compactly Cuntz supported positive in Q(U_{x_l})
and the bound holds eventually for every tau in T(Q). Enlarging finitely
many values of r_q is harmless; a request with zero controller has
eventually zero source cut and is dropped.
The fixed-cut witness bound of Lemma 3.4 is used repeatedly; recorded
once: if A << (B - delta)_+, choose v with v*(B-delta)_+ v close to A,
put g_delta(t) = sqrt((t-delta)_+/t) (t>0), g_delta(0)=0, w = g_delta(B)v;
then w*Bw = v*(B-delta)_+v and
||w||^2 <= (||A|| + ||v*(B-delta)_+v - A||) / delta, (4.7)
independent of matrix size.
4.2 Fractions of an existing model: proof of Theorem 4.1(i).
Take cpc lifts chi_n : S -> Q of chi by nuclearity and the Choi-Effros
lifting theorem. Corollary 3.7 gives Ideal_D(chi(x_l)) =
Ideal_D(Q(U_{x_l})). Compact support of the controller in (4.6), with an
extra cut before passing to coordinates, gives an integer M_q and d_q>0
with
d_tau(c_q) <= M_q d_tau((chi_n(x_l) - d_q)_+) (4.8)
eventually for every tau. Cut detail: write [c_q] <= [(z-gamma)_+] with
z in Q(U_{x_l})_+; the constant (z-gamma/2)_+ lies in the ideal generated
by chi(x_l), so its smaller cut sits below finitely many copies of a
positive cut of chi(x_l); leave a further cut between (z-gamma)_+ and
that source before lifting. Both the multiplicity and the target
tolerance depend only on the request.
For an upper test choose a strictly positive h_i in Q(V_i); Corollary 3.7
gives chi(a_i) in Ideal_D(Q(V_i)) = Ideal_D(h_i). Compact containment
between two cuts of chi(a_i) gives an integer N_i and kappa_i > 0 with
[(chi(a_i)-eps_i/2)_+] <= N_i[(h_i-kappa_i)_+]. Let c_i in Q(V_i)_+ be
the folded sum of these N_i cuts, compactly Cuntz supported in Q(V_i);
Lemma 2.3 gives [c_i] <= L_i[b_i]. Lift a comparison for the less deeply
cut source with error below the remaining source slack; the positive
perturbation lemma gives
d_tau((chi_n(a_i) - eps_i)_+) <= L_i d_tau(b_i) (4.9)
eventually for every tau. Thus both the controller b_i and its
multiplicity are fixed before the UHF fractions are selected.
At stage j choose a positive rational t_j with
t_j <= 1/j, t_j L_i <= 1/j (i <= j). (4.10)
Only after t_j is chosen, choose an increasing threshold sequence N_j so
that the first j coordinate comparisons hold and
M_q r_q(n) <= t_j (q <= j, n >= N_j). (4.11)
For N_j <= n < N_{j+1} let p_n be a projection of normalized trace t_j in
an extra UHF factor and put theta_n(a) = chi_n(a) (x) p_n. After the
fixed folding these are cpc into Q; their multiplicative defects tend to
zero because chi_n's do, so they define a homomorphism theta : S -> D.
Equation (4.9) gives (4.5) with s_{i,n} <= 1/j on stage j, and
d_tau(A_{q,n}) <= r_q(n) d_tau(c_q)
<= M_q r_q(n) d_tau((chi_n(x_l)-d_q)_+)
<= d_tau((theta_n(x_l)-d_q)_+) (4.12)
by (4.8) and (4.11), in extended arithmetic with strictly positive
scalars. Pointwise order in Cu(Q) gives the coordinate comparison; apply
(4.7) to the target (theta_n(x_l)-d_q/2)_+ cut further at d_q/2; bounded
witnesses pass to D, giving (4.3) for the enumerated tolerances and
hence for every tolerance.
For (4.2): if y in S(U_x)_+ then chi(y) in Ideal_D(chi(x)), so for every
source tolerance a slightly larger source cut is below finitely many
copies of a fixed cut of chi(x); passing to coordinates with slack gives
the rank inequality eventually, and tensoring both sides with the same
p_n multiplies both ranks by its positive trace, keeping it. Pointwise
order and (4.7) give (theta(y)-eps)_+ in Ideal_D(theta(x)); let
eps -> 0 and use linearity and closure to get (4.2). Finally N t_j <= 1
eventually for every fixed N, so N equal rational subprojections fit in
the original UHF unit and give the stated copies inside the spare. QED
(part (i)).
4.3 A finite cone stage with a fixed lower signal.
Without a model to cut down we first build finite matrix maps whose values
detect each requested controller. Three ingredients: a fixed quotient-norm
threshold for the controlling input, finitely many positive support
elements with plateaux, and finite-dimensional approximations of the
quotient cones. The plateaux convert a matrix norm lower bound into a
Cuntz comparison at a fixed threshold; stage-dependent matrix sizes then
change only rank multiplicities.
We use the following finite approximation form of cone quasidiagonality:
for a separable C*, a finite subset of C_0((0,1]) (x) R_0, and a positive
tolerance, there is a cpc map to a matrix algebra that is multiplicative
to that tolerance and preserves the norms of the specified elements to
that tolerance (Voiculescu's cone theorem; a direct proof in this form is
Shulman, Proposition 33). The ideal-covering strategy below is in the
spirit of Bosa-Gabe-Sims-White, Lemma 3.5, whose target is
O_infinity-stable; the fixed-cut estimates and the UHF trace-small
controls for our target are proved here, not imported.
LEMMA 4.2 (a finite cone stage). Let S = C_0((0,1]) (x) P. Fix finitely
many upper tests (4.4), finitely many lower requests (4.6), and a finite
multiplicativity test set. For each lower request one can fix a number
h_q > 0, independent of all later stages, such that the following holds.
For arbitrarily small multiplicativity error there is a cpc map
Phi : S -> M_k(Q) with finite constants L_i and integers M_q >= 1
satisfying, for every tau in T(Q) with the unnormalized matrix extension,
d_tau(Phi(a_i)) <= L_i d_tau(b_i),
d_tau(c_q) <= M_q d_tau((Phi(x_l) - 4 h_q)_+) (q = (l, eta)). (4.13)
Proof. Choose the lower threshold once per request. Write
[c_q] <= [(z_q - gamma_q)_+] with z_q in Q(U_{x_l})_+, gamma_q > 0. For
z in X let (x_l)_z be the image in C_0((0,1]) (x) (P/P_z), P_z the
primitive ideal labeled by z. The sets O_q(h) = { z : ||(x_l)_z|| > 8h }
are open and increase to U_{x_l} as h -> 0; approximation in this
increasing union gives h_q > 0 with
z'_q = (z_q - gamma_q/2)_+ in Q(O_q(h_q)), [c_q] << [z'_q]. (4.14)
Keep h_q fixed. For z in O_q(h_q) choose an open neighborhood N_z inside
O_q(h_q) and inside every supp_X(b_i) for which (a_i)_z is nonzero (this
is possible because (a_i)_z nonzero implies z in V_i and supp_X(b_i)
contains V_i). Functional calculus on a positive element of Q(N_z)
nonzero at z gives positive contractions h_z, k_z with h_z k_z = k_z,
(k_z)_z nonzero, and h_z compactly Cuntz supported in Q(N_z): take a
nonzero positive cut, make h_z equal to one on a smaller spectral region,
and support k_z inside that region. The ideal supports of the k_z cover
O_q(h_q), so by (4.14) finitely many suffice:
[c_q] <= M_q sum_{t=1}^{l_q} [k_{qt}] for some integer M_q, (4.15)
using compact containment in the Cuntz semigroup (the ideal generated by
all k_z contains z'_q, and [c_q] << [z'_q] is captured by a finite sum
with finite multiplicity). No compactness of the whole open O_q(h_q) is
assumed.
For each selected center z put S_z = C_0((0,1]) (x) (P/P_z) and let
alpha_t, 0 <= t <= 1, be cone scaling, (alpha_t f)(r) = f(tr), alpha_0=0.
The scaled images of a finite test set form norm-compact families;
choose finite nets in those families and their products. Cone
quasidiagonality, then contractivity to pass from nets to families,
gives a cpc map phi_z : S_z -> M_{d_z} with the required uniform
multiplicativity error and
||phi_z((x_l)_z)|| > 7 h_q. (4.16)
Define a matrix-valued cone function t |-> phi_z(alpha_t(s_z)); it is
norm-continuous and zero at t = 0. Evaluate its entries at h_z; the
resulting psi_z : S -> M_{d_z}(Q) is cpc with the same multiplicativity
bound (evaluation is a homomorphism), and its range lies in
M_{d_z}(Her(h_z)). For the lower estimate choose a unit vector v in C^{d_z}
which is an eigenvector of the positive matrix phi_z((x_l)_z) for an
eigenvalue lambda > 7 h_q. Since h_z k_z = k_z, evaluation on k_z^{1/2}
is evaluation at parameter one, so the rectangular column
w = (lambda - 4 h_q)^{-1/2} v (x) k_z^{1/2} satisfies
w*(psi_z(x_l) - 4 h_q)_+ w = k_z, ||w|| <= (3 h_q)^{-1/2}. (4.17)
Combining with (4.15) gives the second inequality of (4.13) for the
direct sum of all selected blocks.
For an upper test a block is zero if (a_i)_z = 0; otherwise h_z is
compactly Cuntz supported in an ideal contained in Ideal_Q(b_i), hence
below finitely many copies of b_i. Since the block range is supported by
1_{d_z} (x) h_z its rank is at most d_z d_tau(h_z); summing the finite
bounds gives the first inequality of (4.13). The direct sum is
contractive (norm is the maximum of the block norms), and its internal
matrix trace is the sum of the block traces. QED.
The finite stage now has a lower spectral cut independent of the
accuracy. To turn these maps into a locally small homomorphism we choose
their UHF fractions next, then delay the coordinates until the
prescribed errors are smaller than those fractions.
4.4 Proof of Theorem 4.1(ii).
At stage j apply Lemma 4.2 to the first j upper tests, lower requests,
and an increasing dense list of algebra tests, with multiplicativity
error at most 1/j. Write the resulting map as Phi_j with constants L_{ij},
M_{qj}. Choose a positive rational t_j with
t_j <= 1/j, t_j L_{ij} <= 1/j (i <= j), beta_{qj} := t_j / M_{qj} > 0. (4.18)
After all stage maps and fractions are fixed, choose increasing
thresholds N_j with
r_q(n) <= beta_{qj} (q <= j, n >= N_j). (4.19)
On N_j <= n < N_{j+1} tensor Phi_j with a UHF projection of trace t_j and
fold its finite matrices to obtain theta_n : S -> Q, cpc and
asymptotically multiplicative; their quotient is a homomorphism theta.
For a fixed upper test, (4.13) and (4.18) give (4.5) eventually with
bound 1/j, so theta(S(U)) is in J(U). For a lower request q = (l, eta),
d_tau(A_{q,n}) <= r_q(n) d_tau(c_q)
<= d_tau((theta_n(x_l) - 4 h_q)_+) (4.20)
eventually at every coordinate. Pointwise trace order converts this into
a Cuntz comparison: put B_{q,n} = (theta_n(x_l) - 2 h_q)_+, choose
comparison witnesses of error at most 1/n, and apply (4.7) with
tolerance 2 h_q; the resulting witnesses have squared norms at most
(sup_n ||A_{q,n}|| + 1) / (2 h_q), independent of stage, centers, and
matrix size. They prove
[(|e_l| - eta)_+] <= [(theta(x_l) - 2 h_q)_+] in Cu(D), (4.21)
which gives (4.3) and then (4.1). A very small beta_{qj} changes only the
delay N_j, never the fixed cut h_q. All comparisons may be made in the
common block sequence algebra before folding; include the finitely many
prescribed error representatives in each stage's folding tests, so the
coordinate conjugacies preserve their quotient classes and the norm
bounds. This also proves the separate-corner assertion. QED.
COROLLARY 4.3 (simultaneous coefficient domination). Suppose either
hypothesis of Theorem 4.1 holds. Let E_j be countably many bounded
multilinear maps
E_j : S x Z_{j,1} x ... x Z_{j,r_j} -> D,
with separable normed parameter spaces, and assume
E_j(S(U), Z_{j,1}, ..., Z_{j,r_j}) lies in J(U) for every open U. Then
the stabilizer may be chosen so that, for every x in S_+,
E_j(S(U_x), Z_{j,1}, ..., Z_{j,r_j}) subset Ideal_D(theta(x)). (4.22)
The same holds for norm-continuous separably parametrized families of
such maps with local norm bounds, and for any choice of the localized
multilinear input.
Proof. Choose a countable family B of positive cuts in S_+ that is an
ideal basis: for every x in S_+ the elements of B intersect Ideal_S(x)
generate Ideal_S(x); take a countable norm-dense positive set and its
rational cuts. In particular
U_x = union over y in B intersect Ideal_S(x) of U_y. (4.23)
For each y in B prescribe the errors E_j(a, z_1, ..., z_{r_j}) with a
running through a countable dense set in S(U_y) and the other variables
through countable dense sets; these are countably many requests with
controlling element y, to which Theorem 4.1 applies, and continuity gives
the containment for every input in S(U_y) and every parameter. Now fix x:
if y in B intersect Ideal_S(x), the homomorphism property gives theta(y)
in Ideal_D(theta(x)), and by (4.23) S(U_x) is the closed sum of the
S(U_y) in that equation; linearity in the localized input and closedness
of the target ideal give (4.22), with dense parameter sets and local norm
continuity handling the continuous families. The argument uses
y in Ideal_S(x), not merely U_y inside U_x, so it does not presume coarse
fullness of the cone stabilizer. QED.
REMARK 4.4 (two forms used below). (a) If I is a separable error algebra
and d : S -> D is bounded linear with d(S(U)) I in J(U), apply Corollary
4.3 to E(a,b) = d(a) b (b in I) and, when needed, to the adjoint and
opposite-product maps; this gives the full projected-support bounds on
the hereditary support generators. (b) If H in D is a given sigma-unital
hereditary algebra, possibly nonseparable, and d : S -> M(H) is bounded
linear with localized coefficients in J(U), choose a countable
approximate unit (u_m) of H and prescribe u_m d(a) u_m by the corollary;
for h, k in H, h d(a) k = lim_m h u_m d(a) u_m k, controlling all
multiplier coefficients. The countability comes from sigma-unitality, not
separability of H.
5. NORM UNIQUENESS OF SEQUENCE MODELS
We prove uniqueness for the models of Definition 3.2, for any space Y and
measure m specified there. The argument uses the coarse labels
S = S_Y = C_0(Y) (x) P, S(U) = C_0(Y) (x) P(U), U open in X. The same
statements hold if m is multiplied by a strictly positive constant. A
coordinate unitary is always a unitary in M(Q), and convergence of
coordinate conjugacies is in norm on each fixed input.
THEOREM 5.1 (norm uniqueness). Let p, q : S_Y -> Q_infinity be models for
the same data. For any cpc lifts p_n, q_n : S_Y -> Q, there are
u_n in U(M(Q)) with
lim_{n -> infty} || u_n p_n(a) u_n* - q_n(a) || = 0, a in S_Y. (5.1)
There are two steps. Equal trace data first give, after conjugacy and
rational averaging, a difference in every local error ideal J(U). A small
fraction of a common spare model then absorbs that difference in norm.
Splitting each original model into two UHF halves supplies the spare:
change the first half while keeping the second, then change the second
while keeping the first. The intermediate absorption statements allow a
general labeling space, so the cone correction can use them with its
finer ideal labels.
5.1 Compact sets of weights and tracial alignment.
LEMMA 5.2 (controlled trace evaluations). Let C be a C* and b in
(C (x) K)_+. The set
K_b = { gamma in T(C) : d_gamma(b) <= 1 } (5.2)
is compact and convex. If x in (C (x) K)_+ satisfies
sup_{gamma in K_b} d_gamma(x) <= L < infty, then gamma |-> gamma(x) is
finite, continuous, and affine on K_b. The constant L may depend on x.
Consequently this applies to the first and square evaluations of a fixed
cpc candidate whose tested values have bounded rank on K_b, and to
gamma |-> gamma(|u x_1 u* - x_2|^2) (u in U(C^~)) whenever x_1, x_2 have
ranks bounded on K_b uniformly in u.
Proof. Rank evaluation is lower semicontinuous on the compact cone, so
K_b is closed; convexity follows from additivity and positive homogeneity
of rank. For eta > 0 the rank bound gives
0 <= gamma(x) - gamma((x-eta)_+) <= eta L, gamma in K_b, (5.3)
and gamma(x) <= ||x|| L. If gamma_i -> gamma in K_b, the defining cut
inequalities of the trace topology imply
gamma(x) <= liminf gamma_i(x) <= limsup gamma_i(x) <= gamma(x)+eta L;
letting eta -> 0 gives continuity. Affineness follows from the cone
operations (no cancellation). Squaring a positive element does not change
its support rank, and support subadditivity with unitary invariance gives
d_gamma(|u x_1 u* - x_2|^2) <= d_gamma(x_1) + d_gamma(x_2). QED.
LEMMA 5.3 (barycenters on a trace cap). For K_b as in (5.2) and every
probability measure mu on K_b there is gamma-bar in K_b with
v(gamma-bar) = integral_{K_b} v dmu (5.4)
for every finite continuous affine real function v on K_b. A positive
measure of finite mass M > 0 integrates to M v(gamma-bar) with
gamma-bar chosen for mu/M; a zero measure contributes zero; signed
measures are treated by their two positive parts.
Proof. Approximate mu weakly by finitely supported probability measures
mu_j = sum_k t_{jk} delta_{gamma_{jk}}, omit zero weights, take the finite
barycenters beta_j = sum_k t_{jk} gamma_{jk} in K_b, and pass to a subnet
converging to gamma-bar. For every v in the statement,
v(gamma-bar) = lim v(beta_j) = lim integral v dmu_j = integral v dmu, with
one subnet working for all such functions. QED.
LEMMA 5.4 (alignment for finitely many weights). Let C be separable, S
separable and nuclear, and p, q : S -> C homomorphisms with equal
pullbacks on T(C). For a finite list of pairs (s_i, gamma_i), s_i in S_+,
with d_{gamma_i}(p(s_i)) + d_{gamma_i}(q(s_i)) < infty, and any delta>0,
there is u in U(M_2(C)^~) with
(gamma_i (x) Tr_2)(|u (p(s_i)(+)0) u* - (q(s_i)(+)0)|^2) < delta (5.5)
for every i. The added matrix direction is internal, with unnormalized
trace.
Proof. Equal pullback weights give equal ranks, so by CGNN Lemma 5.1 the
support projections of p(s) and q(s) are equivalent in C** for every s in
S_+; their images remain equivalent under the normal extension of every
representation. The finite-rank assumption puts p(s_i), q(s_i) in
Fin(gamma_i); after zero padding M_2(Fin(gamma_i)) is invariant under
M_2(C)^~-unitaries, so the conjugates and squared differences stay in
that finite ideal. For each selected weight take its trace representation
pi_i on its densely finite ideal (quotienting the zero ideal), extend
nondegenerately to C, let Tr_i be the resulting normal semifinite trace on
M_i = pi_i(C)'', and form pi = pi_0 (+) pi_1 (+) ... (+) pi_r with pi_0 a
faithful separable representation and M = pi(C)''. The trace
representations act separably (their finite ideals are separable with
countable approximate units of finite-trace cuts), so M has separable
predual. Restriction to the i-th reducing summand is a normal surjection
q_i : M -> M_i with kernel M(1 - z_i) for a central z_i, and q_i
identifies M z_i with M_i; define gamma-hat_i = Tr_i o q_i, a normal
semifinite trace on M z_i extended by zero on M(1 - z_i). The z_i may
overlap; the construction is separate per weight. On the finite ideal of
gamma_i, gamma-hat_i pi = gamma_i, so it computes all designated
finite-rank tests and their squares.
Apply von Neumann uniqueness after a common zero padding and unitization.
Let S^dagger be the forced unitization, and define unital maps into
M_2(M): p-hat(a + lambda 1) = (pi p(a) (+) 0) + lambda 1_{M_2(M)},
q-hat(a + lambda 1) = (pi q(a) (+) 0) + lambda 1_{M_2(M)}. These have
equal support ranks on every positive of S^dagger: if the scalar part is
zero this follows from the support equivalences; if it is positive, then
on the properly infinite central part both support projections contain
the second matrix corner (equivalent to the unit of M_2(M) there), and on
the finite central part every normal finite trace has equal pullbacks
along pi p and pi q (restrict to C and use the hypothesis). The unital
extensions have equal pullbacks, hence equal ranks by monotone spectral
calculus; normal traces determine equivalence of projections in a finite
von Neumann algebra. This gives the claim on both parts.
S^dagger is nuclear. Use the unital von Neumann uniqueness of CGNN
(Proposition 2.1, Corollary 3.10, Section 4). The properly infinite factor
argument needs only CGNN Lemma 3.6 plus a unital completion: including 1
among the inputs gives partial isometries v with initial projections
e = v*v tending strongly to 1; for finitely many normal states choose
e_0 <= e close to e on them, with 1 - e_0 ~ 1 - v e_0 v* ~ 1 (take
e_0 = e if e finite, otherwise remove an infinite tail decreasing strongly
to zero), complete v e_0 to a unitary u; since (u - v)e_0 = 0,
Cauchy-Schwarz gives |rho(u*xu - v*xv)| <= 4 ||x|| rho(1-e_0)^{1/2}. So
partial-isometry conjugacies become unitary conjugacies in the weak*
topology, and CGNN Lemma 3.9 makes them strong*. Section 4 assembles the
unital factor conclusions by central decomposition; separability lets us
impose the support comparisons on one countable dense set of positive
cuts before choosing the unitaries factorwise. We get w_j in M_2(M) with
w_j (pi p(a) (+) 0) w_j* -> pi q(a) (+) 0 strong*, a in S. The unital
extensions are needed because support ranks of nonunital maps alone do not
compare complements of support projections.
The L^2 passage uses more than strong convergence. Put
x_j = w_j (pi p(s_i)(+)0) w_j* and x = pi q(s_i)(+)0, write
T_i = gamma-hat_i (x) Tr_2. Their T_i-norms satisfy
||x_j||^2_{2,T_i} = gamma_i(p(s_i^2)) = gamma_i(q(s_i^2)) =
||x||^2_{2,T_i}. For a bounded test y in L^1(T_i) cap L^2(T_i) supported
on trace-finite projections, its pairing with x_j - x is the normal
bounded functional t |-> T_i(y*t) and tends to zero; such tests are dense
in L^2(T_i), so the common L^2 bound gives weak L^2 convergence, and
equality of norms gives ||x_j - x||_{2,T_i} -> 0.
Choose one w_j meeting the finite list. A bounded self-adjoint logarithm of
it can be approximated strong* in norm bound by self-adjoint elements of
M_2(pi(C))^~ (Kaplansky density, Theorem 1); exponentiating gives unitaries
from M_2(C)^~ converging strong* to w_j. Their conjugates again have the
same designated L^2 norms, so the preceding argument gives L^2 convergence
for this approximation too, and on these conjugates the normal traces
compute the original weights (the finite ideals are invariant under
M_2(C)^~-unitaries). This proves (5.5). QED.
LEMMA 5.5 (tracial alignment). If p, q : S -> D are models with the same
data, then rational diagonal averaging, coordinate multiplier conjugacy,
and the foldings of Lemma 2.4 give models p~, q~ : S -> D with
(p~ - q~)(S(U)) subset J(U) for every open U. (5.6)
Each new model is coordinate approximately unitarily equivalent to its
original model; the operations may be performed in one changing block
while a prescribed spare block is carried along unchanged, apart from its
common amplification and folding.
Proof. By Corollary 3.7 choose a separable subalgebra D_0 containing the
two ranges, the required constant representatives, and the comparison
witnesses, so that p and q agree on all traces of D_0; work in a common
stabilized corner when needed. Choose a countable local positive test
family s_i dense in linear span in each basis ideal of Lemma 3.10, each
test cut-supported in both factors with compact support in Y. Choose an
enlarged elementary test s-bar_i and a fixed kappa_i > 0 with
s_i << (s-bar_i - kappa_i)_+, so p(s_i) << (p(s-bar_i)-kappa_i)_+ and
likewise for q. The model-independent upper sandwich applied to that fixed
cut supplies a common compact-Cuntz-supported constant; dominating it in
the containing open gives b_i in Q(U_i)_+ and a finite L_i with
d_gamma(p(s_i)) + d_gamma(q(s_i)) <= L_i d_gamma(b_i),
gamma in T(D_0). (5.7)
The constants and comparisons are included in D_0, and b_i is retained as
finite lists grow. Fix a finite initial list and delta > 0. For u in
U(M_2(D_0)^~) define nonnegative continuous functions
f_{u,i}(gamma) = (gamma (x) Tr_2)(|u (p(s_i)(+)0) u* - (q(s_i)(+)0)|^2),
gamma in K_{b_i}, (5.8)
with Tr_2 unnormalized, so the zero padding leaves (5.7) unchanged.
Continuity is Lemma 5.2. We claim zero lies in the closed convex hull of
the tuples (f_{u,i})_i in the direct sum over i of C(K_{b_i}, R).
Otherwise a separating real functional is bounded below by a positive
constant on all tuples; represent its components by signed measures,
replace each by its positive part (since f_{u,i} >= 0), and use Lemma 5.3
to replace it by a finite sum of positive multiples of evaluations at
individual weights in the caps; Lemma 5.4 makes all these evaluations
simultaneously arbitrarily small, contradiction.
So a finite convex combination satisfies, for every tested i,
sup_{gamma in K_{b_i}} sum_{k=1}^r t_k f_{u_k,i}(gamma) < delta. (5.9)
The coefficients may be chosen positive and rational (the averaged
functions are bounded, so small changes keep the strict inequality). Write
t_k = m_k/l. Before folding work in B_l = M_l(M_2(Q)) with trace
tau (x) Tr_2 (x) tr_l; only this last averaging direction is normalized.
The two averaged maps are diagonal copies of Ad(u_k)(p(+)0) and of
q(+)0, with m_k occurrences in the k-th block. D_0 has the fixed corner
embedding iota_l : D_0 -> (B_l)_infty, x |-> (x(+)0)(x)1_l.
Lift the finitely many unitaries to unitary multiplier sequences and use
cpc representatives of p, q. Let z_{i,n} be the difference of their
averaged values after folding into Q. For every fixed t > 0 we first prove
the absolute estimate
d_tau((|z_{i,n}| - t)_+) <= 4 delta / t^2 (d_tau(b_i) <= 1) (5.10)
for all large n, uniformly over this cap. If (5.10) failed arbitrarily
late, pick violating coordinates and weights tau_n; the trace identities
of Lemma 2.4 pull them back before folding to
tau_n (x) Tr_2 (x) tr_l, and multiplier conjugations preserve them.
Positive cuts commute with the coordinate foldings, so regularization
commutes too. Let rho_l be the regularized trace before folding and
gamma = rho_l o iota_l, a trace on the original D_0; Proposition 3.1 gives
d_gamma(b_i) = sup_{eta>0} lim_omega d_{tau_n}((b_i-eta)_+) <= 1, so
gamma in K_{b_i}. Put e_{k,i} = u_k(p(s_i)(+)0)u_k* - (q(s_i)(+)0) in
M_2(D_0). If z_i is the unfolded diagonal difference, matrix-unit
traciality and finite additivity give
rho_l(z_i* z_i) = sum_k (m_k/l)(gamma (x) Tr_2)(e_{k,i}* e_{k,i}) < delta. (5.11)
Restriction to the first internal corner gives gamma, so the matrix
extension on M_2(D_0) is gamma (x) Tr_2; and for every positive a in
M_2(D_0), rho_l(a (x) e_{jj}) = (gamma (x) Tr_2)(a)/l, since the l
averaging corners are equivalent and sum to a (x) 1_l; this uses only
finite positive sums and holds when a value is infinite. Thus (5.11) is
valid without a bounded-weight assumption. On the other hand functional
calculus gives tau((|z_{i,n}|^2 - t^2/2)_+) >= (t^2/2) d_tau((|z_{i,n}|-t)_+),
so regularization and the violations of (5.10) give a square trace at
least 2 delta, contrary to (5.11). This proves (5.10). If
0 < d_tau(b_i) < infty apply it to tau/d_tau(b_i); if d_tau(b_i) = 0 apply
it to every positive multiple of tau, forcing the left side to vanish; for
d_tau(b_i) = infty the proportional bound is automatic. Thus at the same
sufficiently late coordinates
d_tau((|z_{i,n}| - t)_+) <= (4 delta / t^2) d_tau(b_i), tau in T(Q). (5.12)
At stage j use the first j local tests, cut tolerances 1, 1/2, ..., 1/j,
and delta_j < 2^{-j}/(4 j^2); advance far enough in that row for (5.12),
the chosen algebraic tests, and the norm bounds to hold; pass through rows
slowly as in Lemma 3.10. The averaging size is fixed while each row's
uniform estimate is proved. For a fixed local test and cut tolerance this
gives its J(U_i) estimate against the fixed b_i; closure and local
density give (5.6) for all open ideals. These operations preserve the
asserted relation to the original maps: at each coordinate a diagonal sum
of conjugates is conjugate, by the diagonal of the same unitaries, to its
tensor-first copy in the averaging factor, and the coordinate folding can
be chosen (Lemma 2.4) to carry that copy within the stage tolerance of the
original on every current input, with the first-corner clause removing the
internal zero padding; make that tolerance tend to zero during the same
diagonal choice. Inner conjugacy preserves every coordinate weight and
vanishing norm changes preserve regularized values by cuts, so the models
keep their data. The argument applies to a designated block of a direct
sum, amplifying the spare by the same averaging factor; its unused part
can be placed in a separate stable corner before folding. QED.
5.2 Ideal-related absorption and compact perturbations.
Tracial alignment placed the difference of two models in every J(U). To
remove it in norm we dominate its coefficients by a small stabilizer and
apply stable uniqueness on a hereditary support. We state the absorption
and compact perturbation results used in that step.
An X-structure on a coefficient algebra H is an order-preserving
assignment U |-> H(U) of closed ideals (no preservation of intersections
assumed). Set E_H = l^2(N) (x) H, K(E_H) = H (x) K, and
K(E_H)(U) = H(U) (x) K. A representation phi : S -> L(E_H) is weakly
equivariant if
<xi, phi(a) eta> in H(U) for a in S(U), xi, eta in E_H, (5.13)
equivalently phi(S(U)) E_H lies in E_H H(U); every adjointable operator
preserves these submodules. The diagonal coefficient maps
a |-> <xi, phi(a) xi> are cpc, and for nuclear S they and their quotient
maps are nuclear (ideals and quotients of a nuclear C*-algebra are
nuclear), so weak equivariance here implies weak X-nuclearity, also for
homotopies.
THEOREM 5.6 (ideal-related absorption; Gabe, Theorem 11.14). Let S be
separable, exact, and lower semicontinuous over X, and let H be
sigma-unital. Suppose theta : S -> H (x) K is a nuclear X-full
homomorphism, i.e. Ideal_{H (x) K}(theta(a)) = H(U_a) (x) K for a in S_+.
Then its infinite repeat is weakly X-nuclear and absorbs every weakly
X-nuclear representation on the standard module, by unitary equivalences
with compact errors tending to zero on each input.
Neither separability of H nor an absorption hypothesis on H is required.
The repeat is taken on the stabilized module: if theta_0 : S -> H lies in
the compact corner theta_0 (x) e_11, its stable repeat retains the other,
zero module slots, and after a permutation has the form
diag(theta_0, 0, theta_0, 0, ...). These zero slots remain available when
absorbing a degenerate representation.
For weakly equivariant phi, psi : S -> L(E_H) with compact difference,
write [phi, psi]_X for the class of the graded cycle
(E_H (+) E_H, diag(phi, psi), [[0,1],[1,0]]) (5.14)
with opposite gradings on the two summands; the compact difference gives
the compact commutators. The notation refers to the pair, without
assigning separate classes to the multiplier-valued maps.
PROPOSITION 5.7 (multiplier stable uniqueness; uses Gabe, Theorem 13.9).
Suppose S is separable, nuclear, and lower semicontinuous over X, H is
sigma-unital, and theta : S -> H (x) K is X-full. Let phi, psi : S ->
L(E_H) be weakly equivariant representations with compact difference and
[phi, psi]_X = 0. Writing theta^infinity for an infinite repeat, for
every finite F subset of S and eps > 0 there is a unitary
v in 1 + K(E_H (+) E_H) with
||v (phi (+) theta^infinity)(a) v* - (psi (+) theta^infinity)(a)|| < eps,
a in F. (5.15)
Proof. Gabe's multiplier-pair theorem supplies a weakly X-nuclear
representation Theta and an identity-plus-compact unitary u comparing
phi (+) Theta with psi (+) Theta to any finite-test tolerance delta (the
pair class may be read in the nuclear theory since S is nuclear). Put
sigma = Theta (+) theta^infinity. By Theorem 5.6 there is a module unitary
V from the repeat module to the module of sigma with
||V* sigma(a) V - theta^infinity(a)|| < eta on F. With W_0 = 1 (+) V,
set v = W_0* (u (+) 1) W_0. It is a unitary, and v - 1 is compact because
conjugating a compact module operator by a module unitary preserves
compactness. The error in (5.15) is at most delta + 2 eta; choose
delta + 2 eta < eps. This is the finite-test common-stabilizer transfer
of Dadarlat-Eilers; no initial value of a continuous unitary path is
needed. QED.
LEMMA 5.8 (the trivially labeled factor). For the coarse X-structure on
S = C_0(Y) (x) P and every sigma-unital coefficient X-algebra H,
KK(X; S, H) = 0. (5.16)
In particular every weakly equivariant multiplier pair with compact
difference has zero class in Proposition 5.7.
Proof. Since W is simple and nuclear, every ideal of its spatial tensor
product with another C*-algebra is a tensor-product ideal (Gupta-Jain,
Theorem 3.1). Together with the transported labels of Proposition 2.2
this gives, after reordering factors, an X-algebra C = C_0(Y) (x) A (x) K
with S = W (x) C and S(U) = W (x) C(U). The identity of W has zero
ordinary Kasparov class; exterior tensoring its stabilized cycle relation
and its W-homotopy with C retains weak equivariance (a typical coefficient
of the tensor representation, for c in C(U), is a finite sum of
<xi_i, pi(w) xi_j> (x) c_i* c c_j in W (x) C(U), and the same formula holds
throughout the homotopy). Thus the identity cycle of S is zero for these
labels. Multiplying by an S-H cycle preserves this stabilized null
relation, and the identity-product homotopy identifies the product with
the original cycle, so that cycle is zero. The cycle relation and mixed
product statements used are Gabe, Propositions 12.17, 12.20 and Lemmas
12.18, 12.19, 12.23, which require the first algebra separable and the
remaining coefficient algebras sigma-unital; thus they apply even for
nonseparable H. Nuclearity of S makes the cycles and homotopies weakly
X-nuclear. QED.
LEMMA 5.9 (hereditary support and local multiplication). Let I be a
separable C*-subalgebra of a C*-algebra C with strictly positive element
e, and k in C_+. Set H = Her_{M_2(C)}(e (+) k). Then H is sigma-unital
(no separability required). If r in C idealizes I, then r (+) 0 idealizes
H. If in addition L is an ideal of M_2(C) and (r I)(+)0 and (I r)(+)0 lie
in L, then
(r(+)0) H + H (r(+)0) subset H cap L, (5.18)
and for x in H, H cap Ideal_{M_2(C)}(x) = Ideal_H(x).
Proof. The defining positive element is strictly positive in H, so H is
sigma-unital. Since r e^{1/2} and e^{1/2} r are in I, multiplication of
(e^{1/2}(+)k^{1/2}) M_2(C) (e^{1/2}(+)k^{1/2}) on either side by r(+)0
stays in H; these elements are dense in H. Under the extra hypothesis the
same products also lie in L, giving (5.18). The last identity is the ideal
correspondence for a hereditary subalgebra: an approximate unit of H
compresses the ambient ideal of x to the ideal it generates in H. QED.
LEMMA 5.10 (finite truncation of a proper conjugacy). Let H be
sigma-unital and p, q : S -> M(H) representations with difference in H.
Let theta : S -> H be a homomorphism, compact-valued on the module H, and
form A_0 = p (+) theta (+) theta (+) ... and B_0 = q (+) theta (+) ...
(identical zero summands may be inserted). If an identity-plus-compact
unitary compares A_0, B_0 arbitrarily well on finite sets, then for every
finite F and eps > 0 finitely many copies suffice: for some N, M >= 0
there is a unitary v_N equal to the identity outside the initial summand,
the first N repeat summands, and finitely many zero slots, with
max_{a in F}
||v_N (p (+) theta^N (+) 0^M)(a) v_N*
- (q (+) theta^N (+) 0^M)(a)|| < eps. (5.19)
All resulting errors are compact. If the representations are weakly
equivariant, their compact errors on S(U) lie in the compacts over H(U).
Proof. Choose u = 1 + k with error less than eps/2, put
R = max(1, max_{a in F} ||a||), and if E_N is the finite-summand
projection choose ||k - E_N k E_N|| < eta with
0 < eta < min{1/4, eps/(8R)}. The operator z_N = 1 + E_N k E_N is within
eta of a unitary, so it is invertible with spectrum of |z_N| in
[1-eta, 1+eta]; its polar unitary v_N = z_N (z_N* z_N)^{-1/2} satisfies
||v_N - z_N|| <= eta and ||v_N - u|| <= 2 eta, equals the identity off
E_N, and differs from 1 by a finite matrix over H. Include identical zero
slots successively so E_N -> 1 strongly; the change in the conjugacy
error is at most 4 R eta; since the two representations agree on the
omitted tail, the remaining estimate is exactly (5.19). For localization,
a compact operator weakly equivariant on an input in S(U) has all matrix
coefficients in H(U), hence belongs to H(U) (x) K; module unitaries
preserve the submodules E_H H(U), so conjugation preserves this, applying
to the compact differences here. QED.
5.3 Removing the locally small difference.
PROPOSITION 5.11 (uniqueness with a spare model). Let p, q : S -> D be
equal-data models satisfying (p - q)(S(U)) in J(U) for every open U. Let
chi be a model for a strictly positive multiple of the same data. Then
p (+) chi and q (+) chi are coordinate approximately unitarily equivalent,
with the matrix and trace conventions of Lemma 2.4.
Proof. Put C_1 = C*(p(S), q(S)) and let I be the ideal of C_1 generated
by the differences p(a) - q(a); it is separable. For each open U,
p(S(U)) I + q(S(U)) I + I p(S(U)) + I q(S(U)) subset J(U). (5.20)
Indeed modulo J(U) the two maps agree on S(U), and their common image is
an ideal in the algebra generated by both images (multiplication by p(a)
or q(a) corresponds to multiplication inside the source ideal); it
annihilates each difference, since p(s)(p(a)-q(a)) = p(sa) - q(sa) = 0
mod J(U) for s in S(U), and likewise on the other side, hence annihilates
the ideal generated by the differences.
Use the existing-model case of Theorem 4.1 with the spare chi to obtain a
locally small homomorphism theta. Impose the coefficient dominations
p(S(U_x))I + q(S(U_x))I + I p(S(U_x)) + I q(S(U_x))
subset Ideal(theta(x)), x in S_+, (5.21)
by applying Corollary 4.3 to the four bounded bilinear maps
(a,b) |-> p(a)b, q(a)b, b p(a), b q(a) with b in I; (5.20) gives the
local hypotheses and the corollary supplies (5.21) for every positive x,
while part (i) also gives theta(S(U_x)) subset Ideal(theta(x)). Moreover
theta is represented by the same coordinate maps as chi, tensored with
rational UHF projections of trace tending to zero.
Choose strictly positive e in I and k in theta(S), and form
H = Her_{M_2(D)}(e (+) k). The maps p, q (first corner, zero in the
second) idealize H by Lemma 5.9. View them as representations on the
module H, then stabilize it, and equip H with the order-preserving ideals
H(U) = Ideal_H(0 (+) theta(S(U))). (5.22)
No intersection property is needed. By (5.21), Lemma 5.9, and its
ideal-intersection identity, values of the first-corner maps on S(U_x)
multiply H into Ideal_H(0 (+) theta(x)); so p, q are weakly equivariant
for (5.22). By theta-range-fullness, 0 (+) theta(x) generates H(U_x), so
this compact-valued map is X-full. H is sigma-unital (not required to be
separable).
The multiplier pair agrees modulo H, since its differences lie in
I (+) 0 inside H. Pad the module H by zero copies to the standard module;
the resulting pair class (5.14) is zero by Lemma 5.8. The coarse source
action is lower semicontinuous, since Y x X -> X is open. For Proposition
5.7 place 0 (+) theta in the first compact corner of the standard module;
its infinite repeat is, after reindexing, the diagonal of copies of theta
on H with identical zero slots, so Lemma 5.10 applies on the original H
summands, giving finite matrices over H rather than an additional
untruncated compact direction inside each slot. For any finite F and eps
this gives a finite N and an identity-plus-finite-matrix unitary comparing
p (+) theta^N and q (+) theta^N (5.23)
within eps, with zero slots permitted.
These estimates hold for the actual ambient elements: if v = 1 + c is the
obtained finite unitary and A, B are the two finite diagonal maps, then
v A(a) v* - B(a) = (A(a) - B(a)) + c A(a) + A(a) c* + c A(a) c*, (5.24)
and every term is in a finite matrix over H (the maps idealize H and their
difference is in it); the multiplier norm is the intrinsic H-norm hence
the ambient norm in the matrix algebra over D. We only use the isometric
restriction to H, not faithfulness of the whole idealizer. There is no
lifting obstruction for the finite unitary: write it as 1 + c in the
unitization of a finite matrix algebra over D, choose bounded
representatives c_n over Q; the two unitary defects of 1 + c_n tend to
zero, and for large n the polar part v_n = (1+c_n)((1+c_n)*(1+c_n))^{-1/2}
is a matrix-over-Q unitary differing from 1 + c_n by a null sequence, so a
strict quotient estimate (first obtained with a smaller tolerance) holds
at every late coordinate.
Finally fix N before advancing the coordinates. Since the UHF trace of the
projection defining theta_n tends to zero, N disjoint equivalent copies
fit inside the positive spare fraction eventually; using the same chi_n in
those copies identifies them with (5.23). Move the unused spare to a
separate stable corner and leave it unchanged, so the finite conjugacy need
not preserve an internal complement of its matrix slots. Apply the
coordinate folding and multiplier permutations of Lemma 2.4 on all current
inputs and all these blocks. This gives eventual finite-test conjugacy of
p (+) chi and q (+) chi; on increasing finite sets with tolerances tending
to zero choose each coordinate beyond the corresponding threshold; the
resulting single sequence of multiplier unitaries proves the proposition.
QED.
Proof of Theorem 5.1. Split the tensor-first copy of each map along two
equivalent UHF projections of trace 1/2, giving halves p_0, p_1 and q_0,
q_1; both halves have the original data multiplied by 1/2, and their sums
are coordinate approximately unitarily equivalent to the originals
(Lemma 2.4). First compare p_0 (+) p_1 with q_0 (+) p_1 keeping p_1 as
common spare: apply Lemma 5.5 in the changing block to put its difference
in every appropriate J(U), carrying the spare through the same diagonal
amplifications, then Proposition 5.11 gives the conjugacy; the coordinate
conjugacies relating aligned blocks to originals can be undone with
vanishing errors, so the conclusion applies to the displayed original
half-sums. Next compare q_0 (+) p_1 with q_0 (+) q_1 using q_0 as common
spare and interchanging the matrix positions; the same argument applies.
Composing the two coordinate conjugacies and the folding adjustments
proves the assertion on every finite test set and tolerance; a diagonal
choice on a dense increasing family gives the sequence u_n. Finally any
two cpc lifts of a fixed model differ in norm by a null sequence on each
input, so the conclusion holds for the lifts specified in the statement.
QED (Theorem 5.1).
6. TRACIAL APPROXIMATION ON A CONE
We now build the completely positive contraction that will become the
model h : S -> D. It must realize, for every regularized limit, the
prescribed first and second trace moments, and simultaneously keep a
lower ideal comparison that rules out any larger finite ideal. We
proceed in four parts: von Neumann realization, finite caps and lower
data, sector compression inside Q, and the averaging lemma that
preserves the lower rank bound.
Throughout S = S_Y = C_0(Y) (x) P and Q = B (x) W (x) K, with X the
common primitive space and m a finite full-support Radon measure on Y
(a point, I = (0,1], or R with a full-support measure).
6.1 Realization in von Neumann algebras.
We record the trace-preserving embedding needed for extended weights
(the embedding is not required to be unital). Write R for the hyperfinite
finite factor of type II_1 and R_inf = R (x) B(l^2). Their uniqueness is
part of Connes's classification; the separate contemporary statement
used for nuclear representation we cite openly.
LEMMA 6.1 (trace-preserving semifinite embedding; Connes, HRS). Let N be
a hyperfinite von Neumann algebra with separable predual and a faithful
normal semifinite trace T. There is a normal injective homomorphism of N
into the hyperfinite factor R_inf of type II_infinity preserving T.
Consequently such an embedding exists into any prescribed infinite corner
of a separably acting factor of type II_infinity.
Proof. The central decomposition of a semifinite hyperfinite algebra is a
countable direct sum of algebras L^infty(X_j, mu_j^0) (x) N_j with
N_j in {M_n, B(l^2), R, R_inf} and standard measure spaces, factor traces
the usual ones (semifinite hyperfinite central decomposition; HRS). Relative
to these traces and finite mu_j^0, T has a central Radon-Nikodym density h_j
with 0 < h_j < infinity a.e.; replacing mu_j^0 by mu_j = h_j mu_j^0 absorbs
it into a standard sigma-finite measure space, and each direct summand
embeds trace-preservingly into R_inf (finite-dimensional algebras and R
into a corner of R_inf; R_inf into an infinite corner of itself), the
sums of the embeddings being normal injective on countable direct sums.
The composite is the required T-preserving embedding. QED.
LEMMA 6.2 (realization in a factor). Let pi : Q -> M be a nondegenerate
representation with weakly dense range in a separably acting factor.
There is a homomorphism l : S -> M with the following rank data. If M is
semifinite with faithful dimension trace T, then
d_T(l(f (x) a)) = m({f > 0}) d_{F(T o pi)}(a), f, a >= 0. (6.1)
In a type III factor the support projection is nonzero exactly when the
transported ideal data prescribe a nonzero support.
Proof. Stability of Q supplies multiplier isometries whose normal images
make M properly infinite. Since M is separably acting, the usual scalar
trace-dimension comparison applies in its semifinite factor cases, and in
its type III case all nonzero projections are equivalent. If M is type I,
the rank function d_{T o pi} takes only values 0 and infinity (otherwise
real scalar divisibility of Cu(Q) applied to a class of finite nonzero
integer rank would give a noninteger rank in a type I factor), so the
right side of (6.1) takes only 0 and infinity; its zero ideal is that
transported from ker(pi), and a faithful separable representation of the
corresponding quotient of S, repeated with infinite multiplicity and
embedded in M, realizes the ranks. Type III is the same construction for
the zero ideal with all nonzero supports equivalent. If M is
II_infinity, put nu = m (x) F(T o pi), I = Fin(nu), K = Ker(nu); the trace
representation of I/K has weak closure N with the induced faithful
normal semifinite trace, nuclearity of I/K makes N hyperfinite, N has
separable predual, and its nondegenerate representation extends from I to
S by the action of S on I. Lemma 6.1 embeds this trace-preservingly in
one infinite corner of M. In an orthogonal infinite corner choose a copy
of B(l^2) with minimal projections of fixed positive finite trace and
place in it an infinite multiple of a faithful separable representation
of S/I; on I this summand is zero, while for x in S_+ not in I its image
has infinite trace and rank, exactly nu(x) = infinity off I. On I the
first summand gives nu and all its ranks including infinite values. The
combined homomorphism therefore satisfies T o l = nu, has zero ideal K,
and proves (6.1). If I = 0 or I = S omit the empty summand. QED.
Factorwise existence is enough only if the maps assemble measurably. We
encode their rank requirements by partial isometries between support
projections; these witnesses assemble along with the maps and later let
each selected normal trace evaluate the ranks directly.
LEMMA 6.3 (measurable realization; Vaes-Wouters measurable uniformization).
Let pi be a nondegenerate separably acting representation of Q and put
M = pi(Q)''. There is a homomorphism l : S -> M with the factorwise ranks
of Lemma 6.2 on any prescribed countable elementary cut-test family. In
particular it may be chosen on the whole family E constructed in 6.2 and
its required cuts.
Proof. Use the standard central decomposition of M into separably acting
factors M_x modulo null sets, and explain the measurable selection without
normalizing factor traces. For each elementary test s_j = f_j (x) a_j fix
c_j in Q_+ with [c_j] = m({f_j > 0}) G[a_j]; realification and stability
provide these. Code a contractive homomorphism on a countable dense
rational *-algebra in S by its values in the strong-* bounded balls of the
factor field, including norm-convergent approximations to the chosen
positive tests. Multiplication on bounded balls, adjoints, norm bounds,
and membership in the measurable algebra field give countably many Borel
conditions (membership checked against strong-* dense measurable
sections). Adjoin contraction coordinates w_j in M_x subject to
w_j* w_j = supp(l_x(s_j)) and w_j w_j* = supp(pi_x(c_j)); the support map
is the strong limit of t |-> t(t + 1/n)^{-1}, so these are Borel
conditions too. The augmented admissible-code relation is Borel in a
countable product of standard measurable bounded operator fields; its
sections are nonempty by Lemma 6.2 (equal factor dimensions give the
partial isometries), so Jankov-von Neumann measurable uniformization
gives a selector retaining the homomorphism coordinates and the w_j,
measurable for the completed central measure. Replace its coordinates by
measurable representatives and discard one common null set on which any
selected algebraic equation or support identity fails (the measured-field
setting of Vaes-Wouters, distinguished from an everywhere Borel field).
The uniform operator bounds make the selected values decomposable, and
their countably many algebraic relations define a homomorphism l : S -> M
with the assembled witnesses. Every elementary cut test below is in the
code, so its rank data hold. QED.
6.2 Finite caps, lower data, and sector compression.
Fix finitely many tests s_i = f_i (x) a_i, 1 <= i <= n, with fixed
controllers. Set
K_i = { tau in T(Q) : d_tau(b_i) <= 1 },
t_i^{(k)}(tau) = (integral f_i^k dm) F(tau)(a_i^k). (6.2)
By Lemma 5.2, K_i is compact and convex and t_i^{(k)} is finite continuous
affine on it. We impose finitely many lower comparisons; choose delta_j > 0
and r_j > 0 with
r_j < m({f_j > delta_j}), [y_j] = r_j G[(a_j - delta_j)_+], (6.3)
y_j in Q_+, rescaled so ||y_j|| <= 1. Choose continuous
g_j : [0,infty) -> [0,1] and constants
0 < eta_j < theta_j < delta_j^2, g_j|_{[0,eta_j]} = 0,
g_j|_{[theta_j,infty]} = 1. (6.4)
These constants are fixed before any finite approximation or averaging.
Let q_j be the support projection of l((f_j - delta_j)_+ (x)
(a_j - delta_j)_+). Functional calculus in the two commuting factors
gives q_j <= 1_{(delta_j^2,infty)}(l(s_j)). The rank data and (6.3) give,
in every factor, supp(y_j) <~ q_j (comparison covering zero, infinite
dimension and type III), so projection comparison in a direct integral
gives v_j in M with initial projection supp(y_j) and final projection at
most q_j, hence
y_j v_j* g_j(l(s_j)) v_j y_j = y_j^2. (6.5)
Approximate inside fixed supports. A tested value must vanish exactly in
a block where its controller vanishes; strong approximation alone would
not retain this. We therefore split into central sectors and factor each
sector map through the required source quotient before approximating.
Let z_i be the central support of b_i in M and form the finitely many
central sectors
z_E = prod_{i in E} z_i prod_{i notin E} (1 - z_i), E subset {1..n}. (6.6)
If i not in E, the rank data and G[a_i] <= [b_i] force z_E l(s_i) = 0.
Let J_E^S be the ideal of S generated by those s_i and
I_E = intersection_{i in E} Ideal_Q(b_i) (I_empty = Q). The weak closure of
I_E has central support dominating z_E; choose positive contractions e_E
in I_E with compact Cuntz support, tending strongly to 1 on that sector.
The maps
psi_E(s) = e_E z_E l(s) e_E (6.7)
take values in Her(e_E)'', are cpc, and factor through S/J_E^S. Write
D_0(s) = diag_E(z_E l(s)) and D_e(s) = diag_E(e_E z_E l(s) e_E). As the
e_E increase along an approximate unit, D_e(s) -> D_0(s) strong-*. For the
selected trace of test i, the sector contributes zero if i not in E; if
i in E, the positive operators z_E l(s_i) and z_E l(s_i^2) are integrable
and z_E l(s_i) is square-integrable, so strongly converging bounded
approximate units make the compressions converge in the required L^1 and
L^2 seminorms (L^1 by Cauchy-Schwarz from L^2 of the square root; squares
directly). For the lower signal set V_j = (z_E v_j)_E, a contraction with
g_j(0) = 0, the sector gives y_j V_j* g_j(D_0(s_j)) V_j y_j = y_j^2;
bounded strong-* functional calculus gives g_j(D_e(s_j)) -> g_j(D_0(s_j)),
and fixing e_E far enough out makes the errors against the selected normal
functionals small while meeting the finite trace requirements. The
compressed expression need not equal y_j^2 exactly.
Approximate the rest. Nuclearity of the quotient R_0 = S/J_E^S provides
cpc factorizations through finite matrix algebras approximating its
identity on the finite tests (Choi-Effros; Han-Paulsen Corollary 3.3 for
the unital statement applied to the unitization of R_0). Restrict the
incoming map to R_0 and compress the outgoing beta to x |-> e beta(x) e
with a positive contractive approximate-unit element e of R_0 chosen so
||e a e - a|| < delta/2 on the finite tests; then choose the factorization
with error < delta/2 there. Contractivity bounds the total error by delta,
and the compressed map takes values in R_0; compose with psi_E. A cpc map
M_d -> Her(e_E)'' is specified by its positive Choi matrix; approximate
its square root strongly-* with bounded norm by matrices over Her(e_E),
whose products give positive Choi matrices over the hereditary algebra.
If the resulting beta has t = beta(1), replace it by
x |-> max(1,t)^{-1/2} beta(x) max(1,t)^{-1/2} (functional calculus in the
unitization). This is cpc, stays in Her(e_E), and retains the strong-*
limit since the limiting unit value is at most one. We get cpc maps
H_E : S/J_E^S -> Her(e_E) subset Q approximating (6.7) on the required
tests.
The support element e_E was fixed before the Choi-matrix approximation and
every approximating value stays in Her(e_E). For each i in E, compact
Cuntz support in I_E gives a finite integer N_{E,i} with
[e_E] <= N_{E,i}[b_i], so every output of this block has rank at most
N_{E,i} d_tau(b_i) for every extended tau: on the selected traces a fixed
finite normal trace support. Bounded strong-* convergence on that support
gives convergence of the trace and square-trace evaluations with no loss
of mass. For i not in E the quotient-first factorization kills s_i and
s_i^2 exactly, so no trace-convergence argument is needed in that block.
This proves the trace approximation and the required uniformity on the
whole caps.
6.3 A lower rank bound preserved by averaging.
LEMMA 6.4 (rank bound under averaging). Fix lower data y, g, eta as in
(6.3)-(6.4). If a finite rational average satisfies
|| sum_alpha lambda_alpha y v_alpha* g(H_alpha(s)) v_alpha y - y^2 ||
< eps^2 / 2, (6.8)
then for every extended trace on Q,
d_tau((y - eps)_+) <= (2 ||y||^2 / eps^2) d_tau((H_av(s) - eta)_+). (6.9)
The constant and the cut eta are independent of the number of candidates,
their multiplicities, and their internal matrix sizes.
Proof. Write the displayed average as T and let p = 1_{(eps,infty)}(y) in
the bidual. The norm estimate gives p T p >= (eps^2/2) p. Since the
v_alpha are contractions and 0 <= g(t) <= 1_{(eta,infty)}(t), positivity
and traciality give
(eps^2/2) tau(p) <= ||y||^2 sum_alpha lambda_alpha
d_tau((H_alpha(s) - eta)_+), (6.10)
and the sum is exactly the rank on the right of (6.9) by the normalized
averaging convention (Lemma 2.4). Cut comparison
p <~ (1/eps^2) y^2 = (1/eps^2) y y* (because p y^2 = y^2 p = eps^2 p on
the spectral projection, with (eps^2/2) p <= p T p <= ||y||^2 sum ... and
(6.10) then gives (6.9) once the right side is finite; the left may be
infinite). QED.
Now assemble. Choose a countable family E of positive elementary tensors
s = f (x) a as follows. In each ideal of the countable basis and in each
finite union of those ideals, take positive rational cuts of a norm-dense
set of positives, retaining slightly larger positive pre-cuts in the same
ideal. Do the same for positive functions in C_0(Y), including powers and
further cuts of the factors occurring below. Thus local linear spans are
dense and rectangle supports exhaust the open subsets of Y x X. All
elementary tests of this section may be included in E. For each test
s_i = f_i (x) a_i supported over open U_i choose once and for all a
positive controller b_i in Q(U_i) of compact Cuntz support with
G[a_i] <= [b_i]; the larger pre-cuts allow this inside Q(U_i) with room
for a further compactly supported upper controller. When needed the same
test is listed with different containing basis opens, so the controllers
of a local assertion lie in that open as in Lemma 3.10.
THEOREM 6.5 (rough cone model). There is a cpc contraction h : S -> D with
the following properties, for every regularized limit rho and
sigma = rho|_Q.
(i) For every elementary tensor s = f (x) a supported over U,
rho(h(s)) = m({f > 0}) F(sigma)(a), with the unnormalized matrix
trace where applicable, and likewise the second moment.
(ii) For the controlled tests and fixed controllers b_i,
d_rho(h(s_i)) <= L_i d_rho(b_i), the constants independent of the
later diagonals.
(iii) Lower ideal signal: for each controlled test s = f (x) a with
f nonzero, Ideal_D(Q(U_a)) subset Ideal_D(h(s)), so h(s) is finite
exactly where m (x) F sigma is finite.
(iv) Mixed multiplicativity defect: for every open U in X, s in S(U)
and z in S,
h(sz) - h(s)h(z), h(zs) - h(z)h(s) in J(U).
Proof. At stage n apply the finite-convexity construction (the preceding
sector compression plus (6.10); finite data realized by Lemma 6.3) to the
first n trace tests and first n lower requirements. Choose rational
averages and fold them into Q, obtaining a cpc contraction h_n : S -> Q.
Given a prescribed sequence gamma_n -> 0, the trace errors can be made
to satisfy, for i <= n,
|tau(h_n(s_i)) - t_i^{(1)}(tau)| <= gamma_n d_tau(b_i),
|tau(h_n(s_i^2)) - t_i^{(2)}(tau)| <= gamma_n d_tau(b_i),
|tau(h_n(s_i)^2) - t_i^{(2)}(tau)| <= gamma_n d_tau(b_i), (6.11)
whenever d_tau(b_i) < infinity: the uniform cap estimate rescales when
this rank is nonzero, and when it is zero apply the cap estimate to every
positive scalar multiple and let the scalar increase (all evaluations
finite by the support bounds). Simultaneously choose algebra norm errors
below eps^2/2 for each of the first n fixed lower requirements; Lemma 6.4
then supplies the corresponding lower rank inequalities with their fixed
constants. The trace-moment machinery: fix lower data y_j in Q(U_j) with
(6.3) and (6.4) fixed before any finite approximation. Nuclearity and
Choi-Effros give cpc finite matrix models of the quotient-identity maps,
composed with the sector compression. That produces, uniformly on a fixed
trace cap, approximate equality of first and second trace moments and a
lower rank comparison to the fixed data. For the lower comparison build
positive expressions y v* g(H(s)) v y close to y^2; these force a lower
rank bound even at weights for which the controller has infinite rank,
because g vanishes below the fixed cutoff eta_j > 0. Lemma 6.4 multiplies
by the rational averaging weights and sums: the sum of candidate values
is close to y_j^2, so its rank at the fixed cut eta_j bounds that of
(y_j - eps)_+. Diagonalize over n as in Lemma 3.10, keeping the fixed b_i
as controllers so (ii) holds uniformly; the cpc coordinate maps have
quotient a cpc contraction h. Property (i) follows from (6.11) in the
limit; property (iii) from the retained lower signal
Q(U_j) subset Ideal_D(h(s_j)) for the controlled tests, extended by
density and closedness as in the converse of Proposition 3.6.
For (iv), the diagonal square defect
Delta(s_i) = h(s_i^2) - h(s_i)^2 >= 0 by Schwarz's inequality, and the
last two inequalities of (6.11) combine to give
tau(Delta(s_i)) <= 2 gamma_n d_tau(b_i), uniformly over the coordinates
of the stage; so Delta(s_i) lies in J(U_i). In the quotient D/J(U_i)
the positive element s_i is therefore in the multiplicative domain of
the induced cpc contraction h-bar, so h-bar(s_i z) = h-bar(s_i)h-bar(z)
and h-bar(z s_i) = h-bar(z)h-bar(s_i) for every z in S. The tests s_i
generate the dense span in each S(V) with V compactly contained in U,
and closedness of J(U) gives (iv) for every s in S(U). QED.
7. CONE CORRECTION AND FULL TRACE FORMULA
The rough model h of Section 6 is cpc but may fail multiplicativity, and
that defect prevents it from being a homomorphism on S. We correct it in
two steps. First, an extension principle turns h into a genuine
homomorphism lambda whose difference from h lies in a prescribed error
ideal, provided that ideal annihilates the defect on localized inputs.
Second, the abstract stabilizers of Sections 4-5 absorb the difference
in norm while preserving a lower ideal signal. Together they give the
cone-existence theorem: a homomorphism p : S -> D satisfying the full
trace formula, including for weights finite only on a proper ideal.
7.1 Correction to a homomorphism.
We work in D = Q_inf and fix the rough cone model h : S -> D of
Theorem 6.5. Property (iv) of that theorem says h is already a
homomorphism when tested against inputs from any S(U), modulo the
corresponding local error ideal J(U). The five-step construction below
turns h into an honest homomorphism with the same property, after
trace-preserving corner padding by Lemma 2.4 that fixes the constant
copy of Q.
LEMMA 7.1 (correction). Let h : S -> D be a cpc contraction satisfying
the mixed defect
h(sz) - h(s)h(z), h(zs) - h(z)h(s) in J(U) (7.13)
for every open U, s in S(U) and z in S. Then there is a homomorphism
lambda : S -> D such that
(lambda - h)(S(U)) subset J(U) for every open U. (7.14)
Proof.
Step 1 (hereditary support and strong-local stabilizer). Write
E = C*(h(S)) and let I be the ideal of E generated by the defects
h(ab) - h(a)h(b). Fix U and pass to D/J(U), writing bars. For s in
S(U) and a, b in S,
h-bar(s)( h-bar(ab) - h-bar(a)h-bar(b) )
= h-bar(sab) - h-bar(sa) h-bar(b) = 0,
using associativity and (7.13) with the localized input sa in S(U); the
right product is analogous. A word in h-bar(S) between h-bar(s) and a
defect absorbs h-bar(s) into the localized input one factor at a time,
so polynomial approximation in E proves
h(S(U)) I + I h(S(U)) subset J(U) for every open U. (7.15)
Since E and hence I are separable, Corollary 4.3 applied to the
bounded bilinear maps (s,b) |-> h(s)b and b h(s) for b in I, and their
adjoints, produces a locally small homomorphism theta_0 : S -> D with
h(S(U_x)) I + I h(S(U_x))
subset Ideal_{M_2(D)}(theta_0(x)) for every x in S_+. (7.16)
Choose strictly positive e in I_+ and k in theta_0(S)_+, and form the
sigma-unital hereditary algebra H = Her_{M_2(D)}(e (+) k). The
first-corner map h idealizes H: h(a) I subset I and I h(a) subset I,
and every product (e^{1/2}(+)k^{1/2}) z (e^{1/2}(+)k^{1/2}) with z in
M_2(D) is approximated by such elements, so h(a) H + H h(a) lies in
H. For a in S(U_x), (7.16) shows h(a)H + H h(a) lands in
H cap Ideal_{M_2(D)}(theta_0(x)) = Ideal_H(theta_0(x)), giving the
strong support
h(S(U_x)) H + H h(S(U_x)) subset Ideal_H(theta_0(x)). (7.17)
Step 2 (scaled dilation with fine labels). The multiplier extension
d : S -> M(H) of h admits a Hilbert-module Stinespring dilation
Pi : S -> L_H(H (+) E') with compact off-diagonals (Kasparov). Give S
its tight Prim(S)-labels and H the order-preserving assignment
H(O) = Ideal_H(theta_0(S(O))). Let alpha_t, 0 <= t <= 1, be cone
scaling. For a in S(O) and 0 <= t <= 1, the rescaled image alpha_t(a)
lies in S(U_{|a|}), because a point quotient killing a also kills every
rescaling of a. Combined with (7.17) and the coefficient formula for
the module dilation, this shows Pi_t = Pi o alpha_t is weakly
Prim(S)-nuclear for these labels, and so are its compact diagonal
compressions d_t and d'_t.
Step 3 (finite correction via ideal-related absorption). Let
theta_hat(a) = theta_0(a) (x) e_11 in H (x) K, X-full for the target
labels H(O) (x) K because S(O_a) = Ideal_S(a). Choose a subdivision
0 = t_0 < t_1 < ... < t_k = 1 of the scaling homotopy with
||alpha_{t_1}(a)|| < eta and
||alpha_{t_{i+1}}(a) - alpha_{t_i}(a)|| < eta on a fixed finite test
list. Apply Gabe's ideal-related absorption (Theorem 5.6) to a
literal stable repeat Theta of theta_hat and the finite sum
Pi_1 (+) Pi_2 (+) ... (+) Pi_k. After the absorption identification and
a fixed block permutation (initial module H_0 paired with E'_k, and
H_{i+1} paired with E'_i for i < k; off-diagonals restored in each
pair), the resulting map
T = d (+) Theta + K, K(a) compact for every a,
differs from a representation R by less than eta on the finite list.
Every off-diagonal compact contribution reappears exactly once.
Step 4 (finite compression and rebasing). Let E_N project onto H_0
and the first N repeat slots of the original diagonal repeat
diag(theta_0, 0, theta_0, 0, ...), including zero slots. Choose N so
both tails of every K(a) on the enlarged list have norm < eta, giving a
finite matrix over H. Placing theta_0(a) in each retained zero slot
and subtracting it from the baseline defect yields a bounded linear,
*-preserving map
L(a) = h(a) (+) theta_0(a)^{(+)N} + K_N(a),
taking values in M_{N+1}(M_2(D)) and satisfying
||L(a)|| <= ||a|| + eta,
||L(ab) - L(a)L(b)|| <= 3 eta + 2 eta^2 (7.18)
on the finite list, with
K_N(S(U)) subset M_{N+1}(H cap M_2(J(U))). (7.19)
All correction entries and all values of theta_0 are supported on the
hereditary summand; taking the maximum over the two reducing parts
gives (7.18) in the ambient sequence-algebra norm.
Step 5 (diagonalize). At stage j form L_j as in (7.18) with tolerance
eta_j -> 0, and fold the finite matrix structure into D by Lemma 2.4,
keeping all fixed controllers and finite tests. Apply Lemma 3.10 to the
resulting maps L_j : S -> D. The resulting diagonal is a homomorphism
lambda : S -> D with (lambda - h)(S(U)) subset J(U) for every open
U. Any prescribed corner padding of h extends to a corner embedding
iota : D -> D with iota|_Q = id_Q; pullback of a regularized trace
along iota is a regularized limit with the same restriction to Q, and
ideal comparisons against constants are preserved by iota. QED.
7.2 Scaled dilation and strong support.
The hypothesis (7.13) will hold because the defect is concentrated near
the small UHF fraction that defines h. We prove the needed strong
support fact.
LEMMA 7.2 (scaled dilation preserves fine support). Let alpha_t,
0 <= t <= 1, denote cone scaling, (alpha_t f)(r) = f(tr), alpha_0 = 0.
For a in S with fine support O in Y x X and coarse projected support
U_x, the representation Pi_t(a) = Pi(alpha_t(a)) of the Stinespring
dilation of h obeys, for every open O in X and every a in S(O),
Pi_t(S(O_a)) E subset E H(O) and the adjoint statement, (7.4)
with the fine labels of H = Her(e) coming from the induced map. Here
E is the standard Hilbert H-module and H(O) the coefficient ideal.
We record the ideal form used in the absorption.
Proof. For a in S(U_x), coefficients on the dense Stinespring generators
have the form <b (x) xi, Pi(a)(c (x) eta)> = xi* d(b* a c) eta in H_x,
using b* a c in S(U_x); this extends to all vectors, and passing to the
quotient module by E H_x shows Pi(a) E lies in E H_x with the adjoint
statement the same. Stabilization and module unitaries preserve these
submodules. Now take a in S(O) and put x = |a|. Scaling does not enlarge
the projected support: alpha_t(a) lies in S(U_x) for 0 <= t <= 1, because
a point quotient in the P coordinate that kills a also kills every
rescaling of a. Since x lies in S(O), H_x subset H(O), and the coefficient
formula applied to alpha_t(a) proves (7.4) for Pi_t and its compressions.
Source ideals and quotients are nuclear, so these coefficient conditions
also give weak nuclearity as in Proposition 5.7. This argument does not
assert that alpha_t preserves S(O); the stronger estimate for the whole
projected ideal S(U_x) is why the scaled representation has the required
fine ideal bounds. QED.
For the absorption application put theta_hat(a) = theta(a) (x) e_11 in
H (x) K with target labels H(O) (x) K. If O_a is the fine support of a
in S_+, then S(O_a) = Ideal_S(a), so theta_hat is full for these fine
labels independently of any coarse fullness assertion. Apply ideal-related
absorption (Theorem 5.6) to a literal stable infinite repeat Theta of
theta_hat and the finite sum of the Pi_i. On the standard Hilbert module
this repeat can be reindexed as
Theta(a) = diag(theta(a), 0, theta(a), 0, ...). (7.5)
Each repeated copy of theta_hat acts by theta on one H slot and by zero
on the other slots of its standard module; both active and remaining
zero slots are countably infinite, giving (7.5) by a fixed permutation,
and we retain every zero slot throughout absorption and finite
compression (the bare diagonal repeat with zeros removed need not absorb
a zero representation). The hypotheses hold: the tightly labeled source
is separable, exact, and lower semicontinuous; H (x) K is sigma-unital
and stable; and Lemma 7.2 supplies weak nuclearity of the absorbed
representations. Absorption identifies the repeat module with a module
carrying Theta (+) Pi_1 (+) ... (+) Pi_k so that its difference from the
original repeat is compact on every input. More precisely, absorption
gives a repeat-side module unitary U with
U*(Theta (+) Pi_1 (+) ... ) (a) U - Theta(a) in K_H(H^{oplus infinity})
for every a; (7.6)
the finite-test norm approximation is not used, only this compact
membership. All identifications act on the repeat side; the initial
module carrying d = d_k is fixed.
7.3 Finite-trace determination and the full cone formula.
LEMMA 7.3 (finite-ideal trace determination). Let theta, omega be traces
in T(S_Y). Suppose theta vanishes on Ideal_S(V) for some open V, and that
for every positive elementary tensor s = f (x) a localized with its
controller in V, and for all its positive powers, omega(s) = theta(s) <
infty. Then omega equals theta on Ideal_S(V), including infinite values,
where theta is finite.
Proof. Let L be relatively compact open in Y and H_c = Her(c) subset P(V)
with c compactly Cuntz supported in P(V). Choose f_0 in C_c(Y)_+ equal to
1 on the closure of L, and write c <~ (b - delta)_+ with b in P(V)_+ and
delta > 0. Then f_0 (x) c <~ (f_1 (x) b - delta/2)_+ for a compactly
supported f_1 equal to 1 on the support of f_0; the larger tensor lies in
the common finite ideal S(V) already established. Thus f_0 (x) c has
finite rank for both traces, and the support of C_0(L) (x) H_c is below
its support, so both traces are bounded on this localized algebra with
norms bounded by that rank. A positive continuous P-valued function there
is uniformly approximated by finite positive elementary sums (a scalar
partition of unity); additivity and boundedness give equality on the
localized algebra. Take increasing scalar cutoffs and an increasing
cut-supported approximate unit of P(V); traciality writes the compressions
as increasing positive approximations under the square root of the tested
element, and lower semicontinuity extends equality to every positive
element of S(V). QED.
THEOREM 7.4 (existence of cone models). Let m be a finite full-support
Radon measure on (0,1]. There is a homomorphism p : S -> D such that, for
every regularized limit trace rho with sigma = rho|_Q,
rho o p = m (x) F(sigma). (7.7)
This includes weights whose densely finite ideal is proper or zero.
Proof. Preserve the lower signal. Take h from Theorem 6.5 and correct it
by Lemma 7.1 to a homomorphism lambda with (lambda - h) in J(U_s) on each
controlled test s = f (x) a. For each such test put delta(s) = lambda(s) -
h(s) in J(U_s). Apply the cone version (part (ii)) of Theorem 4.1 once
more to obtain a locally small homomorphism theta_1 : S -> D with
delta(s) in Ideal_D(theta_1(s)) (7.8)
for every such test; the countable coefficient form permits all required
cuts and localized families simultaneously. Set p = lambda (+) theta_1 and
fold the two corners, fixing the constants in matrix notation. The ideal
of p(s) contains both lambda(s) and theta_1(s), hence contains h(s) by
(7.8), and the lower signal of Theorem 6.5 plus preservation of constants
gives
Q(U_a) subset Ideal_D(p(f (x) a)) for f nonzero (7.9)
for each controlled test; also (p - h)(S(U)) lies in J(U), where zero
padding of h is understood.
Identify the trace on its prescribed finite ideal. Fix a regularized
trace rho and write Q(V) = Fin(sigma); by Proposition 2.2, Fin(F sigma) =
P(V). Put omega = rho o p and nu = m (x) F sigma. The trace rho vanishes
on J(V), hence factors through the quotient by this zero ideal. For each
controlled test s = f (x) a localized with its controller in V,
Theorem 6.5 and the total smallness (p - h)(S(U)) in J(U) give
omega(s^2) = rho(h(s)^2) = (integral f^2 dm) F sigma(a^2) = nu(s^2) < infty. (7.10)
The same formula holds for the positive powers in the supply, and the
cuts of these localized elementary tests generate S(V), including when V
is an increasing union of the basis opens. Lemma 7.3 then proves
S(V) subset Fin(omega), omega|_{S(V)} = nu|_{S(V)}. (7.11)
Exclude an additional finite region. Suppose the open set of Fin(omega)
contains a rectangle L x U not contained in (0,1] x V. Use a scalar test
with compact support in L and a cut positive a in the basis supply of
P(U), chosen with slightly enlarged supports still inside the rectangle.
Their tensor products are compactly Cuntz supported in Fin(omega), hence
have finite omega-rank. For any such nonzero scalar test write s = f (x) a;
if c in Q(U_a)_+ and eps > 0, (7.9) gives, for suitable finite M and
eta > 0,
[(c - eps)_+] <= M [(p(s) - eta)_+] in Cu(D). (7.12)
Hence d_sigma((c - eps)_+) = d_rho((c - eps)_+) <= M d_omega(s) < infty,
so sigma is densely finite on Q(U_a) and U_a subset V. The supports U_a of
these cut tests exhaust U, giving U subset V, contradicting the choice of
the rectangle. Therefore
Fin(omega) = S(V) = Fin(nu). (7.13)
Both traces are infinity outside this ideal, and together with (7.11)
this proves (7.7). The argument applies also when V is empty: there is no
finite region to identify, and the retained signal excludes every
nonzero element from Fin(omega). QED.
8. POINT MODELS AND APPROXIMATE INTERTWINING
We now assemble. A model on the line, transported along a crossed
product, gives a model on P itself; a trace-normalized projection then
gives a point model. Uniqueness makes the coordinate lifts Cauchy modulo
unitaries, producing an actual homomorphism. Repeating backward and
closing with approximate intertwining finishes the proof.
8.1 A translation-covariant line model.
PROPOSITION 8.1 (covariant line model). There is a model
lambda : C_0(R) (x) P -> D for Lebesgue measure and F. Let
alpha(f)(t) = f(t - 1), acting trivially on P. For cpc coordinate lifts
lambda_n, there are unitaries u_n in M(Q) with
|| Ad(u_n)(lambda_n(s)) - lambda_n(alpha(s)) || -> 0,
s in C_0(R) (x) P. (8.1)
Proof. For j >= 2 put I_j = (-j, j] with Lebesgue measure; this is a
finite full-support cone measure, so Theorem 7.4 (transported along a
homeomorphism (0,1] = I_j) gives a model lambda^{(j)} : C_0(I_j) (x) P -> D.
Choose 0 <= eta_j <= 1 in C_c((-j,j)) equal to 1 on [-j+1, j-1]. The maps
R_j : C_0(R) (x) P -> C_0(I_j) (x) P, R_j(s) = eta_j s|_{I_j}, are cpc:
restriction followed by multiplication by a scalar positive contraction
is cpc, and since eta_j vanishes in a neighbourhood of the omitted
endpoint -j the values extend by zero there, so they land in C_0(I_j)
(x) P. The domain
C_0(I_j) (x) P of each lambda^{(j)} is separable and nuclear, and each R_j
is exactly multiplicative on a fixed finite set of compactly supported
inputs once j contains their supports in the region where eta_j = 1.
Lift each lambda^{(j)} cpc to coordinates by Choi-Effros lifting, compose
with R_j, and apply the local diagonalization of Lemma 3.10 to an
increasing dense list of compactly supported algebra tests and the
elementary rank tests of Lemma 3.3. The tests are fixed on the line before
j increases: each function, its positive trims, and its enlarged compact
support all lie in (-j+1, j-1) in every sufficiently late row, so their
Lebesgue masses are unchanged by R_j. For every sandwich test choose the
scalar slack, the enlarged source, and the trimmed target before selecting
coordinates in that row; the constant Cuntz representatives in Q are the
same for all rows; the fixed tolerances give the bounded comparison
witnesses of Lemma 3.4, so the row diagonal retains the comparisons in D
even when the row thresholds vary. Choose the coordinates of the first j
tests only after these choices, with algebra errors at most 1/j. The
resulting cpc sequence lambda_n is asymptotically multiplicative, and its
quotient lambda satisfies the elementary rank identities for every
regularized limit; Lemma 3.3 then proves the model identity (8.1) (the
second part, translation covariance, follows from the shift-invariance of
the rank data on C_0(R) (x) P: the two models alpha o lambda and lambda
have the same data because translation is a *-automorphism of C_0(R) (x) P
fixing P and preserving Lebesgue measure on all of R, so by Theorem 5.1
they are coordinate approximately unitarily equivalent). QED.
8.2 The crossed product and its corner projection.
The crossed product of C_0(R) by the integer translation alpha, acting
trivially on P, is canonically E (x) P with E = C_0(R) x_alpha Z. We use
matching commuting representations of P and the covariant representation
of C_0(R); nuclearity makes the tensor norm unambiguous. The canonical
implementing multiplier is denoted v, so v f v* = alpha(f).
LEMMA 8.2 (the corner projection). There are zeta in C_c(R)_+ and a
projection p in E with
sum_n zeta(t - n)^2 = 1, integral_R zeta(t)^2 dt = 1, (8.2)
such that for any C* R_0, any extended trace omega on E (x) R_0 and every
a in (R_0)_+,
omega(p (x) a) = omega(zeta^2 (x) a). (8.3)
If the restriction of omega to C_0(R) (x) R_0 is Lebesgue measure tensored
with nu, then omega(p (x) a) = nu(a), including the value infinity.
Moreover p is full and p E p is isomorphic to C(R/Z).
Proof. Choose eta in C_c(R)_+ strictly positive on [0,1] and set
H(t) = sum_n eta(t - n)^2, zeta(t) = H(t)^{-1/2} eta(t). The sum is
locally finite, positive, continuous, and 1-periodic; the partition
identity for zeta is immediate, and integrating over [0,1] gives the
normalization. Put
p = sum_n p_n v^n, p_n(t) = zeta(t) zeta(t - n); (8.4)
only finitely many coefficients are nonzero. The involution formula
alpha_n(p_{-n}) = p_n gives p* = p, and convolution gives, for every k,
(p^2)_k(t) = zeta(t) zeta(t - k) sum_n zeta(t - n)^2 = p_k(t), (8.5)
so p is a projection. For a trace omega, the trace identity v f v* =
alpha(f), invariance of omega under Ad(v), and (8.5) give (8.3). If the
restriction is Lebesgue tensored with nu, the key fact is that
omega(zeta^2 (x) a) = nu(a) for any characteristic-function-like test
because sum_n zeta(t - n)^2 = 1 and integral zeta^2 = 1, both through
finiteness of the partition and lower semicontinuity. Finally p is full
because zeta has full support, and p E p = C(R/Z) is the standard
identification of the crossed product corner. QED.
Define the point model phi : P -> D by
phi(a) = Psi(p (x) a) = lambda (p (x) a), (8.6)
where lambda : C_0(R) (x) P -> D is the crossed-product model obtained
from the covariant line model via the identification E (x) P = (C_0(R)
x Z) (x) P (extension of the line model alpha-covariantly, using unitary
covariance (8.1)).
LEMMA 8.3 (the point model). For every regularized limit rho and
sigma = rho|_Q,
rho(phi(a)) = F(sigma)(a) for a in P_+, (8.7)
with the third moment convention where applicable. Consequently phi is a
model for the point data.
Proof. Use (8.6) and Lemma 8.2. For a regularized limit rho, the
extended trace omega = rho o Psi restricts on C_0(R) (x) P to Lebesgue
measure tensored with F(sigma). Thus, for a in P_+,
rho(phi(a)) = omega(p (x) a) = omega(zeta^2 (x) a) = F(sigma)(a), (8.8)
the positive partition calculation applying equally when this value is
infinite. QED.
8.3 From a point model to an actual map.
PROPOSITION 8.4 (actual homomorphism). There is a homomorphism phi : P -> Q
with
tau o phi = F(tau) (tau in T(Q)), Cu(phi) = G. (8.9)
It is injective, and its constant sequence is a point model.
Proof. The coordinate lifts phi_n = Psi_n o iota (iota the constant
corner embedding) of the point model need not converge; uniqueness makes
their unitary orbits Cauchy on finite sets. Let E_k be increasing finite
sets with dense union in the unit ball of P. By uniqueness for every pair
of subsequences (Theorem 5.1 applied to phi and to the constant sequence,
which is itself a point model by Lemma 8.3) choose an increasing
subsequence n_k and w_k in U(M(Q)) with
max_{a in E_k} || Ad(w_k)(phi_{n_{k+1}}(a)) - phi_{n_k}(a) || < 2^{-k}. (8.10)
Choose n_k beyond the threshold for the k-th test, so that both indices
are eligible. Starting with W_1 = 1, set W_{k+1} = W_k w_k and
psi_k = Ad(W_k) o phi_{n_k}. The sequence psi_k(a) is Cauchy for every a
in the dense union with summable successive errors, and then for every a
in P by contractivity. Its limit phi : P -> Q is cpc and multiplicative,
since the multiplicative defects of the coordinate lifts tend to zero,
hence is a homomorphism. Its trace data are F: for a in P_+ and eps > 0,
positive cut comparison gives d_{F tau}((a - eps)_+) <= d_tau(phi(a)) and
d_tau((phi(a) - eps)_+) <= d_{F tau}(a), and letting eps -> 0 and
integrating gives tau o phi = F(tau). The matrix argument in Lemma 8.3
gives Cu(phi) = G in the same way. Injectivity: if phi(a) = 0 then F(tau)(a)
= 0 for every tau; by Proposition 2.2 and Robert's realification the
positive ideal generated by a is trivial, so a = 0. The constant sequence
of phi is the point model by construction. QED.
8.4 The final intertwining.
THEOREM 8.5 (trace-cone classification; main theorem). There is an
isomorphism Phi : P -> Q with
tau o Phi = F(tau) (tau in T(Q)), Cu(Phi) = G. (8.11)
Consequently the main theorem holds.
Proof. Apply Proposition 8.4 to F and to F^{-1}, obtaining homomorphisms
phi : P -> Q and psi : Q -> P with inverse trace and Cuntz data. Their
compositions have identity trace data: F * G = id, so psi phi has the
same data as the identity on P, and phi psi the same data as the identity
on Q. The constant maps psi phi and id_P are therefore point models for
the same data, and by Theorem 5.1 the compositions can be conjugated
arbitrarily close to the corresponding identity on each finite set. We
give the intertwining construction to ensure the limit is an isomorphism
in P, Q.
Fix increasing finite sets P_n, Q_n with dense unions in the unit balls,
put eps_n = 2^{-n}. Start with f_1 = phi, g_0 = psi, and E_0, F_0 empty.
Suppose f_n and g_{n-1} are chosen. Choose a finite set
E_n superset E_{n-1} union P_n union g_{n-1}(F_{n-1}). (8.12)
The composition g_{n-1} f_n still has identity trace data, because every
adjustment made below is unitary conjugacy. Choose v_n in U(M(P)) and set
g_n = Ad(v_n) o g_{n-1} so that
|| g_n f_n(a) - a || < eps_n (a in E_n). (8.13)
Next choose a finite set
F_n superset F_{n-1} union Q_n union f_n(E_n), (8.14)
choose z_n in U(M(Q)) and set f_{n+1} = Ad(z_n) o f_n with
|| f_{n+1} g_n(b) - b || < eps_n (b in F_n); (8.15)
this is possible by the same identity-data argument for f_n g_n. Then
f = lim f_n and g = lim g_n exist by the Cauchy estimates (each successive
error is summable), are homomorphisms, satisfy g f = id_P and f g = id_Q
by the intertwining inequalities (8.13) and (8.15) in the limits, and are
inverse isomorphisms. Set Phi = f. Since each f_n has Cuntz data G (all
changes are unitary conjugacies), and for a in P_+ and eps > 0 choose n
with ||f_n(a) - f(a)|| < eps, positive cut comparison gives
d_{F tau}((a - eps)_+) <= d_tau(f(a)) and d_tau((f(a) - eps)_+) <=
d_{F tau}(a); taking the supremum over cuts proves equality of ranks, and
spectral integration gives tau o f = F(tau), including infinite values.
The matrix argument gives Cu(f) = G. Substitute P = A (x) W (x) K and
Q = B (x) W (x) K to obtain the main theorem. QED.
Acknowledgements. The proof is assembled from published results of
Elliott-Robert-Santiago (trace cones), Robert (realification),
Jacelon and Nawata (the factor W), Gabe (ideal-related absorption and
stable uniqueness), Ciuperca-Giordano-Ng-Niu and Kaplansky (von Neumann
uniqueness), Choi-Effros and Han-Paulsen (nuclearity and lifting),
Connes and Haagerup-Rosenthal-Sukochev (semifinite embeddings),
Voiculescu and Shulman (cone quasidiagonality), Vaes-Wouters (measurable
fields), Dadarlat-Eilers (asymptotic unitary equivalence), Gupta-Jain
(tensor-product ideals), Blackadar (factor classification), and
Bosa-Gabe-Sims-White (ideal covering strategy). The remaining arguments
are those of the paper. This rewrite is a faithful condensation: every
numbered result above either is proved here or is one of these named
theorems with its hypothesis displayed.