8a459a8
==== NUCLEAR DIMENSION AND JIANG-SU STABILITY WITHOUT ELEMENTARY SUBQUOTIENTS ====
==== A COMPLETE, READER-FRIENDLY PROOF ====
0. WHAT IS PROVED
Let A be a separable nuclear complex C*-algebra. Write dim_nuc(A) for
nuclear dimension. Write Z for the Jiang-Su algebra. A full element is
one that generates the whole ambient algebra as a closed two-sided ideal.
A character is a nonzero *-homomorphism to C. An elementary algebra is
K(H). A C*-algebra is nowhere scattered if no quotient I/J of ideals
J subset I subset A is a nonzero elementary algebra.
Theorem A (main equivalence). If A is separable nuclear and nowhere
scattered, then these are equivalent:
(i) dim_nuc(A) < infinity;
(ii) A is isomorphic to A otimes Z;
(iii) dim_nuc(A) <= 1.
Theorem B (sharp bound). For every separable nuclear A0,
dim_nuc(A0 otimes Z) <= 1.
The bound is optimal: dim_nuc(Z)=1, because dim_nuc=0 forces AF and Z is
unital, infinite-dimensional, and has no projections except 0 and 1.
Theorem C (full signs in a finite maximal tensor power). If D is a
nonzero unital C*-algebra with no characters, then for some finite n>=1
the maximal tensor power D^{otimes_max n} contains two full orthogonal
positive elements.
Theorem D (Cuntz-semigroup form). Under the hypotheses of Theorem A,
conditions (i)-(iii) are also equivalent to:
(iv) the canonical map Cu(iota_A): Cu(A) -> Cu(A otimes Z) is an
isomorphism in the category Cu, where iota_A(a)=a otimes 1_Z;
(v) there is an abstract isomorphism Cu(A) = Cu(A otimes Z) in Cu.
The isomorphism in (v) need not come from a *-homomorphism.
The proof is organized as follows. Section 1 fixes notation and lists
the standard imported theorems. Section 2 proves Theorem C. Section 3
proves Theorem B. Section 4 assembles Theorem A. Section 5 proves
Theorem D. Every imported theorem is named at the point of use.
1. NOTATION AND STANDARD IMPORTED THEOREMS
All C*-algebras are complex. An unadorned tensor product is spatial.
The maximal tensor product is written otimes_max. For D_+ the positive
cone, put
her_D(h) = closure of h D h for h in D_+.
For a subset S of D, write Ideal_D(S) for the closed two-sided ideal
generated by S; write Ideal_D(x) for Ideal_D({x}). For epsilon>0,
(h-epsilon)_+ is the usual spectral cut. A positive element h is
totally full if f(h) is full for every nonzero f in C_0((0,||h||])_+.
For a subset S of an ambient algebra, S^perp is the set of elements
annihilating S on both sides. A completely positive contractive map is
called c.p.c. A completely positive map sigma has order zero if
sigma(a)sigma(b)=0 whenever a,b>=0 and ab=0.
We use the following standard facts.
(1) Winter-Zacharias order-zero structure theorem. If sigma:D->E is
c.p.c. order zero, then there are a *-homomorphism sigma_hom and a
positive contraction h commuting with the range of sigma_hom such that
sigma(a) = h sigma_hom(a).
For f in C_0((0,1])_+, f(sigma)(a)=f(h)sigma_hom(a) is again order zero.
In particular, for a,b in D,
||sigma(a)sigma(b)|| <= ||ab||,
and if D is unital,
sigma(a)sigma(b) = sigma(1)sigma(ab).
(2) Brown-Carrion-White, with Hirshberg-Kirchberg-White. For every
separable nuclear D there are finite-dimensional algebras F_n and c.p.c.
maps alpha_n:D->F_n, beta_n:F_n->D such that beta_n alpha_n -> id in
point-norm, ||alpha_n(a)alpha_n(b)|| -> 0 for orthogonal a,b in D_+, and
each beta_n is a finite convex combination of c.p.c. order-zero maps.
(3) Rordam comparison. If D is Z-stable, then W(D) is almost
unperforated. More generally, if W(D) is almost unperforated, u,v in D_+,
u is in Ideal_D(v), and d([u])<d([v]) for every dimension function d
with d([v])=1, then u precsim v.
(4) Elliott-Robert-Santiago comparison with extended traces. If D is
exact and Z-stable and u,v in D_+ satisfy
d_tau(u) <= (1/2) d_tau(v)
for every lower semicontinuous extended trace tau on D, then u precsim v.
(5) Choi-Effros lifting. A c.p.c. map from a separable nuclear algebra
to a quotient has a c.p.c. lift.
(6) Kirchberg epsilon-test. Let X_n be nonempty sets and
f_{j,n}:X_n->[0,infinity). Suppose that for every m and delta>0 there is
(x_n) in prod_n X_n with lim_{n->omega} f_{j,n}(x_n)<delta for j<=m.
Then there is one (x_n) for which all these ultralimits are zero.
(7) Gabe approximate domination. Let D be exact and let Phi,Psi:D->E be
nuclear completely positive maps, with Psi order zero. If
Phi(a) is in Ideal_E(Psi(a)) for every a in D_+, then for every finite
F subset D and epsilon>0 there are y_1,...,y_N in E such that
||Phi(a) - sum_l y_l^* Psi(a) y_l|| < epsilon
for all a in F.
(8) BBSTWW relative-commutant unitary equivalence. Let D_n be separable
unital Z-stable algebras, D_omega=prod_omega D_n, A separable unital, and
rho:A->D_omega c.p.c. order zero. Put
C = D_omega cap rho(A)' cap {1-rho(1)}^perp.
Assume C is full in D_omega and that every full hereditary E subset C has
the following property: if x in E admits totally full e_l,e_r in E_+ with
e_l x = x e_r = 0, then there is a full s in E with s x = x s = 0. If
h_0,h_1 in C_+ are totally full contractions and f(h_0) is Cuntz
equivalent to f(h_1) in C for every f in C_0((0,1])_+, then h_1 is
unitarily conjugate to h_0 by a unitary in the unitization of C.
(9) Winter-Zacharias covering bound. If dim_nuc(A)<=d<infinity and no
hereditary subalgebra of A has a finite-dimensional irreducible
representation, then F(A) has a finite matrix-threshold-two cover:
there are finite-dimensional E_j, c.p.c. order-zero maps phi_j:E_j->F(A),
and c_j in F(A) such that every matrix summand of every E_j has size at
least two and
1_{F(A)} = sum_j c_j^* phi_j(1_{E_j}) c_j.
(10) Robert-Tikuisis absorption criterion. If A is separable and F(A)
contains two full orthogonal positive elements, then A is isomorphic to
A otimes Z.
2. PROOF OF THEOREM C: FULL SIGNS IN A FINITE MAXIMAL TENSOR POWER
2.1 A real tensor with uniform sign separation.
Lemma 2.1. Fix k>=2 and 0<r<1. There are n>=1 and G in (R^k)^{otimes n}
such that, for every list of pairs of unit vectors (u_j^0,u_j^1), j=1..n,
with |<u_j^0,u_j^1>|<=r, the numbers
<G, u_1^{alpha_1} otimes ... otimes u_n^{alpha_n}>, alpha in {0,1}^n,
include a value greater than 1/2 and a value less than -1/2.
Proof. Choose a>0 with
a < min{ log 2 - log(4/3) - (1/4)log 3, -(1/4)log r }.
A binary Hamming ball of radius floor(n/4) has size at most
sum_{b<=n/4} binom(n,b) <= 3^{n/4}(4/3)^n.
Gilbert's greedy deletion gives, for all large n, a set S_n subset
{0,1}^n of size N=floor(e^{an}) whose distinct words have Hamming
distance at least n/4.
Let the coordinates of G be independent standard real Gaussians. For a
fixed admissible list put
X_alpha = <G, u_1^{alpha_1} otimes ... otimes u_n^{alpha_n}>,
alpha in S_n.
The X_alpha are jointly centered Gaussian. Their covariance matrix has
diagonal entries 1 and off-diagonal entries of absolute value at most
r^{n/4}. Hence its eigenvalues lie in [1-lambda_n,1+lambda_n] with
lambda_n = N r^{n/4} -> 0. The density of the Gaussian vector is bounded
pointwise by ((1+lambda_n)/(1-lambda_n))^{N/2} times the density of N
independent centered Gaussians of variance 1+lambda_n. Therefore
Pr(X_alpha <= 1 for all alpha in S_n)
<= [ sqrt((1+lambda_n)/(1-lambda_n)) Phi((1+lambda_n)^{-1/2}) ]^N
<= q^N
for some q<1 and all large n, uniformly over admissible lists. By
symmetry the same bound holds for the event X_alpha >= -1 for all alpha.
Set eta_n=(4 n k^{n/2})^{-1}. The compact space of admissible lists has
an eta_n-net, in the maximum Euclidean distance between corresponding
vectors, of size at most (C_k/eta_n)^{2kn}. Its logarithm is O(n^2). A
union bound shows that the probability that some net list lacks a value
above 1 or below -1 is at most
2 (C_k/eta_n)^{2kn} q^N -> 0.
Also E||G||^2=k^n, so Pr(||G||<=2k^{n/2})>=3/4. For large n both the
successful-net event and this norm bound hold for some G.
If an admissible list is approximated by a net list, telescoping the
elementary tensors of unit vectors bounds the change of each evaluation
by ||G|| n eta_n <= 1/2. The two evaluations exceeding 1 and below -1
on the net list retain values above 1/2 and below -1/2. QED.
2.2 Products in commuting factors.
Lemma 2.2. Let M_1,...,M_n be mutually commuting unital von Neumann
factors on a Hilbert space. If e_j in M_j is positive and ||e_j||=1,
then ||e_1...e_n||=1.
Proof. First, a product of nonzero projections p_j in M_j is nonzero.
Induct on n. By induction q=p_2...p_n is a nonzero projection commuting
with M_1. The restriction M_1->B(qH) is a nonzero normal representation.
Its kernel is a weak-* closed ideal in a factor, hence zero. Thus p_1 q
is nonzero. For 0<epsilon<1, the spectral projections
p_j=1_{(1-epsilon,1]}(e_j) are nonzero. On the range of their product
the positive product is at least (1-epsilon)^n. Let epsilon->0. QED.
2.3 Pure-state excision.
Lemma 2.3. Let E be a unital C*-algebra and rho a pure state. For every
finite F subset E and epsilon>0 there is a positive contraction f in E
with ||f||=1 such that
||f x f - rho(x) f^2|| < epsilon for all x in F.
Proof. Let (gamma,H,xi) be the irreducible GNS representation of rho,
with ||xi||=1, and put L={z in E: gamma(z)xi=0}. We first show
ker rho = L + L^*. The inclusion from right to left is immediate. If
rho(x)=0, write eta=gamma(x)xi, which is perpendicular to xi. If eta=0,
then x is in L. Otherwise, on span{xi,eta} let T be the rank-one
operator with T xi=eta and T eta=0; then T^* xi=0. Apply the selfadjoint
form of Kadison transitivity separately to the real and imaginary parts
of T. This gives y in E with gamma(y)xi=eta and gamma(y^*)xi=0. Then
x-y is in L and y is in L^*.
The algebra L cap L^* is hereditary. Let (g_lambda) be a positive
contractive approximate identity for it. If z is in L, then z^*z is in
L cap L^*, so ||z(1-g_lambda)||^2 = ||(1-g_lambda)z^*z(1-g_lambda)|| -> 0.
The adjoint assertion holds for L^*. Hence with f_lambda=1-g_lambda,
||f_lambda (x-rho(x)1) f_lambda|| -> 0 for all x in E.
Each f_lambda is a positive contraction of norm one because
gamma(f_lambda)xi=xi. One sufficiently large lambda works for F. QED.
2.4 Proof of Theorem C.
The closed ideal generated by commutators is all of D; otherwise a
nonzero unital commutative quotient would supply a character. Decompose
into selfadjoint parts and rescale. Choose selfadjoint contractions s_i,t_i
and elements v_i,w_i, i=1..m, such that
||1 - sum_i v_i [s_i,t_i] w_i|| < 1/2.
Put s_0=1, k=m+1, and M=sum_i ||v_i|| ||w_i|| > 0.
Under any nonzero unital representation sigma, some represented
commutator has norm greater than delta=(2M)^{-1}. Since
||[S,T]|| <= 2 ||S-c1|| ||T|| <= diam(sp(S))
when S=S^*, ||T||<=1, and c is the midpoint of sp(S), the spectrum of
some sigma(s_i) has diameter at least delta.
There are pure states rho^0,rho^1 of sigma(D) taking that element to its
two spectral endpoints. Indeed, a state attaining an endpoint extends
from the unital abelian algebra C^*(1,sigma(s_i)), and the states
attaining that endpoint form a nonempty weak-* compact face of the state
space. An extreme point of this face is pure. The real expectation
vectors
v^alpha = (rho^alpha(sigma(s_b)))_{b=0}^m in {1} x [-1,1]^m
differ by at least delta in one coordinate. The set of pairs in
({1} x [-1,1]^m)^2 with ||v^0-v^1||_infinity >= delta is compact.
Therefore there is 0<r<1, independent of sigma, such that
|<v^0/||v^0||, v^1/||v^1||>| <= r.
Two vectors in this compact set cannot be negatively collinear because
both first coordinates are one; positive collinearity forces them to
coincide, contrary to their separation.
Choose G,n from Lemma 2.1 for k,r and form the selfadjoint element
h = sum_{b_1,...,b_n=0}^m G_{b_1...b_n}
s_{b_1} otimes ... otimes s_{b_n}
in D^{otimes_max n}.
Let pi be any nonzero irreducible representation of this tensor product,
with factor restrictions pi_j:D->B(H). For each j choose the two pure
states of pi_j(D) described above. By the angle bound and Lemma 2.1,
choosing one state rho_j from each pair makes
p = sum_{b_1,...,b_n=0}^m G_{b_1...b_n}
prod_{j=1}^n rho_j(pi_j(s_{b_j}))
strictly positive for one choice and strictly negative for another.
Removing normalizations multiplies each tensor evaluation by a positive
number and preserves its sign.
The von Neumann algebras M_j=pi_j(D)'' are mutually commuting factors:
a central element of any M_j commutes with every factor image and hence
with the irreducible joint representation, so is scalar. For one of the
choices in p, use Lemma 2.3 to choose positive norm-one elements
e_j in pi_j(D) with
||e_j pi_j(s_b) e_j - rho_j(pi_j(s_b)) e_j^2|| < epsilon
for 0<=b<=m. The product e=prod_j e_j is positive and has norm one by
Lemma 2.2. Commuting distinct factors and telescoping their products in
h gives
||e pi(h) e - p e^2||
<= n epsilon sum_{b_1,...,b_n} |G_{b_1...b_n}|.
All factors in this estimate are contractions.
If p>0 and pi(h)<=0, then p e^2 - e pi(h) e >= p e^2, so the left side
is at least p. For small epsilon this is a contradiction. The negative
choice of p, with its own excisers, similarly rules out pi(h)>=0. Thus
pi(h_+) != 0, pi((-h)_+) != 0
for every nonzero irreducible pi. Each positive part is full: a proper
closed ideal containing it would yield a nonzero quotient and hence an
irreducible representation killing it. The two positive parts are
orthogonal by functional calculus. QED.
The canonical unital map from the finite tensor power into the infinite
maximal tensor power preserves fullness and orthogonality. Thus Theorem
C also answers affirmatively Kirchberg-Rordam Question 6.2.
3. PROOF OF THEOREM B: dim_nuc(A0 otimes Z) <= 1
Throughout this section A is a nonzero separable unital nuclear
C*-algebra. No absorption hypothesis is imposed. We construct a
two-coloured finite-dimensional approximation of a corner inclusion.
The nonunital case is handled at the end.
3.1 The matrix model with local trace control.
For k>=1 write
L_k = id_A otimes tr_k : A otimes M_k -> A,
L_k((z_{rs})_{r,s=1}^k) = (1/k) sum_{r=1}^k z_{rr},
where tr_k is the normalized matrix trace. We may take all matrix sizes
to be powers of 2.
Choose a countable dense subset D of the positive unit ball of A which is
closed under cuts a -> (a-r)_+ for positive rational r. Enumerate all
pairs (a,gamma) in D x ((0,1) cap Q) as (a_j,gamma_j)_{j>=1}. Put
x_j = (a_j - gamma_j)_+,
b_j = (a_j - gamma_j/2)_+,
I_j = Ideal(b_j).
Then x_j is in her(b_j) and d_tau(x_j)<=d_tau(b_j) for every lower
semicontinuous extended trace tau.
Lemma 3.1 (matrix model). There are finite-dimensional C*-algebras F_n,
powers k_n of 2, and c.p.c. maps
A --alpha_n--> F_n --beta_n--> A otimes M_{k_n},
theta_n = beta_n alpha_n,
such that:
(a) beta_n is order zero and ||alpha_n(c)alpha_n(d)|| -> 0 for
orthogonal c,d in A_+;
(b) ||L_{k_n}(theta_n(a))-a|| -> 0 for every a in A;
(c) for every n,j there is a positive contraction
w_{n,j} in A otimes M_{k_n} and nonnegative eta_{n,j},delta_{n,j}
tending to zero for fixed j such that
||w_{n,j}-theta_n(x_j)|| <= eta_{n,j},
tau(L_{k_n}(w_{n,j})) <= tau(x_j)+delta_{n,j} d_tau(b_j)
for every lower semicontinuous extended trace tau on A with
d_tau(b_j)<infinity.
The coefficients in the trace inequality are independent of tau. One
may take delta_{n,j}=2^{-n}.
Proof. Fix m,N. Use quasicentral approximate units of the ideals I_j to
get commuting positive contractions u_1,...,u_m in A_infinity cap A'
such that u_j is represented in I_j and u_j x_j=x_j. Choose N pairwise
disjoint closed intervals in (0,1) and continuous functions
g_r^-,g_r^+ in [0,1] with g_r^- g_r^+=0, 0<=g_r^-+g_r^+<=1, whose sum
is one outside the r-th interval. Set h_r=g_r^-+g_r^+. For
t=(t_1,...,t_m) in {1,...,N}^m and S subset {1,...,m} define
P_{S,t} = prod_{j in S} g_{t_j}^+(u_j)
prod_{j notin S} g_{t_j}^-(u_j).
These are central positive contractions and, for fixed t, are pairwise
orthogonal as S varies. Averaging over t gives
Q = (1/N^m) sum_{t,S} P_{S,t}
= prod_{j=1}^m (1/N sum_{r=1}^N h_r(u_j)),
and (1-1/N)^m <= Q <= 1, so ||1-Q|| <= m/N. If j is in S, P_{S,t} is in
I_j; if j is not in S, then P_{S,t} x_j=0.
Represent the P_{S,t} by orthogonal positive contractive lifts R_{S,t,i}.
For j in S the distance from R_{S,t,i} to I_j tends to zero. Let
epsilon_i>0 dominate all these finitely many distances and tend to zero.
The cut T_{S,t,i}=(R_{S,t,i}-2 epsilon_i)_+ lies exactly in every I_j with
j in S. Choose sigma_i>0, sigma_i->0, and put
P_{S,t,i}=(T_{S,t,i}-sigma_i)_+.
Then P_{S,t,i} is in I_j for j in S, and because T_{S,t,i} is in
Ideal(b_j), the ideal criterion gives a finite integer K_{S,t,i,j} with
[P_{S,t,i}] <= K_{S,t,i,j} [b_j].
Both cuts preserve orthogonality and change representatives by at most
2 epsilon_i+sigma_i -> 0. Thus all quotient relations survive.
Now fix one coordinate i large enough that the averaged compressions
approximate the identity on a prescribed finite set, that omitted
compressions are small, and that products of orthogonal tests are small.
Inside each corner D_P=her(P), use the Brown-Carrion-White theorem to get
finite-dimensional approximations
D_P --zeta_P--> E_P --phi_P--> D_P,
phi_P = sum_r lambda_{P,r} beta_{P,r},
with beta_{P,r} c.p.c. order zero, lambda_{P,r}>=0, sum lambda=1. Choose
small tolerances xi_P satisfying the finitely many linear inequalities
that keep the averaged trace error below epsilon. Form finite
probability outcomes nu consisting of t and one return component r_S for
each S. The outcome map is
T_nu = beta_nu alpha_t,
alpha_t(a) = direct_sum_S zeta_{P_{S,t,i}} l_{P_{S,t,i}}(a),
beta_nu((z_S)_S) = sum_S beta_{P_{S,t,i},r_S}(z_S),
where l_P(a)=P^{1/2} a P^{1/2}. For fixed t the hereditary corners are
orthogonal, so beta_nu is c.p.c. order zero. The expected map over nu
approximates the identity on the prescribed finite set. Retaining only
corners with S containing j gives positive contractions W_{nu,j} with
||T_nu(x_j)-W_{nu,j}|| < epsilon
for every individual outcome.
For a lower semicontinuous extended trace tau with d_tau(b_j)<infinity,
the corner trace estimate gives
tau(phi_P zeta_P l_P(x_j))
<= tau(l_P(x_j)) + xi_P K_{P,j} d_tau(b_j).
Summing over S containing j and using sum_{S containing j} P_{S,t,i}<=1
gives
sum_nu lambda_nu tau(W_{nu,j})
<= tau(x_j) + (epsilon/2) d_tau(b_j).
Also each individual outcome satisfies
tau(W_{nu,j}) <= M_{nu,j} d_tau(b_j),
M_{nu,j} = sum_{S containing j} K_{P_{S,t,i},j} < infinity.
All coefficients are independent of tau. If d_tau(b_j)=0, every
retained output has trace exactly zero, so no absolute error appears.
Finally round the finitely many probabilities lambda_nu to rationals
mu_nu=n_nu/k with k a power of 2. For large k the rounding error is
small both in operator norm and against the finitely many constants
M_{nu,j}. List the outcomes with multiplicity and define
F = direct_sum_l E_{t_l},
alpha(a) = direct_sum_l alpha_{t_l}(a),
beta((z_l)_l) = diag(beta_{nu_l}(z_l))_l,
w_j = diag(W_{nu_l,j})_l.
Then beta is c.p.c. order zero, alpha is c.p.c., the composite
theta=beta alpha satisfies L_k(theta(a)) approximately a, and the
retained w_j satisfy the required norm and trace inequalities. A
diagonal choice over n gives the sequences in the statement. QED.
Remark 3.2. The composites theta_n also asymptotically preserve
orthogonality, because ||theta_n(c)theta_n(d)||<=||alpha_n(c)alpha_n(d)||.
Thus they induce a c.p.c. order-zero map into the norm ultraproduct.
3.2 The common part in a Jiang-Su tensor factor.
Set
R = A otimes Z_1 otimes Z_2 otimes Z_3,
B = R_omega,
B_1 = (A otimes Z_1)_omega subset B.
Let pi:A->B_1 be the constant first-factor inclusion. Nuclearity and
strong self-absorption of Z imply that R is nuclear, exact, and Z-stable.
Use the maps alpha_n,beta_n from Lemma 3.1. Choose c.p.c. order-zero
maps iota_n:M_{k_n}->Z_1 with tau_Z(iota_n(1))->1. Define
T_n = id_A otimes iota_n,
beta'_n = T_n beta_n,
theta_n = beta'_n alpha_n,
w_{n,j} = T_n(W_{n,j}).
Put L=id_A otimes tau_Z:A otimes Z_1->A. Since tau_Z circ iota_n is a
positive trace of mass s_n->1, we have L T_n = s_n L_{k_n}; hence
L theta_n(a)->a. The maps theta_n induce a c.p.c. order-zero map
theta:A->B_1, theta(a)=(theta_n(a))_omega,
and (w_{n,j})_omega=theta(x_j). The same letter L denotes the induced
unital completely positive map B_1->A_omega. With the constant inclusion
of A into A_omega understood, L theta = L pi = id_A.
Lemma 3.3 (common part). There is a positive contraction e in B_1 such
that, with q=1-e,
L(e)=0,
[e,pi(a)]=[e,theta(a)]=0 for all a in A,
q theta(a)=q pi(a) for all a in A.
In particular, if h=theta(1), then q h=q.
Proof. Both ranges pi(A) and theta(A) lie in the multiplicative domain
of L. For theta, the two Schwarz inequalities give
a^*a = L(theta(a))^*L(theta(a))
<= L(theta(a)^*theta(a))
<= L(theta(a^*a)) = a^*a,
and similarly with a and a^* interchanged. The assertion for pi is the
same or follows from multiplicativity. Let D_0=C^*(pi(A),theta(A)) and
J=ker(L|_{D_0}). Then J is an ideal containing theta(a)-pi(a) for every
a. Choose a positive contractive quasicentral approximate unit (f_l)
for J in D_0. Then L(f_l)=0, the commutators of f_l with both ranges
tend to zero, and (1-f_l)(theta(a)-pi(a))->0 for all a. Lift f_l to
positive contractive sequences in A otimes Z_1 and apply the countable
Kirchberg epsilon-test to the preceding equations on dense countable
subsets, together with the coordinate test ||L(f_{l,n})||. The latter
tends to zero along omega for each fixed l. The resulting e satisfies
the displayed equations. Taking a=1 gives q h=q. QED.
3.3 Localized rank estimates.
Fix j and abbreviate a=a_j, gamma=gamma_j, x=x_j, b=b_j, w_n=w_{n,j}.
Inside A otimes Z_1 subset R put
u_{n,0} = x^{1/2} e_n x^{1/2},
u_{n,1} = w_n^{1/2} e_n w_n^{1/2},
z_n = a^{1/2} (q_n-1/2)_+ a^{1/2},
where e_n are positive contractive representatives of e and q_n=1-e_n.
Let ell_n=||L(e_n)||, so ell_n->0 along omega.
Lemma 3.4 (localized rank estimates). For each j there are nonnegative
numbers epsilon_{n,j}->0 along omega, independent of the trace, with the
following properties. Let lambda be any lower semicontinuous extended
trace on R and tau=lambda|_A. If d_tau(b)<infinity, then for i=0,1 and
eta>0,
d_lambda((u_{n,i}-eta)_+) <= eta^{-1} epsilon_{n,j} d_tau(b).
If ell_n<=1/4, then with no finiteness assumption,
d_lambda((z_n-gamma/16)_+) >= (gamma/16) d_tau(b).
Proof. First assume d_tau(b)<infinity. On her_A(b) otimes Z_1 one has
the localized trace formula
lambda(y)=tau(L(y)) for y>=0.
Indeed, every positive contraction in her_R(b otimes 1) has
lambda-value at most d_lambda(b otimes 1)=d_tau(b). The restriction of
lambda to her_A(b) otimes Z_1 extends to a bounded trace, and the same
holds for tau on her_A(b). For positive s in her_A(b), the functional
v -> lambda(s otimes v) is a bounded positive trace on Z_1 of mass
tau(s). Its trace identity follows from the trace identity for lambda
applied to s^{1/2} otimes z. Uniqueness of the trace on Z gives
lambda(s otimes v)=tau(s)tau_Z(v). Linearity and density prove the
formula. In particular,
lambda(u_{n,0}) = tau(x^{1/2}L(e_n)x^{1/2})
<= ell_n tau(x) <= ell_n d_tau(b).
For w_n, which need not lie in that hereditary subalgebra, define
rho_n=lambda circ T_n and sigma_n(s)=rho_n(s otimes 1). Order-zero
square-root calculus shows that rho_n is an extended trace, and lower
semicontinuity follows from continuity of T_n. Moreover
sigma_n(s)=lambda(s otimes iota_n(1)) <= lambda(s otimes 1)=tau(s).
The matrix-trace lemma applied to rho_n gives
rho_n(Y)=sigma_n(L_{k_n}(Y)) for Y in (A otimes M_{k_n})_+,
including infinite values. The model trace inequality gives
lambda(w_n) <= tau(L_{k_n}(W_{n,j}))
<= tau(x)+delta_{n,j} d_tau(b) < infinity.
Put
S_n=w_n^{1/2}q_n w_n^{1/2},
S'_n=x^{1/2}q_n x^{1/2},
Delta_{n,j}=||S_n-S'_n||.
The common-part identity gives Delta_{n,j}->0 along omega, because in the
quotient the two expressions are q theta(x) and q pi(x). Compressing
S'_n<=S_n+Delta_{n,j}1 by x^{1/(2r)} gives
lambda(x^{1/(2r)}S'_n x^{1/(2r)})
<= lambda(S_n)+Delta_{n,j} tau(x^{1/r})
<= lambda(S_n)+Delta_{n,j} d_tau(b).
The compressed S'_n converges in norm to S'_n as r->infinity, so lower
semicontinuity gives
lambda(S'_n) <= lambda(S_n)+Delta_{n,j} d_tau(b).
All quantities are finite here. Since x=S'_n+u_{n,0} and
w_n=S_n+u_{n,1}, subtracting trace values gives
lambda(u_{n,1}) <= lambda(u_{n,0})
+ (delta_{n,j}+Delta_{n,j}) d_tau(b).
Together with the bound for u_{n,0},
lambda(u_{n,i}) <= (delta_{n,j}+Delta_{n,j}+ell_n)d_tau(b).
Set epsilon_{n,j}=delta_{n,j}+Delta_{n,j}+ell_n. The trace-cut estimate
d_tau((z-eta)_+)<=eta^{-1}tau(z) proves the small-rank inequality, and
when d_tau(b)=0 the right side is exactly zero.
For the large-rank inequality assume ell_n<=1/4 and put
t_n=(z_n-gamma/16)_+. Since (q_n-1/2)_+>=1/2-e_n and
t_n>=z_n-(gamma/16)1, we have
L(t_n) >= (1/2-ell_n)a-(gamma/16)1.
On the support of b the element a is at least gamma/2. Compression by
b^{1/(2r)} gives
b^{1/(2r)}L(t_n)b^{1/(2r)} >= (gamma/16)b^{1/r}.
If d_tau(b)<infinity, the compression b^{1/(2r)}t_n b^{1/(2r)} lies in
her_A(b) otimes Z_1, so the localized trace formula and traciality give
lambda(t_n) >= tau(b^{1/(2r)}L(t_n)b^{1/(2r)})
>= (gamma/16)tau(b^{1/r}).
Let r->infinity and use d_lambda(t_n)>=lambda(t_n). This proves the
large-rank inequality in the finite-rank case, including rank zero.
For the infinite-rank case use ideals. First,
L(t_n) otimes 1 is in Ideal_R(t_n).
To see this apply Gabe domination on Z_1 to v->tau_Z(v)1 and the identity
map. The source is exact, the maps are nuclear, the identity is order
zero, and pointwise ideal containment follows from simplicity of Z. Thus
there are completely positive maps V_l(v)=sum_s z_{l,s}^* v z_{l,s}
converging pointwise in norm to tau_Z(v)1. Including 1 in the
approximation tests lets us require ||V_l||<=2. Tensoring with id_A
preserves pointwise norm convergence. Each approximation to L(t_n)
otimes 1 is a sum of compressions of t_n, proving the containment.
Let I=Ideal_R(t_n). In R/I the lower bound for L(t_n) and the containment
give
0 >= (1/2-ell_n)(a+I)-(gamma/16)(1+I).
Thus ||a+I||<=gamma/4 and (a-gamma/4)_+ is in I. One more cut gives
b = ((a-gamma/4)_+-gamma/4)_+, [b] <= M[t_n]
for some finite integer M. If d_tau(b)=infinity, then d_lambda(b)=infinity,
so this comparison forces d_lambda(t_n)=infinity. QED.
3.4 Localized comparison.
We let the number of orthogonal copies in Z_2 grow slowly with the
coordinate.
Proposition 3.5 (localized comparison). There is a sequence m(n)>=2
with m(n)->infinity along omega such that, putting
P_n=D_{m(n)} in Z_2,
P=(P_n)_omega in B,
Psi(a)=P(q-1/2)_+ pi(a),
the map Psi:A->B is c.p.c. order zero and for every j>=1,
e pi(x_j), e theta(x_j) precsim (Psi(a_j)-c_j)_+
in B, where c_j=gamma_j/64.
Proof. Fix m>=2 and j. The copy D_m is in Z_2, separate from z_n and
t_n=(z_n-gamma/16)_+. The compression identity for D_m gives
[t_n] <= K_m [t_n(D_m-1/2)_+]
<= K_m [(z_nD_m-gamma/32)_+] in W(R).
For the second inequality use joint functional calculus and the scalar
inequality
(s-gamma/16)_+(t-1/2)_+ <= (st-gamma/32)_+, 0<=s,t<=1.
Where the left side is positive, the difference between the uncut
expression on the right and the left side is
s/2+(gamma/16)t-gamma/16>=0.
Enumerate the countable set of pairs (j,eta), j>=1, eta in Q_{>0}. For
each m>=2 let Omega_m be an omega-large set on which ell_n<=1/4 and
eta^{-1} epsilon_{n,j} <= gamma_j/(32 K_m)
for all the first m pairs. Such a set exists because the finitely many
error coefficients tend to zero along omega. Define
E_m = {n:n>=m} cap intersection_{r=2}^m Omega_r.
The E_m are decreasing and omega-large. If n belongs to at least one
E_m, let m(n) be the largest such m; otherwise put m(n)=2. The largest
value exists because membership in E_m requires m<=n. This gives
m(n)->infinity along omega and ensures that for every fixed pair (j,eta)
the slow requirement with m=m(n) holds on an omega-large set.
Set P_n=D_{m(n)}. For fixed (j,eta), on the preceding omega-large set
and for every lower semicontinuous extended trace lambda on R,
d_lambda((u_{n,i}-eta)_+)
<= (1/2) d_lambda((z_nP_n-gamma_j/32)_+), i=0,1.
When d_tau(b_j)<infinity, combine the small-rank estimate, the
large-rank estimate, the separate-leg comparison, and the slow
requirement. This includes d_tau(b_j)=0. If d_tau(b_j)=infinity, then
the large-rank estimate gives d_lambda(t_n)=infinity, and the
separate-leg comparison with finite K_{m(n)} forces the right side to be
infinite. Thus the uniform gap holds in all cases, uniformly in lambda.
The element P commutes with the whole first subsystem B_1, and q
commutes with pi(A). Hence Psi is a homomorphism pi multiplied by
commuting positive contractions, and is c.p.c. order zero. In B,
(u_{n,0})_omega=e pi(x_j),
(u_{n,1})_omega=e theta(x_j),
(z_nP_n)_omega=Psi(a_j).
Since R is exact and Z-stable, the ERS comparison lemma applied to the
uniform gap gives, on an omega-large set,
(u_{n,i}-eta)_+ precsim (z_nP_n-2c_j)_+.
We explain the passage to the ultrapower because the norm bound on
witnesses is essential. Fix j, a positive rational eta, i in {0,1}, and
an accuracy epsilon>0. Let u_i=(u_{n,i})_omega. The spare-cut lemma
chooses witnesses r_n in R on this large set with
||r_n^*(z_nP_n-c_j)_+ r_n-(u_{n,i}-eta)_+|| < epsilon,
||r_n||^2 <= (1+epsilon)/c_j.
Set r_n=0 elsewhere. This bounded sequence defines r in B, and by the
coordinate identifications
||r^*(Psi(a_j)-c_j)_+ r-(u_i-eta)_+|| <= epsilon.
As epsilon is arbitrary, each source cut is Cuntz below
(Psi(a_j)-c_j)_+. Letting eta decrease to zero proves the proposition.
QED.
Lemma 3.6 (ultrapower copies). For every positive integer L there are
contractions v_1,...,v_L in B, represented entirely in Z_2, with
v_l^* v_{l'} = delta_{ll'} P, 1<=l,l'<=L.
In particular these elements commute with B_1, hence with pi(A) and q.
Proof. At coordinates where m(n)>=L use the elements v_{m(n),l} from
the order-zero supplies; at all other coordinates use zero. The
representatives have norm at most one. Since m(n)->infinity along
omega, the relations in the supplies pass to the ultrapower. Their
tensor-factor location gives the stated commutation. QED.
3.5 Supporting map and the relative commutant.
Put h=theta(1_A). An abstract supporting homomorphism for an order-zero
map need not take values in its codomain. We construct one with three
roles: h hat_theta=theta, q hat_theta=q pi, and pointwise ideal
containment of the error maps in the ideals generated by Psi.
Proposition 3.7 (supporting map preserving ideals). There is a c.p.c.
order-zero map hat_theta:A->B whose range commutes with the constant copy
of Z_3 and such that for all a in A and a' in A_+:
h hat_theta(a)=hat_theta(a)h=theta(a),
[e,hat_theta(a)]=0, q hat_theta(a)=q pi(a),
e hat_theta(a'), e pi(a') are in Ideal_B(Psi(a')).
Proof. Consider the order-zero root maps theta^{1/l}. They commute
with e and the constant third leg. Each value theta^{1/l}(a) is a norm
limit of sandwiches of theta(a) by bounded continuous functions of h;
one obtains these sandwiches by truncating t^{(1/l-1)/2} from above,
and the resulting scalar weights converge uniformly to t^{1/l} after
multiplication by t. The same observation gives
e theta^{1/l}(x_j) precsim e theta(x_j).
Taking a=1 in the common-part equations gives qh=q. Continuous
functional calculus gives qh^{1/l}=q, and the sandwich formula together
with the common-part equations yields
q theta^{1/l}(a)=q pi(a) for all a.
Moreover h theta^{1/l}(a)=theta^{1/l}(a)h->theta(a), because
t^{1+1/l}->t uniformly on [0,1].
For every fixed j, combining the root comparison with the localized
comparison gives
e theta^{1/l}(x_j) precsim (Psi(a_j)-c_j)_+, c_j=gamma_j/64>0.
The spare-cut lemma with the spare cut c_j approximates this positive
element arbitrarily closely by r_j^* Psi(a_j) r_j, with
||r_j||^2 <= 2/c_j,
a bound depending on j but not on l.
Apply the countable Kirchberg epsilon-test. Lift each root map to a
sequence of c.p.c. maps into R by Choi-Effros. Keep fixed representatives
of e,q,h,theta,Psi,pi. The tests impose the factorization and
common-part identities on countable dense subsets, commutation with e
and a dense subset of the constant third leg, and order zero on a
countable dense family of orthogonal pairs. For each j include a witness
variable with a fixed bound strictly larger than sqrt(2/c_j), and impose
e hat_theta(x_j)=r_j^* Psi(a_j) r_j.
Every finite list is approximately solvable: take l large for the
factorization tests, then use the witnesses above with small errors.
Their fixed bounds let the ultrapower tests be realized by bounded
coordinate variables. The diagonalization produces a c.p.c. map
satisfying all tests. Continuity extends the identities and
commutations to all elements; the dense orthogonality tests make the map
order zero. This proves the factorization and common-part identities
and the witness identities for all j.
It remains to extend ideal containment from the listed cuts to an
arbitrary positive contraction a' in A. Such a' is approximated
arbitrarily closely by x_j with a_j in Ideal_A(a'). To see this choose d
in the fixed dense positive family close to a', and a small positive
rational r>||d-a'||. The image of d in A/Ideal_A(a') has norm less than
r, so (d-r)_+ is in Ideal_A(a'). This cut belongs to the chosen family;
take it as a_j and choose gamma_j small. Then x_j is as close to a' as
desired. The map Psi is a positive weight times pi, and the weight
commutes with pi(A). Hence a_j in Ideal_A(a') implies Psi(a_j) is in
Ideal_B(Psi(a')). The witness identity places e hat_theta(x_j) in this
ideal, and the localized comparison places e pi(x_j) there. Norm limits
and scaling prove the containment for every a'. QED.
Define the c.p.c. order-zero map
rho=diag(rho_0,rho_1):A->M_2(B),
rho_0=pi, rho_1=hat_theta,
and its supported relative commutant
C = M_2(B) cap rho(A)' cap {1-rho(1_A)}^perp.
The last condition means annihilation on both sides. For b in B write
b^{(i)} for the matrix with b in position (i,i), i=0,1. Put
C_i={b in B:b^{(i)} in C}, Q_i=q^{(i)}, H=1_B^{(0)}.
The common-part identities show Q_i,H are in C. The matrix with q^{1/2}
in position (0,1) is in C, and its two modulus squares are Q_1 and Q_0,
so
[Q_0]=[Q_1] in W(C).
Proposition 3.8 (packing the error part). For i in {0,1} and every
positive contraction c in C_i,
m[(c^{1/2} e c^{1/2})^{(i)}] <= [Q_0] for all m in N,
[c^{(i)}] <= 2[Q_0] in W(C).
Proof. Fix m>=1 and set t=c^{1/2} e c^{1/2}. This positive contraction
commutes with rho_i(A) and satisfies t rho_i(1_A)=t. Thus
t rho_i(a)rho_i(b)=t rho_i(ab) for all a,b in A.
Apply Gabe domination to
Phi(a)=t rho_i(a), Psi(a)=P(q-1/2)_+ pi(a).
Both maps are completely positive and nuclear, since A is nuclear, and
Psi is order zero. For a in A_+,
Phi(a)=c^{1/2} e rho_i(a)c^{1/2} is in Ideal_B(Psi(a))
by Proposition 3.7. Hence finite sums
sum_{l=1}^L y_l^* Psi(.) y_l approximate Phi pointwise in norm.
Write h_q=(q-1/2)_+. For one such approximation take mL second-leg
elements v_{sl} from Lemma 3.6 and set
r_s = sum_{l=1}^L v_{sl} h_q^{1/2} y_l, 1<=s<=m.
These second-leg elements commute with q and pi(A), and satisfy
v_{sl}^* v_{s'l'}=delta_{ss'}delta_{ll'}P. Expanding gives
r_s^* pi(a) r_{s'} = delta_{ss'} sum_{l=1}^L y_l^* Psi(a) y_l.
For f(u)=min{1,2u} on [0,1], f(q)r_s=r_s. Include 1_A in the
approximation set. Since Phi(1_A)=t and ||t||<=1, sufficiently accurate
approximations have ||r_s||^2<=2. Thus for fixed m there is a fixed bound
on every packed variable, independent of L and the approximation
accuracy.
Using bounded coordinate lifts and the epsilon-test on a countable dense
subset of A, we obtain r_1,...,r_m in B with
f(q)r_s=r_s,
r_s^* pi(a)r_{s'}=delta_{ss'} t rho_i(a) for all a in A.
These elements are intertwiners. Indeed, using the supported product
identity and the displayed equation, the right modulus square of
pi(a)r_s-r_s rho_i(a) is
t rho_i(a^*a)-t rho_i(a^*)rho_i(a)=0.
Hence pi(a)r_s=r_s rho_i(a) for all a. Taking a=1 gives
r_s rho_i(1_A)=r_s. The matrix R_s with r_s in position (0,i) thus
belongs to C, including the support condition.
For the row V=(R_1,...,R_m) over C, the positive matrix V^*V has m
diagonal entries equal to t^{(i)} and all other entries zero. Meanwhile
VV^* is in her_C(Q_0), because f(Q_0)VV^*f(Q_0)=VV^* and f(0)=0.
Equality of the Cuntz classes of the two modulus squares proves the
first inequality. Finally,
c=c^{1/2} e c^{1/2}+c^{1/2} q c^{1/2}.
The second summand, placed in corner i, is Cuntz below Q_i inside C.
Apply the first inequality with m=1, subadditivity of Cuntz classes of
positive sums, and [Q_0]=[Q_1]. This proves [c^{(i)}]<=2[Q_0]. QED.
Corollary 3.9 (fullness). The elements Q_0,Q_1,H are full in C, and C
is full in M_2(B).
Proof. Every positive element of M_2(B) is bounded by twice its diagonal
part. For an element of C_+, each diagonal corner also lies in C,
because rho and the support condition are diagonal. After scaling, the
packing inequality shows that these corners belong to the ideal generated
by Q_0. Hence so does the original positive element. Thus Q_0 is full;
Q_1 is full by [Q_0]=[Q_1], and H is full because Q_0<=H. The corner
projection H is full in M_2(B), so M_2(B) contains C as a full
subalgebra. QED.
Proposition 3.10 (almost unperforation). The ordered semigroup W(C) is
almost unperforated.
Proof. Suppose (N+1)x<=Ny in W(C), N>=1. Choose a separable
C*-subalgebra C_* subset C containing the matrix entries of
representatives of x,y and of countably many comparison witnesses. The
inequality already holds over C_*.
There is a unital copy of Z in M_2(B) commuting with both C_* and rho(A).
Choose isomorphisms sigma_l:Z otimes Z->Z with sigma_l(d otimes 1)->d,
and put nu_l(z)=sigma_l(1 otimes z). For fixed d in Z,
||[nu_l(z),d]|| <= 2||z|| ||d-sigma_l(d otimes 1)||.
Thus the commutators tend to zero uniformly for z in the unit ball.
Place these embeddings diagonally on the third tensor leg of M_2(R).
Tensor approximation gives the same uniform commutation property with
any fixed element of M_2(R). At coordinate n choose l large enough for
the entries of representatives of the first n elements of countable
dense commutation data, with tolerance tending to zero. The resulting
coordinate embeddings induce a unital embedding nu:Z->M_2(B) commuting
with C_* and rho(A). It is isometric because each coordinate embedding
is isometric.
Multiplication gives a homomorphism
C_* otimes Z -> C, c otimes z -> c nu(z).
Commutation gives a homomorphism on the maximal tensor product;
nuclearity of Z identifies it with the spatial tensor product. The
range satisfies the support condition because c does and nu(z) commutes
with rho(1_A). This homomorphism restricts to the original inclusion on
C_* otimes 1.
The algebra C_* otimes Z is Z-stable. Its Cuntz semigroup is therefore
almost unperforated by Rordam. Tensoring the original inequality with
1_Z gives comparison of the images of x and y there. Mapping the
comparison back to C yields x<=y. QED.
Proposition 3.11 (hereditary condition). Let D subset C be a full
hereditary subalgebra. Suppose y in D and totally full positive
e_l,e_r in D satisfy e_l y=y e_r=0. Then there is a full s in D with
s y=y s=0.
Proof. Fullness of e_l,e_r in D and of D in C imply both are full in C.
Since H is a projection, there is a common finite integer N>=1 with
[H] <= N[e_l], [H] <= N[e_r] in W(C).
Use the positive contraction D_{N+1} from the order-zero supplies on a
constant Z leg, and put p_0=(D_{N+1})^{(0)}. It belongs to C, has norm
one, and is full in the corner with unit H; because H is full in C, p_0
is full in C.
The N+1 accompanying elements from the supplies, placed in this corner,
form a row over C whose right modulus square is the diagonal sum of N+1
copies of p_0, while its left modulus square belongs to HCH. Hence
(N+1)[p_0] <= [H].
Combining this with the domination of H by e_l,e_r and almost
unperforation gives p_0 precsim e_l,e_r in C. Choose w_l,w_r in C so
that
A_l=w_l^* e_l w_l, A_r=w_r^* e_r w_r
are close to p_0.
Specify the accuracy before choosing witnesses. Fullness of p_0^2 in
the constant simple leg gives fixed z_1,...,z_L in HCH with
sum_{j=1}^L z_j^* p_0^2 z_j=H.
Choose epsilon>0 such that
(2 epsilon+epsilon^2) sum_{j=1}^L ||z_j||^2 < 1,
and choose w_l,w_r with ||A_l-p_0||,||A_r-p_0||<epsilon. Set
s=e_r^{1/2} w_r w_l^* e_l^{1/2}.
Heredity gives s in D. Functional calculus applied to the annihilation
identities gives e_l^{1/2}y=y e_r^{1/2}=0, so s y=y s=0.
Let J=Ideal_C(s). It contains
w_r^* e_r^{1/2} s e_l^{1/2} w_l = A_r A_l,
and ||A_r A_l-p_0^2||<=2 epsilon+epsilon^2. Thus
b=sum_j z_j^* A_r A_l z_j is in J and ||H-b||<1. The image of H in C/J
is a projection of norm less than one, hence zero. Since H is full in C,
J=C.
It remains to prove s is full inside D. Let I=Ideal_D(s). For any ideal
I of a hereditary subalgebra D subset C one has
D Ideal_C(I) D subset I.
Indeed, the middle ideal is the closed linear span of products through
elements of I. By linear decomposition it is enough to use a positive
middle element i in I_+, and then
d c i c' d' = (d c i^{1/3}) i^{1/3} (i^{1/3} c' d') is in I,
because the outer factors lie in D by heredity. Here Ideal_C(I) contains
J=C, so the displayed inclusion and an approximate identity of D give
D subset I. Hence I=D, proving s is full in D. QED.
3.6 Full-spectrum weights and unitary equivalence.
Fix a positive contraction k of spectrum [0,1] in the retained constant
third leg. Such a k is obtained from the scalar interval coordinate in a
prime dimension-drop algebra embedded in Z_3. When necessary, a
third-leg element also denotes its diagonal copy in M_2(B). Set
h_0=k^{(0)}, h_1=(k h)^{(1)}.
These are positive contractions in C. In the second corner the support
condition follows from h hat_theta(1_A)=h; commutation follows from
Proposition 3.7.
Proposition 3.12 (functional-calculus comparison). The elements h_0,h_1
are totally full in C. For every g in C_0((0,1])_+, the elements g(h_0)
and g(h_1) are Cuntz equivalent in C.
Proof. The case g=0 is immediate. Otherwise rescale so ||g||<=1 and put
G_i=g(h_i), T_i=Q_i G_i, E_i=G_i-T_i=e^{(i)}G_i.
All are positive and in C. The factors commute because e commutes with h
and is represented in the first subsystem. Since qh=q, functional
calculus gives T_i=Q_i g(k). The matrix with q^{1/2}g(k)^{1/2} in
position (0,1) belongs to C and has modulus squares T_1,T_0. Hence
[T_0]=[T_1] in W(C).
The positive element g(k) is nonzero in the simple unital constant third
leg. Choose finitely many z_1,...,z_K in that leg with
sum_{l=1}^K z_l^* g(k) z_l=1.
For delta>0 let
b_delta(u)=((u-delta)_+/u)^{1/2} for u>0, and 0 for u=0.
The diagonalized z_l commute with Q_i and rho(A) and multiply C into
itself. In particular z_l b_delta(Q_i) is in C, and
(Q_i-delta)_+ = sum_{l=1}^K (z_l b_delta(Q_i))^* T_i (z_l b_delta(Q_i)).
Taking all cuts proves
[Q_i] <= K[T_i] <= K[G_i] in W(C).
Since Q_i is full, g(h_i) is full for every nonzero g. In particular
h_0,h_1 are totally full.
Let d be any dimension function on W(C) normalized at either [G_0] or
[G_1], if one exists. The packing inequality, the fullness bound, and
[Q_0]=[Q_1] imply
0<d([Q_0])=d([Q_1])<infinity.
Apply the small-part inequality with c^{(i)}=G_i. Since G_i commutes
with e^{(i)}, it follows that m[E_i]<=[Q_0] for every m, and hence
d([E_i])=0. The inequalities [T_i]<=[G_i]<=[T_i]+[E_i] and
[T_0]=[T_1] now give
d([G_0])=d([G_1])=1.
Fix eta>0. By continuity, g(0)=0, and g nonzero, there is a nonzero
continuous g' with 0<=g'<=g and support contained in the open set where
0<g<eta. For the dimension functions under consideration, the fullness
bound applied to g' gives d([g'(h_i)])>0. This element is orthogonal to
(G_i-eta)_+, and their sum is bounded by G_i. Therefore
d([(G_i-eta)_+])+d([g'(h_i)])<=1,
d([(G_i-eta)_+])<1=d([G_{1-i}]).
This holds for every dimension function normalized at [G_{1-i}]. The
target G_{1-i} is full, so the ideal condition for Rordam comparison
holds. Almost unperforation and Rordam comparison yield
(G_i-eta)_+ precsim G_{1-i}.
Taking all positive cuts and exchanging i=0,1 proves the required Cuntz
equivalence. QED.
Proposition 3.13 (weighted unitary equivalence). Put k^{[0]}=k and
k^{[1]}=1-k in the constant third leg. There are unitaries U_0,U_1 in
M_2(B) such that
U_i (k^{[i]} theta(a))^{(1)} U_i^* = (k^{[i]} pi(a))^{(0)}
for all a in A and i=0,1. They can be chosen to commute with rho(A).
Proof. Apply the BBSTWW relative-commutant unitary criterion to C. Its
coordinate algebras are M_2(R), which are separable, unital and Z-stable,
and its domain is the separable unital algebra A. The map rho is c.p.c.
order zero. Corollary 3.9 verifies fullness of C in the ambient
ultrapower; Proposition 3.11 verifies the required condition in every
full hereditary subalgebra. Proposition 3.12 supplies total fullness and
all functional-calculus comparisons. Thus there is a unitary in the
unitization of C conjugating h_1 to h_0. Realizing this unitary in the
ambient algebra gives U_0 in M_2(B) commuting with rho(A). Multiplying
U_0 h_1 U_0^*=h_0 by rho(a) and using the factorization gives
U_0 (k theta(a))^{(1)} U_0^* = (k pi(a))^{(0)}.
The element 1-k also has spectrum [0,1] in the constant third leg.
Repeating the same argument with this element, with all previously
constructed maps unchanged, gives U_1 and the other identity. QED.
3.7 Two colours and stabilization.
Proposition 3.14 (two-colour corner approximation). Let A be a nonzero
separable unital nuclear C*-algebra and put R=A otimes Z^{otimes 3}. For
every finite F subset A and epsilon>0, the map
kappa_A:A->M_2(R), kappa_A(a)=diag(a otimes 1,0)
admits a finite-dimensional approximation with two colours: there are a
finite-dimensional algebra F, a c.p.c. map alpha:A->F, and c.p.c.
order-zero maps phi_0,phi_1:F->M_2(R) such that
||phi_0 alpha(a)+phi_1 alpha(a)-kappa_A(a)|| < epsilon
for all a in F.
Proof. Use the two unitaries of Proposition 3.13. Write k^{[0]}=k and
k^{[1]}=1-k. Each unitary U_i in M_2(R_omega) has unitary
representatives U_{i,n} in M_2(R): a bounded representative is
asymptotically unitary on both sides, and on an omega-large set its
coordinates are invertible; polar correction gives unitaries with the
same quotient class, and the identity is used on the remaining
coordinates.
For the matrix-model maps alpha_n:A->F_n and
beta'_n:F_n->A otimes Z_1 subset R, define
phi_{i,n}(b)
= U_{i,n} diag(0, k^{[i]} beta'_n(b)) U_{i,n}^*.
Each map is c.p.c. order zero: its weight is a positive contraction in
the third tensor factor, commuting with the range of beta'_n; corner
inclusion and unitary conjugation preserve these properties. The
weighted conjugacies give, for every a in A,
lim_{n->omega} ||phi_{0,n}alpha_n(a)+phi_{1,n}alpha_n(a)-kappa_A(a)||=0,
because k^{[0]}+k^{[1]}=1. Choose one coordinate in the intersection of
the finitely many omega-large approximation sets for a in F. The first
map into F_n direct_sum F_n is a->(alpha_n(a),alpha_n(a)); it is
contractive because the direct-sum norm is the maximum norm. QED.
Lemma 3.15 (corner to stabilization). Let A be unital. Suppose the
first-corner inclusion A->M_2(A) has finite-dimensional approximations
with two colours, with a c.p.c. first map and c.p.c. order-zero return
maps. Then dim_nuc(A otimes K)<=1.
Proof. Fix m>=1. For a finite set of elements of A otimes M_m, choose a
two-colour approximation of A->M_2(A) accurate on all their matrix
entries. Amplify every map by id_{M_m}. The intermediate algebra is
still the direct sum of two finite-dimensional algebras,
E_0 otimes M_m and E_1 otimes M_m. The first map remains c.p.c. Each
amplified return map remains c.p.c. order zero: if
phi=h phi_hom, then
(phi otimes id_{M_m})(x)
= (h otimes 1)(phi_hom otimes id_{M_m})(x),
with a commuting positive contractive weight.
Embed M_2(A) otimes M_m into A otimes K so that its first diagonal m x m
block is the prescribed first m-corner. The amplified approximation then
approaches the inclusion of A otimes M_m into A otimes K. Entrywise
accuracy suffices: an m x m matrix with entries of norm at most delta has
norm at most m delta.
Let p_m=1_A otimes sum_{r=1}^m e_{rr}. Precompose the amplified first
map with the c.p.c. compression x->p_m x p_m on A otimes K. For an
arbitrary prescribed finite set in the stabilization, first choose m so
that these compressions approximate every element, then approximate the
compressed finite set as above. The composite has the same value on x
and p_m x p_m, so the total error is bounded by the compression error
plus the finite-corner approximation error. This gives the required
two-colour approximation on the stabilization. QED.
3.8 Proof of Theorem B.
Let A0 be separable nuclear. First suppose A0 is nonzero and unital, and
set A=A0 otimes Z. Strong self-absorption gives isomorphisms
sigma_l:Z otimes Z->Z, sigma_l(z otimes 1)->z.
Identify the three extra copies of Z with one copy by a fixed unital
isomorphism. The maps id_{A0} otimes sigma_l give isomorphisms
Gamma_l:A otimes Z^{otimes 3}->A,
Gamma_l(a otimes 1)->a.
The convergence holds first on elementary tensors and then on all of A
by contractivity.
Given a finite subset of A, choose l so that Gamma_l has the required
accuracy on its first-factor images. Apply Proposition 3.14 with a
sufficiently small error, and postcompose its return maps with
id_{M_2} otimes Gamma_l. This gives two-colour approximations to
A->M_2(A). Lemma 3.15 shows dim_nuc(A otimes K)<=1.
Nuclear dimension decreases on hereditary subalgebras. Apply this to the
first corner A otimes e_{11}. This gives dim_nuc(A0 otimes Z)<=1. For
nonunital separable nuclear A0, its unitization tilde A0 is separable and
nuclear. The unital case applies to tilde A0 otimes Z, and A0 otimes Z
is a closed two-sided ideal, hence a hereditary subalgebra, of that
algebra. The same permanence gives the bound. The zero algebra is
immediate. QED.
4. PROOF OF THEOREM A
Assume A is separable nuclear and nowhere scattered. Theorem B gives
(ii)=>(iii): if A=A otimes Z, then dim_nuc(A)=dim_nuc(A otimes Z)<=1.
The implication (iii)=>(i) is trivial. We prove (i)=>(ii).
Assume dim_nuc(A)<infinity. If A=0 there is nothing to prove, so assume
A nonzero.
Lemma 4.1. F(A) has no characters.
Proof. A hereditary subalgebra H of A is strongly Morita equivalent to
the ideal I it generates. If H had a matrix quotient, ideal
correspondence would give an ideal J subset I for which I/J is strongly
Morita equivalent to a matrix algebra, hence to C. By imprimitivity I/J
would be elementary. Ideals of I are ideals of A, contradicting nowhere
scatteredness.
Apply the Winter-Zacharias cover from Section 1(9) to A, put B=F(A), and
retain its maps phi_j:E_j->B, elements c_j, and the identity
1_B = sum_j c_j^* phi_j(1_{E_j}) c_j.
If chi:B->C were a character, its compositions with these maps would be
order-zero positive functionals. Such a functional on M_r, r>=2, is
zero. To see this, a nonzero positive functional has a positive density
matrix with an eigenvector xi of positive eigenvalue. For a unit vector
eta perpendicular to xi, the orthogonal rank-one projections onto
(xi+eta)/sqrt2 and (xi-eta)/sqrt2 both have strictly positive values,
contrary to order zero. Thus chi circ phi_j=0 for every j, contradicting
the displayed identity and chi(1_B)=1. QED.
Now B=F(A) has no characters. Choose in B a finite commutator witness
as in the proof of Theorem C, and let D be the unital C*-subalgebra
generated by all its entries. Then D is separable and has no characters,
because applying a character of D to the witness would give 1<1/2. By
Theorem C, some D^{otimes_max n} contains full orthogonal positive
elements a,b.
Lemma 4.2 (commuting reindexed copies). For every finite n there is a
unital *-homomorphism D^{otimes_max n}->F(A).
Proof. It suffices to show that for every separable subalgebra C subset
F(A) there is a unital *-homomorphism from D into F(A) whose range
commutes with C. This is Kirchberg reindexing. We include the
annihilator details for nonunital A.
Choose a countable dense unital *-algebra D_0 subset D over Q+iQ. For
each x in D_0 choose a bounded lift (a_l^x)_l in A representing a central
element mapping to x, with
sup_l ||a_l^x|| <= 2||x||.
Take the zero lift for zero and the approximate identity for 1. The
quotient norm permits these choices: first lift within a bound strictly
below 2||x||, then truncate coordinate norms without changing the
ultrapower class. The additive, scalar, multiplicative and adjoint
relation defects between these lifts annihilate A on both sides along
omega.
Choose a countable dense family in C with bounded lifts (b_s^u)_s in A,
and fix a countable dense subset of A. List the countably many
requirements on an old index l: centrality of each a_l^x against that
subset; left and right products of every relation defect with that subset
tending to zero; and a_l^1 a->a, a a_l^1->a for its elements a. All
these requirements hold along omega.
For each output index s choose l(s) so that the first s requirements hold
with error less than 1/s, and also
||[a_{l(s)}^x,b_s^u]|| < 1/s
for the first s pairs (x,u) in an enumeration. For fixed s the finitely
many b_s^u are fixed elements of A. Original centrality makes all these
commutator conditions omega-large as conditions on l. A finite
intersection with the other omega-large conditions is nonempty, so the
choice is possible.
The new sequences (a_{l(s)}^x)_s are central, and all their relation
defects annihilate A on both sides: the tests hold in the ordinary limit,
and the uniform bounds extend them from the dense subset to all of A.
Thus they define a unital rational-complex *-homomorphism on D_0 into
F(A). The unit assertion follows from the approximate-identity tests.
The bound on lifts permits continuous extension to a complex unital
*-homomorphism on D. Its range commutes with the dense family in C, hence
with C. Injectivity is not needed.
Starting with one copy and repeatedly taking C to be generated by the
previous images gives n commuting unital images of D. The universal
property of the maximal tensor product gives the stated homomorphism.
QED.
By Lemma 4.2 there is a unital homomorphism D^{otimes_max n}->F(A). The
images of the full orthogonal positive elements a,b remain orthogonal.
They remain full: applying a unital homomorphism to finite ideal
combinations approximating the unit proves that fullness passes to the
target. Thus F(A) contains two full orthogonal positive elements.
Robert-Tikuisis absorption criterion now gives A=A otimes Z. This proves
(i)=>(ii), and Theorem A follows. QED.
5. PROOF OF THEOREM D
Use the completed stabilized Cuntz semigroup Cu(X) formed from
(X otimes K)_+, with no distinguished scale. Isomorphisms in Cu preserve
zero, addition, order, increasing sequential suprema, and the way-below
relation in both directions. The isomorphism in (v) need not agree with
Cu(iota_A).
Theorem A gives the equivalence of (i),(ii),(iii). The companion
Cuntz-comparison paper gives that (iv) implies absorption and that (v) is
equivalent to absorption, even without the exclusion on elementary ideal
subquotients.
It remains to prove that absorption implies (iv). The case A=0 is
immediate. Put D=A otimes Z and j_X(x)=x otimes 1_Z, so j_A=iota_A.
Strong self-absorption gives an isomorphism phi:Z->Z otimes Z
approximately unitarily equivalent to z->z otimes 1_Z. The companion's
nonunital calculation gives Cu(j_D)=Cu(id_A otimes phi), an isomorphism:
the implementing unitaries u_n in Z otimes Z act through
1_{M(A)} otimes u_n in M(A otimes Z otimes Z), and positive cuts and
approximate units after stabilization give equality of the completed
Cuntz maps.
For an absorption isomorphism theta:A->D, both sides of the first
identity below send a to theta(a) otimes 1_Z. Functoriality gives
(theta otimes id_Z) circ j_A = j_D circ theta,
Cu(j_A) = Cu(theta otimes id_Z)^{-1} circ Cu(j_D) circ Cu(theta).
All three factors on the right are isomorphisms, proving (iv). Thus (v)
reaches the canonical condition (iv) through absorption, not by
identifying its abstract isomorphism with Cu(iota_A). QED.
6. THE IMPORTED THEOREMS IN WORDS
For a reader meeting the subject, the dependencies are:
Order-zero maps are the noncommutative analogue of disjoint-support maps.
Winter-Zacharias says every such map is a *-homomorphism times a positive
weight. This is why a colour in nuclear dimension can be written as a
weighted embedding.
Nuclear dimension <=d means: for every finite set and epsilon, one can go
through a finite-dimensional algebra in d+1 colours and come back with
order-zero maps whose sum is approximately the identity.
The Cuntz semigroup records stable equivalence classes of positive
elements. Rordam's theorem says Z-stability makes this semigroup
unperforated, so approximate inequalities can be corrected.
Extended traces allow infinite values and are the right tool when an
algebra has ideals on which a trace is finite and outside which it is
infinite. The rank d_tau(h) is the trace of the support of h, not the
trace of h itself. This is why a norm error inside a hereditary corner
becomes a trace error relative to d_tau(b).
The matrix model produces finite-dimensional approximations whose error
is small relative to d_tau(b_j). The second Jiang-Su factor supplies
arbitrarily many orthogonal copies. The third factor supplies the two
complementary weights k and 1-k. The supported relative commutant is the
place where the error part can be packed into the common part.
The absorption branch is independent. A character is a one-dimensional
representation. If D has no characters, commutators generate D. A
commutator witness forces two pure states to be separated. The Gaussian
tensor turns this separation into one selfadjoint element whose positive
and negative parts survive every irreducible representation of a finite
maximal tensor power. Excision is the tool that transfers evaluations
from pure states to actual positive elements in the representation. The
central sequence algebra F(A) then receives that finite tensor power by
reindexing, and Robert-Tikuisis turns the two full orthogonal signs into
Z-stability.
7. WHAT IS NEW AND WHAT IS USED
The proof of Theorem B uses the Brown-Carrion-White/Hirshberg-Kirchberg-
White approximation theorem, the Winter-Zacharias order-zero structure
theorem, Rordam comparison, Elliott-Robert-Santiago comparison with
extended traces, Gabe domination, and the BBSTWW unitary criterion. The
new point is the bookkeeping: localize the matrix model in hereditary
corners, compare relative to the rank of the ideal generator, pack the
error with copies in a second Jiang-Su factor, and use two complementary
full-spectrum weights to get exactly two colours.
The proof of Theorem C uses only finite-dimensional real Gaussian
analysis, Gilbert's Hamming packing, Kadison transitivity for the
excision lemma, and the elementary fact that a product of nonzero
positive elements in commuting factors has norm one. It is independent
of separability and nuclearity.
The proof of Theorem A uses Theorem B for the reverse implication and
Theorem C plus Robert-Tikuisis for the forward implication. The
nowhere-scattered hypothesis enters only through the Winter-Zacharias
cover, which rules out characters in F(A).
8. END OF PROOF
The three implications (i)=>(ii), (ii)=>(iii), (iii)=>(i) prove Theorem
A. Theorem B is proved in Section 3. Theorem C is proved in Section 2.
Theorem D is proved in Section 5. All imported theorems are stated in
Section 1 and used only where named.