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THE KIRCHBERG-RORDAM CHARACTER CRITERION
A self-contained guide and complete proof
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This document has two halves. PART 0 is a long, friendly guide
that explains, in high-school language, every object and every
question that appears in the proof. Read it first. PARTS 1-6
give the full proof, every lemma, every proposition, every
statement, in the original order. The guide is meant to let a
careful sixteen-year-old read the proof without getting lost.
Throughout, every object is introduced at first use and the
notation table at the end of PART 0 lists each symbol.
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PART 0. A GUIDE FOR A CURIOUS READER
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0.1 Numbers, matrices, functions
You already know three flavours of useful math:
(a) the complex numbers C, where you can add, multiply, and
conjugate;
(b) the n-by-n complex matrices M_n(C), where you can add,
multiply, transpose, and take adjoints;
(c) the continuous functions on a compact space X, say the
closed interval [0,1] or a circle, where you can add,
multiply, and complex-conjugate pointwise.
The three flavours behave the same in a way. Each is a set A
with operations
+ addition,
. multiplication,
complex scalar multiplication,
H adjoint: conjugate a number, conjugate transpose a
matrix, complex-conjugate a function,
|| . || a norm that measures size,
1 a distinguished element: the number 1, the identity
matrix, the constant function 1.
0.2 What a Cstar-algebra is
A Cstar-algebra is a set A with all those operations, obeying
the rules you already expect from the three examples.
(i) A is a complex vector space with zero 0 and unit 1.
(ii) Multiplication is associative, complex-bilinear, and
1 times a equals a.
(iii) Adjoint is conjugate-linear, does nothing twice
((a^H)^H = a), and reverses products:
(ab)^H = b^H a^H.
(iv) ||ab|| <= ||a|| ||b|| and ||1|| = 1.
(v) ||a^H a|| = ||a||^2. This "Cstar identity" is what
makes a Cstar-algebra hard to mess with.
The Cstar identity is special. Together with an extra
completeness axiom (every Cauchy sequence converges), it
implies a spectacular theorem of Gelfand and Naimark: every
Cstar-algebra is, up to isomorphism, a closed Cstar-subalgebra
of the bounded operators on some Hilbert space. We do not use
Gelfand-Naimark in this paper, but it is the reason every
example below is "really" an algebra of operators.
If it already feels like a lot, that is normal. This is a
research paper. Read PART 0 slowly. The rest of this document
is here for the moment you have read it once.
0.3 Characters
A character of A is a map chi: A -> C that
preserves + and . (linear over C and multiplicative),
preserves the adjoint: chi(a^H) = conj(chi(a)),
is nonzero.
Because 1 is a multiplicative identity, any nonzero
multiplicative map sends 1 to 1: chi(1) = 1 automatically. So
characters are always unital.
Examples:
C has exactly one character, chi(z) = z. Any unital
multiplicative map C -> C must be the identity.
M_n(C) has NO characters once n >= 2. Proof, in a single
paragraph. Let e_ij be the matrix with 1 in row i, column j
and 0 elsewhere. These are "matrix units": e_ij e_kl =
delta_jk e_il. Take a character chi. Orthogonal projections
e_ii sum to 1 and are self-adjoint. A character sends
self-adjoint elements to real numbers, and sends projections
to numbers c with c^2 = c, so each chi(e_ii) is 0 or 1, and
their sum, chi(1), equals 1. So exactly one chi(e_ii) = 1,
say for i = j, and the others are 0. But then for k not equal
to j, note e_jj = e_jk e_kj so 1 = chi(e_jj) = chi(e_jk)
chi(e_kj). Also e_kk = e_kj e_jk so 0 = chi(e_kk) =
chi(e_kj) chi(e_jk). Both expressions are the same complex
number chi(e_jk) chi(e_kj). Contradiction: 1 = 0. So
M_n(C) has no character once n >= 2.
C(X) for X compact Hausdorff: every character is evaluation
at a point x0 of X, chi(f) = f(x0). This is Gelfand duality.
The last example is a translation. Points of X correspond to
characters of C(X). For a noncommutative Cstar-algebra A, we
can think of characters as the "1-dimensional points" of A.
Having no characters means: no 1-dimensional points.
0.4 Representations and irreducible representations
A representation of A on a Hilbert space H is a map pi from A
to the bounded operators B(H) that preserves +, ., H, and the
norm (a Cstar-homomorphism). In symbols, pi is a unital
Cstar-homomorphism A -> B(H).
A representation is irreducible if the only closed subspaces of
H invariant under all pi(a) are 0 and H itself.
A 1-dimensional irreducible representation is exactly the same
thing as a character: B(C) = C.
So "A has no characters" = "A has no 1-dimensional irreducible
representations". It is possible for A to be nonzero and to
have no characters; M_n(C) is the basic example.
0.5 Tensor products of Cstar-algebras
If A and B are Cstar-algebras, the algebraic tensor product
A (circle-dot) B is the complex vector space generated by
expressions a (circle-dot) b, with
multiplication (a (circle-dot) b) (a' (circle-dot) b') =
(a a') (circle-dot) (b b')
and adjoint (a (circle-dot) b)^H = a^H (circle-dot) b^H.
Constants sit on the diagonal.
That algebraic tensor product is a *-algebra but not yet a
Cstar-algebra; there are many ways to complete it to one. Two
canonical choices matter here.
Maximal tensor product A @max B. The maximal norm of an
element x of A (circle-dot) B is
||x||_max = sup over pairs (pi, sigma) of representations of
A and B of the operator norm of (pi @ sigma)(x).
Here pi @ sigma is the natural map on the algebraic tensor
product: (a (circle-dot) b) -> pi(a) (x) sigma(b) on the tensor
Hilbert space of the two Hilbert spaces. Closing the algebraic
tensor product in || . ||_max gives a Cstar-algebra A @max B.
It is the largest possible completion.
Minimal (spatial) tensor product A @min B. Fix any faithful
representation pi: A -> B(H) and any faithful representation
sigma: B -> B(H'), and take the norm of (pi @ sigma)(x) on
H (x) H'. This gives the minimal norm, and the completion
A @min B. It does not depend on the choices of pi and sigma
(this is a theorem of Fell); A @min B is the smallest possible
completion.
By construction ||x||_min <= ||x||_max. When equality holds
for every x and every B, A is called nuclear. This paper does
NOT assume nuclearity.
Intuition. The maximal tensor product lets the two factors
"see" each other as much as possible: any sequence of
commuting-obstruction data in any representation of A and B
contributes to the norm. The minimal lets each factor see only
its own Hilbert space.
Notation. In this document we write "@" for the tensor product
of Cstar-algebras. Inside the technical body "@" defaults to
"@max" unless "@min" is written out, exactly as in the
original paper. If the two factors agree (for the examples we
care about, a Cstar-algebra is nuclear, or a finite power of a
nuclear one), then @max = @min and no ambiguity arises.
0.6 Cuntz comparison, in one paragraph
Let B be a unital Cstar-algebra. For self-adjoint positive
elements a, b in B_+ (a >= 0 means a has nonnegative spectrum)
we write
a precsim b,
read "a is Cuntz dominated by b" or "a is Cuntz below b", if for
every epsilon > 0 there is a contraction x in B with
|| a - x^H b x || <= epsilon.
This is a genuine comparison theory with a semigroup attached:
[a] is the class of a in the Cuntz semigroup Cu(B). The rules
we actually use are three, and they are easy to check:
(1) [v^H v] = [v v^H] for any rectangular matrix v.
(2) [b_1 + b_2] = [b_1] + [b_2] if b_1,b_2 are positive and
orthogonal, b_1 b_2 = 0.
(3) comparison and addition survive any Cstar-homomorphism
and external tensoring by a fixed positive element,
even under the maximal tensor product.
Rule (1) is proven by a smoothing argument: put
x_gamma = v (b + gamma)^{-1/2} with b = v^H v. Then
x_gamma^H (v v^H) x_gamma = b^2 (b + gamma)^{-1}, which tends
to b as gamma goes to 0. The reverse inequality uses v^H in
place of v. Rule (2) is the Cuntz sum of the "row"
(b_1^{1/2}, b_2^{1/2}). Rule (3) is straightforward: apply a
homomorphism to the witnesses, or tensor them with the unit
of the other factor.
There is one further trivial-looking observation that we use
repeatedly:
[a] <= [1_B] for every a in B_+,
via the exact witness x = a^{1/2}: x^H 1_B x = a.
0.7 Free ultrafilters and the central sequence algebra
Let N = {1, 2, 3, ...}.
A filter on N is a collection of subsets of N that is upward
closed (contains every superset of a member) and closed under
finite intersections, and contains N but not the empty set. A
filter is an ultrafilter if it is maximal. An ultrafilter on N
is free (non-principal) if it does not contain any finite set.
Intuition. A free ultrafilter omega picks out "almost all n"
in a perfectly consistent way: for every subset S of N,
exactly one of S or its complement belongs to omega.
Free ultrafilters exist by the Axiom of Choice; you cannot
write one down explicitly. That is fine, we only ever use the
two properties above.
Now let A be a Cstar-algebra. The algebra of bounded sequences
in A is
l^infty(N, A) = { (a_n) : sup_n ||a_n|| < infinity },
itself a Cstar-algebra under pointwise operations and the sup
norm. Fix a free ultrafilter omega. Let
J_omega = { (a_n) : lim_{n -> omega} ||a_n|| = 0 },
the ideal of "sequences that vanish in norm along omega".
Here lim_{n -> omega} t_n = 0 means: for every epsilon > 0, the
set of n with t_n < epsilon belongs to omega.
The norm ultrapower of A along omega is
A_omega = l^infty(N, A) / J_omega.
Identify A inside A_omega by constant sequences. The norm
central sequence algebra is
F_omega(A) = A_omega (intersection) A',
i.e. the operators in the ultrapower that commute with every
constant sequence of A. Concretely: an element of F_omega(A)
is a bounded sequence (x_n) that asymptotically commutes with
every element of A, in the sense [x_n, a] -> 0 in norm along
omega for every a in A, modulo sequences with norm -> 0 along
omega.
Central means: commutes with A up to vanishing errors. Sequence
algebra because we built it out of l^infty. Ultrapower
because we quotient by the ideal of sequences going to 0 along
omega.
The Kirchberg-Rordam question is: what does F_omega(A) know
about A?
0.8 The Jiang-Su algebra Z
The Jiang-Su algebra Z is a specific Cstar-algebra. It is:
nonzero, unital, separable, has NO characters, and has NO
proper closed two-sided ideals (it is simple), and its
K-theory is K_0(Z) = Z, K_1(Z) = 0.
The Jiang-Su algebra was constructed by Xinhui Jiang and
Hongbing Su in 1999. Its most striking property is
self-absorbing: Z @min Z is isomorphic to Z, and Z @min Z @min
Z, etc. Z is to Cstar-algebras roughly what "the group of
integers" is to abelian groups, but with subtler structure.
A Cstar-algebra A is Z-stable if A is isomorphic to A @min Z.
Z-stability is one of the standard regularity hypotheses in the
Elliott classification programme for simple amenable unital
Cstar-algebras. Intuitively, an algebra that absorbs Z has no
rigidity left that Z itself does not capture.
The Kirchberg-Rordam question was: is "F_omega(A) has no
characters" equivalent to Z-stability of A? This paper
answers YES.
0.9 The dimension-drop algebra I(2,3)
Let
M_2 @ M_3 = M_6 = the six-by-six complex matrices.
The dimension-drop algebra I(2,3) is the Cstar-algebra of
continuous functions f: [0,1] -> M_6 such that
f(0) in the subalgebra M_2 @ 1_3 (which is isomorphic to M_2),
f(1) in the subalgebra 1_2 @ M_3 (which is isomorphic to M_3).
So I(2,3) is a bundle of matrix algebras over the interval,
whose fiber over t in (0,1) is M_6 but whose fibers at the two
endpoint drop down to M_2 and M_3. This is the "dimension
drop".
The irreducible representations of I(2,3) are:
evaluation at 0: f -> f(0) in M_2, dimension 2.
evaluation at t with 0 < t < 1: f -> f(t) in M_6, dim 6.
evaluation at 1: f -> f(1) in M_3, dimension 3.
So all irreducible dimensions are at most 6. No irreducible
representation of dimension 1 exists, i.e. I(2,3) has NO
characters.
The Kirchberg-Rordam result uses I(2,3) as a probe: if a finite
tensor power of D contains a unital copy of I(2,3), then F_omega
of D has the subhomogeneous characterless subalgebra required
by the K-R theorem, and D is Z-stable.
0.10 The main theorem
Every word in the following theorem has now been defined.
MAIN THEOREM. Let A be a nonzero unital separable complex
Cstar-algebra. Let omega be any free ultrafilter on N. Then
F_omega(A) has no characters
if and only if
A is isomorphic to A @min Z.
This is the Kirchberg-Rordam character criterion. It applies
to all such A, with no nuclearity, simplicity, trace, dimension,
or comparison assumption.
The paper proves it in two stages, plus an application of
known input from Kirchberg-Rordam.
0.11 What the paper actually proves
The core of the paper is this finite construction, which is
new and uses no separability, nuclearity, simplicity, or
tracial assumptions:
FINITE THEOREM. Let D be a nonzero unital complex
Cstar-algebra without characters. Then there is an integer
m >= 1 and a unital Cstar-homomorphism
I(2,3) -> D @max D @max ... @max D (m factors).
The exponent m depends on D; no uniform bound is claimed.
Once the finite theorem is in hand, the main theorem is an
easy sandwich using two classical Kirchberg-Rordam results,
one on the existence of commuting copies of a subalgebra, one
on the characterization of Z-stability by a subhomogeneous
characterless subalgebra of F_omega(A). A short corollary
about infinite minimal tensor powers follows by applying a
Dadarlat-Toms result.
So the paper's real work is the finite theorem. The proof
runs in two stages:
STAGE A (Section 3 of the body). Find a full, square-zero
element w in a finite maximal tensor power P of D, i.e. some
w in P with w^2 = 0, ||w|| = 1, and w full (generates P as
a closed two-sided ideal). The main tool: labels in D @ D @
... that stay linearly independent under every irreducible
representation.
STAGE B (Section 4). Turn w into a unital homomorphism
from the "matrix cone" C = {f in C([0,1], M_2): f(0) in
C 1_2}, then into a homomorphism from I(2,3). The main tool:
a continuous simplex of embeddings of M_2 that carries the
defect of a certain orthogonal pair to a small region of a
cube, so that a few carefully chosen elements can absorb the
defect.
The two stages feed the classic Rodam-Winter dimension-drop
criterion, which is proven in full in Section 2 (so the reader
need not go hunting through the literature). Section 5
assembles the finite theorem, the main theorem, and the
corollary on infinite tensor powers.
After that comes Section 6 with the references.
0.12 Notation table
Symbols, in order of appearance:
C complex numbers
R real numbers
N {1, 2, 3, ...}
M_n(C) n x n complex matrices
M_n M_n(C), abbreviated
C(X) continuous complex-valued functions on X
A, B, D, P unital Cstar-algebras
1 multiplicative identity of a Cstar-algebra
0 zero element
a^H adjoint of a
||.|| Cstar norm
a >= 0 a is self-adjoint with nonnegative spectrum
A_+ positive elements of A
a (circle) b tensor product in the plain-vector-space sense
@ tensor product of Cstar-algebras, max by default
@min, @max minimal / maximal tensor product
D @ D @ ... @ D (m times) maximal tensor power of D
A_omega norm ultrapower l^infty(N,A)/J_omega
F_omega(A) norm central sequence algebra
Z the Jiang-Su algebra
Z-stable A is isomorphic to A @min Z
omega a free ultrafilter on N
precsim Cuntz comparison: a precsim b iff x^H b x ~ a
[a] Cuntz class of a
Cu(B) Cuntz semigroup of B
Her(b) closed two-sided ideal generated by b
supp(b) support projection of b, in B** (the double dual)
B** the bidual (enveloping von Neumann algebra)
I(2,3) the dimension-drop algebra
C the matrix cone {f in C([0,1],M_2): f(0) in C 1_2}
iota the function t -> t inside C
e_ij elementary matrix unit
M_2 @ M_3 M_6
[a]+[b] Cuntz addition: class of b_1 + b_2 for orthogonal b_1, b_2
subhomogeneous every irreducible representation has bounded dimension
full the closed two-sided ideal it generates is the whole
algebra; equivalently it is nonzero in every
irreducible representation
cpc completely positive and contractive
cpo completely positive and order zero
J_omega ideal of sequences vanishing in norm along omega
One convention that saves a lot of words: a Cstar-homomorphism
that is injective is automatically isometric. So "unital
embedding" and "unital injective Cstar-homomorphism" mean the
same thing, and the phrases are used interchangeably below.
Everything is defined. Read on.
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PART 1. PRELIMINARIES
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The guide (Part 0) defines every object used below. This part
fixes conventions, states the two abstract criteria the paper
uses, and does the little fact-checking that would otherwise
clutter the main line.
1.1 Notational conventions
All algebras are complex, all are unital unless stated, and all
are Cstar-algebras with unit 1. Abbreviations:
a^H adjoint of a.
a >= 0 a is self-adjoint and its spectrum is contained
in [0,infinity); a is "positive".
A_+ the set of positive elements of A.
1_B the unit of B.
B** the double dual of B (sometimes called the
enveloping von Neumann algebra of B).
Tensor products. In the whole construction, and in the finite
result we prove, the default tensor product is MAXIMAL. We
shorten
D @max D @max ... @max D (n factors)
to "D^(x n)" or "D @...@ D (n times)". MINIMAL tensor products
are only ever used explicitly, always written "@min". The
algebraic tensor product in the vector-space sense is written
"(circle)" over C and "@_R" over R.
Representations. A representation is a Cstar-homomorphism into
B(H). It is irreducible if H has no closed invariant subspace
other than 0 and H. All irreducible representations are taken
to be nonzero and nondegenerate. Nondegenerate means pi(A)H is
dense in H. On a unital algebra a nondegenerate representation
is automatically unital: pi(1) must equal 1_H. That is why
everywhere below "irreducible representation" means "unital
irreducible representation".
1.2 Fullness
Definition. An element x of a Cstar-algebra B is FULL if the
closed two-sided ideal generated by x is all of B.
Lemma 1.1. Let B be a unital Cstar-algebra. An element x of
B is full if and only if pi(x) is nonzero in every irreducible
representation pi of B.
Proof. If J is the closed two-sided ideal generated by x, and
J is not all of B, then B/J is a nonzero Cstar-algebra. It has
a nonzero irreducible representation (this is a standard fact;
every nonzero Cstar-algebra has one), which forms a nonzero
irreducible representation of B vanishing on J and hence on x.
Conversely, if x is not full, then J is proper and the
representation just described is an irreducible representation
of B annihilating x. So x is not detected by irr reps.
Therefore x is full if and only if pi(x) is nonzero in every
irreducible representation. QED
So "full" is the representation-theoretic notion of "nonzero
everywhere". We will use this fact twice, and each time the
proof will be: exhibit an element that survives every
irreducible representation.
1.3 Cuntz comparison
For the comparison theory of positive elements, work in finite
matrix algebras over B; more precisely, allow a and b to sit in
finite corners of M_n(B) for various n, and view them all inside
a common M_N(B) via corner embeddings.
Definition. For positive elements a, b of finite matrix
algebras over B, write
a precsim b
(read: "a is Cuntz below b", or "a is Cuntz dominated by b") if,
for every epsilon > 0, there exists a finite matrix x with
|| a - x^H b x || <= epsilon.
The Cuntz class of a is written [a]. Write [a] <= [b] for
a precsim b. Addition is induced by direct sum:
[b_1] + [b_2] := [diag(b_1, b_2)],
where diag is the block-diagonal matrix. This is the Cuntz
semigroup Cu(B).
The paper only ever uses three elementary rules and one obvious
consequence. We state and prove each.
RULE (1). For every rectangular matrix v with entries in B,
[v^H v] = [v v^H].
Proof. Put b = v^H v, c = v v^H. We show b precsim c and
c precsim b.
For b precsim c, choose, for gamma > 0,
x_gamma = v (b + gamma)^{-1/2}.
Then
x_gamma^H c x_gamma
= (b+gamma)^{-1/2} v^H (v v^H) v (b+gamma)^{-1/2}
= (b+gamma)^{-1/2} b^2 (b+gamma)^{-1/2}
= b^2 (b + gamma)^{-1}.
Apply functional calculus on the spectrum of b. For t >= 0,
t^2/(t + gamma) - t = -gamma t/(t + gamma),
so its absolute value is at most gamma, uniformly in t. Hence
||b^2 (b+gamma)^{-1} - b|| <= gamma, and x_gamma^H c x_gamma
approaches b as gamma -> 0.
For c precsim b, run the same argument with v^H in place of v:
the matrices y_gamma = v^H (c+gamma)^{-1/2} satisfy
y_gamma^H b y_gamma = c^2 (c+gamma)^{-1} -> c. So
c precsim b. Hence [b] = [c]. QED
RULE (2). If a_1,...,a_l are positive and pairwise orthogonal,
b_i b_j = 0 for i not equal to j, then
[a_1 + ... + a_l] = [a_1] + ... + [a_l].
Proof. Consider the l-by-1 column matrix whose entries are the
square roots a_1^{1/2}, ..., a_l^{1/2}:
u = (a_1^{1/2},...,a_l^{1/2})^T.
Then u^H u = a_1 + ... + a_l (a single scalar in B), and u u^H
is the l-by-l diagonal matrix diag(a_1,...,a_l) with entries in
B. Rule (1) applied to u gives
[a_1 + ... + a_l] = [diag(a_1,...,a_l)] = [a_1]+...+[a_l].
QED
RULE (3). Comparison and addition are preserved by every
Cstar-homomorphism, and by tensor product with a fixed positive
element, including in maximal tensor products.
Proof. For a Cstar-homomorphism phi: witness x with
x^H b x ~ a gives phi(x)^H phi(b) phi(x) ~ phi(a). Direct
sums commute with homomorphisms.
For tensoring with a fixed positive c: witness x with
x^H b x ~ a in B gives
(x (circle) c^{1/2})^H (b (circle) c) (x (circle) c^{1/2})
= (x^H b x) (circle) c ~~ a (circle) c,
with error at most the original error times ||c||. Direct sums
again commute with the tensor. Everything works in maximal
tensor products because we use the unit of the other factor.
QED
One immediate consequence:
[a] <= [1_B] for every a in B_+,
via the exact witness x = a^{1/2}, for which x^H 1_B x = a.
That is the entire Cuntz toolkit. No cancellation, no stable
rank hypothesis, no comparison property is invoked.
1.4 The two algebras we reach for: matrix cone and dimension-drop
Matrix cone.
C := { f in C([0,1], M_2) : f(0) in C 1_2 }.
As a set, C is the algebra of continuous M_2-valued functions on
the interval, with the single restriction that at 0 the matrix
must be a complex multiple of the identity. C is a unital
Cstar-algebra containing the constants. It is the unitization
of C_0((0,1], M_2).
The special element
iota in C, iota(t) = t,
is the function "the identity matrix multiplied by t" and will
play a central role: the element iota e_11 of C has a single
nonzero matrix entry, the (1,1) matrix entry, which equals the
identity on the (1,1) coordinate.
Dimension drop.
I(2,3) := { f in C([0,1], M_2 @ M_3) :
f(0) in M_2 @ 1_3,
f(1) in 1_2 @ M_3 }.
This is the continuous-matrix-function Cstar-algebra over the
interval with fibers M_2 = M_2 @ 1_3 at t = 0, M_6 = M_2 @ M_3
in the interior, and M_3 = 1_2 @ M_3 at t = 1.
Lemma 1.2 (irreducible representations of I(2,3)). Every
irreducible representation of I(2,3) is evaluation at some point
t of [0,1], and its dimension is
2 at t = 0,
3 at t = 1,
6 for 0 < t < 1.
In particular I(2,3) is subhomogeneous with bound 6, and has no
characters.
Proof. Let pi: I(2,3) -> B(H) be an irreducible representation.
Consider the central subalgebra
C([0,1]) 1 := { scalar functions t -> g(t) 1_{M_6} }
(intersection) I(2,3).
Its image under pi is central in pi(I(2,3)), and pi is
irreducible, so the image is C 1_H by Schur's lemma. Hence the
restriction of pi to C([0,1]) 1 is a unital character, i.e.
evaluation at some t_0 in [0,1].
Let J be the closed ideal of I(2,3) generated by functions
vanishing at t_0. Every element of J is annihilated by pi
because pi's restriction to C([0,1]) 1 already accounts for the
t_0 evaluation. So pi factors through the quotient
I(2,3)/J.
The quotient is the fiber at t_0:
t_0 = 0: M_2 = M_2 @ 1_3 (dimension 2)
0 < t_0 < 1: M_6 = M_2 @ M_3 (dimension 6)
t_0 = 1: M_3 = 1_2 @ M_3 (dimension 3).
An irreducible representation of M_n(C) has dimension n and is
unique up to equivalence (all irreducible representations of a
full matrix algebra are the defining one). So pi has exactly
the claimed dimension. QED
Since the largest irreducible dimension is 6, I(2,3) is
subhomogeneous with bound 6. Since no irreducible dimension is
1, I(2,3) has no characters. The same reasoning shows that any
quotient of I(2,3) (with a nonzero unit) is subhomogeneous and
characterless: irreducible representations of the quotient pull
back to irreducible representations of I(2,3), and characters
would pull back to characters.
1.5 The Rodam-Winter dimension-drop criterion
We will reach I(2,3) from below via the following criterion of
Rodam and Winter. The statement is elementary to read, but the
proof (Part 2) is the technical heart of the paper's final
assembly.
Proposition 1.3 (Rodam-Winter). Let B be a unital Cstar-
algebra. Let a_1, a_2 in B be orthogonal positive contractions,
that is a_1, a_2 >= 0, ||a_1|| <= 1, ||a_2|| <= 1, and a_1 a_2
= 0. Suppose there exists v in B with
a_1 = v^H v, a_2 = v v^H,
and suppose there exists epsilon > 0 with
1_B - a_1 - a_2 precsim (a_1 - epsilon)_+.
Then there is a unital Cstar-homomorphism
I(2,3) -> B.
This is [RW] Proposition 5.1, (ii) => (iv), with n = 2. It is
proved in full in Part 2 below using Rodam and Winter's method
(scalar endpoint fiber, support projections, weighted flip).
Stable rank one hypotheses that appear in [RW] are used only
for (i) => (ii) in that paper. They are NOT needed for the
implication stated here.
Two remarks on the statement. First, orthogonality of a_1, a_2
ensures a_1 + a_2 <= 1_B, so the element
1_B - a_1 - a_2
is positive and the comparison is a statement about a positive
defect being dominated by a positive cutoff of a_1. Second,
the map from I(2,3) is not just any map: it is unital and a
Cstar-homomorphism. That is exactly what the dimension-drop
criterion produces.
1.6 Central-sequence inputs from Kirchberg-Rordam
Fix a nonzero unital separable Cstar-algebra A and a free
ultrafilter omega on N. The norm central sequence algebra
F_omega(A) was constructed in PART 0. We use two classical
facts.
Proposition 1.4 (Kirchberg-Rordam). Let A be a nonzero unital
separable Cstar-algebra.
(1) If D is a unital separable Cstar-subalgebra of F_omega(A)
containing the ambient unit 1_{F_omega(A)}, then for
every m >= 1 there is a unital Cstar-homomorphism
D @max D @max ... @max D (m times) -> F_omega(A).
(2) A is isomorphic to A @min Z if and only if F_omega(A)
contains, with the ambient unit, a separable
subhomogeneous Cstar-subalgebra without characters.
Part (1) is the "commuting copies lemma". The version most
convenient here is [Fixed231] Lemma 9.3; its primary source is
[KR] Proposition 2.2 and Corollary 2.3. The companion version
has an annihilator ideal that vanishes for unital A: an element
annihilating every element of A annihilates its unit. So its
F(A) is exactly the norm central sequence algebra F_omega(A)
used here. No nuclear-dimension hypothesis appears. We
reprove (1) below to make the quantifiers explicit.
Part (2) is [KR] Theorem 2.5. The word "subhomogeneous" means:
the dimensions of the irreducible representations are all at
most some common finite bound. This input does NOT replace
norm central sequences with a tracial ultrapower, and does NOT
impose nuclearity on A.
Proof of Proposition 1.4, part (1).
Fix a separable Cstar-subalgebra E of F_omega(A). We want a
unital Cstar-homomorphism from D into F_omega(A) (intersection)
E'. Then we will iterate this to get any finite number of
mutually commuting unital copies of D, and use the universal
property of the maximal tensor product.
Choose a countable dense unital *-subalgebra D_0 of D over
Q + iQ. For each x in D_0, choose a bounded lift (a_l^x)_l in
A with sup ||a_l^x|| <= 2 ||x||. Use the zero lift for x = 0
and the constant unit lift for x = 1. The bounded lift is
constructible: choose any representing sequence in
l^infty(N,D)/J_omega and truncate coordinate norms (which does
not change the class in A_omega).
For every test point in a countable enumeration of the additive,
rational-scalar, multiplicative, and adjoint relations in D_0,
the relation defect of the columns (a_l^x)_l tends to zero
along omega. That is: (a_l^x)_l is asymptotically a unital
Q+iQ-*-homomorphism D_0 -> A along omega.
Choose a countable dense subset of A, and choose bounded lifts
(b_s^j)_s of a countable dense subset of E. Here j indexes the
dense subset of E and s indexes the coordinates of the lift.
Enumerate all relation-defect tests and all centrality tests
(bringing in a_*^x against b_*^j). For each output index s,
choose a single index l(s) so that
(relation defect at l(s)) <= 1/s for the first s tests, and
||[a_{l(s)}^x, b_s^j]|| <= 1/s for the first s pairs (x, j).
Each of the finitely many requirements holds on an omega-large
set of old indices l. For fixed s and fixed x in D_0, each
b_s^j is a fixed element of A, so the original centrality of
F_omega(A) applies. The intersection over the finitely many
requirements is therefore nonempty.
The new sequences (a_{l(s)}^x)_s are central in the ordinary
sense (they commute with all of A up to error along omega, which
extends to A), and they satisfy the algebraic relations of D_0
in the ordinary limit, hence along the free ultrafilter omega.
Their classes in F_omega(A) (intersection) E' therefore define
a unital Q+iQ-*-homomorphism D_0 -> F_omega(A) (intersection)
E'. The uniform bound 2||x|| extends this to a complex unital
Cstar-homomorphism D -> F_omega(A) (intersection) E'.
Now take D_copy^(1) = the given copy of D in F_omega(A).
Taking E_1 = C*(D_copy^(1), 1) and applying the above, we get
a unital Cstar-homomorphism D -> F_omega(A) (intersection)
E_1' with image D_copy^(2). Iterating finitely many times,
we get m mutually commuting unital images D_copy^(1), ...,
D_copy^(m) of D inside F_omega(A). The universal property of
the maximal tensor product (maximal exactly because we require
only commuting images, no injectivity) assembles these into
D @max D @max ... @max D (m times) -> F_omega(A).
Injectivity of the individual reindexed copies is irrelevant:
maximality of the tensor product is not a claim of injectivity,
it is a claim of "largest completion under all commuting pairs
of representations". QED (part 1)
Parts (1), (2) and the dimension-drop criterion are now in
hand. Part 2 proves Proposition 1.3. Part 3 constructs the
full square-zero element that feeds into Proposition 1.3 after
one more step (Part 4).
----------------------------------------------------------------
PART 2. THE RODAM-WINTER DIMENSION-DROP CRITERION
----------------------------------------------------------------
This part proves Proposition 1.3 in full. We follow Rodam
and Winter's method while making the pieces explicit: the
support projections, the scalar endpoint fiber, and the
weighted flip identity. The implication is stated for an
arbitrary unital Cstar-algebra B.
Throughout, B** (the double dual) is the enveloping von
Neumann algebra. Support projections and polar decompositions
are taken in B**; where they enter an honest element of B we
will say so explicitly.
----------------------------------------------------------------
Lemma 2.1 (Exact cut under a Cuntz comparison).
----------------------------------------------------------------
Let B be a unital Cstar-algebra. Suppose a, b in B_+ satisfy
a precsim b. Then, for every delta > 0, there is z in B with
z^H z = (a - delta)_+ and z z^H in Her(b).
Here Her(b) is the closed two-sided ideal generated by b.
Proof. Compression of stabilized comparison witnesses to their
(1,1) entries gives witnesses inside B itself. (Stabilized
comparison witnesses live in M_n(B) for some finite n; use
b^{1/2} and approximating finite matrices to recover witnesses
in B.)
Choose x in B and eta with 0 < eta < delta and
|| a - x^H b x || < eta.
Put
e = (a - delta)_+, y = b^{1/2} x e^{1/2}, T = y^H y.
Then
T = e^{1/2} x^H b x e^{1/2} >= e^{1/2} (a - eta) e^{1/2}
>= (delta - eta) e
(the last because on the range of e we have a >= delta, so
a - eta >= delta - eta >= 0), while on the other hand
T <= || x^H b x || e.
Thus e is supported in the range of T, and T is supported in
the range of e, i.e. T and e have the same support projection.
Let y = u T^{1/2} be the polar decomposition in B**, and put
z = u e^{1/2}.
Then z^H z = e^{1/2} u^H u e^{1/2} = e (because u^H u = supp(T)
and e <= T/(delta-eta) implies e is supported on the range of
T). Our job is to show z in B and z z^H in Her(b).
Approximate z by
z_tau = y (T + tau)^{-1/2} e^{1/2}
for tau > 0. Each z_tau is in b^{1/2} B by construction. Put
f_tau(t) = 1 - ( t/(t+tau) )^{1/2}. Then
z - z_tau = u f_tau(T) e^{1/2}, and using u^H u f_tau(T) =
f_tau(T) (because f_tau(0) = 0),
|| z - z_tau ||^2
<= || f_tau(T) e f_tau(T) ||
<= || f_tau(T) T f_tau(T) || / (delta - eta)
= sup_{t >= 0} t f_tau(t)^2 / (delta - eta)
<= tau / (delta - eta) -> 0.
(Here we used e <= T/(delta - eta), and the sup is at most tau
by a direct computation with u = (t/(t+tau))^{1/2} and s = u^2:
t f_tau(t)^2 = tau * s (1 - s^{1/2})/(1 + s^{1/2}) <= tau.)
Thus z is in the norm closure of the b^{1/2} B's, i.e. z in B.
Finally z z^H = u e u^H. This is a norm limit of the elements
u T^{1/2}(T+tau)^{-1/2} ... of b^{1/2} B b^{1/2} = Her(b) as
tau -> 0. So z z^H in Her(b). QED
----------------------------------------------------------------
Gram construction of order-zero maps.
----------------------------------------------------------------
Let B be a unital Cstar-algebra and let x_1, ..., x_q in B be
contractions with
x_i x_j^H = delta_{ij} h for some positive contraction h.
Then the formula
alpha(e_ij) = x_i^H x_j
defines a completely positive, contractive, ORDER-ZERO map
alpha: M_q -> B. (Order-zero means alpha sends orthogonal
positive elements to orthogonal positive elements.)
Why. Complete positivity follows from the Gram structure of
the matrix (x_i^H x_j). On the other hand, the positive
elements x_i^H x_i of B are pairwise orthogonal (because
x_i^H x_i x_j^H x_j = x_i^H h x_j = delta_ij x_i^H x_i h), so
their sum
H := sum_i x_i^H x_i
is a positive contraction. Multiplication gives
alpha(e_ij) alpha(e_kl) = delta_{jk} H alpha(e_il),
which is exactly the order-zero multiplication identity. In
particular alpha preserves orthogonality of positive elements.
The supporting homomorphism bar alpha: M_q -> B** is defined so
that alpha(x) = H bar alpha(x) = bar alpha(x) H, with
bar alpha(1_q) = supp(H). Its matrix units are the strong
limits of alpha(e_ij)(H + tau)^{-1} as tau -> 0; the
multiplication identity and the bound alpha(e_ij)^H alpha(e_ij)
<= H^2 verify boundedness, the matrix-unit relations, and the
support claim.
----------------------------------------------------------------
Lemma 2.2 (Commuting order-zero maps give a dimension-drop map).
----------------------------------------------------------------
Let B be a unital Cstar-algebra. Let
alpha: M_3 -> B, beta: M_2 -> B
be completely positive, contractive, order-zero maps with
commuting ranges. Suppose
alpha(1_3) + beta(1_2) = 1_B.
Then there is a unital Cstar-homomorphism I(2,3) -> B.
Proof. Consider the canonical order-zero maps from the two
matrix algebras into I(2,3):
alpha_0(y)(t) = t (1_2 @ y), beta_0(x)(t) = (1-t)(x @ 1_3),
where the right-hand sides are viewed as functions [0,1] ->
M_2 @ M_3. Both alpha_0 and beta_0 are cpc order-zero.
The generated algebra of the images of alpha_0 and beta_0
contains the scalar function t (from the (1,1) entry of
alpha_0 with y = e_11), and products of entries generate the
ideal C_0((0,1), M_2 @ M_3). At the endpoints the values
generate M_2 + M_3 = M_2 (+) M_3. Together these generate all
of I(2,3).
Set
E := C*(alpha(M_3), beta(M_2), 1_B).
The element alpha(1_3) is central in E: it commutes with
alpha(M_3) (both in the alpha range) and with beta(M_2) by
hypothesis. In any irreducible representation of E, this
central element acts as a scalar
r in [0,1].
If 0 < r < 1, then r^{-1} alpha and (1-r)^{-1} beta give
commuting unital homomorphisms M_3 -> E, M_2 -> E whose
products generate a representation of M_2 @ M_3. If r = 0, only
the M_2 map survives; if r = 1, only the M_3 map survives.
Substitution. Any element x of I(2,3) is a noncommutative
*-polynomial in alpha_0(y) and beta_0(x). Substituting alpha
and beta gives a bounded element of E, and in every irreducible
representation of E this element evaluates to the value at a
point r in [0,1]. Hence
|| x ||_{I(2,3)} = sup_{r in [0,1]} || x(r) ||_{M_2 @ M_3},
|| p(alpha, beta) ||_B <= sup_{r in [0,1]} || x(r) ||_{M_2 @ M_3}.
So substitution is well defined and contractive on the generated
algebra, and it extends from the dense generated algebra to the
required unital Cstar-homomorphism I(2,3) -> B. QED
----------------------------------------------------------------
Proposition 2.3 (= Proposition 1.3, RW dimension-drop criterion).
----------------------------------------------------------------
Let B be a unital Cstar-algebra. Let a_1, a_2 in B be
orthogonal positive contractions with
a_1 = w^H w, a_2 = w w^H
for some w in B (this w is the "v" of Proposition 1.3, renamed
to avoid clash with the contraction v built below), and
1_B - a_1 - a_2 precsim (a_1 - epsilon)_+
for some 0 < epsilon < 1. Then there is a unital Cstar-
homomorphism I(2,3) -> B.
The proof takes eight steps.
STAGE 1. Smoothing.
Put
f(t) = min{ t/epsilon, 1 }, s_1 = f(a_1)^{1/2}.
Do the polar decomposition w^H = u_0 a_2^{1/2}, and put
s_2 = u_0 f(a_2)^{1/2}.
Then s_2 is in B: it equals w^H q(a_2), where the continuous
function q(t) = (f(t)/t)^{1/2} for t > 0 is assigned the value
epsilon^{-1/2} at t = 0.
Compute:
s_i s_j^H = delta_{ij} s_1^2, d := s_1^H s_1 + s_2^H s_2
= f(a_1) + f(a_2).
The off-diagonal products s_i s_j^H vanish for i != j because
a_1 a_2 = 0. So the s_i satisfy the Gram conditions with
h = s_1^2. Let rho: M_2 -> B be the corresponding order-zero
map, rho(e_ij) = s_i^H s_j, with rho(1_2) = d. Let sigma be
its supporting homomorphism, and p := supp(d) = sigma(1_2).
Put c = 1 - a_1 - a_2. Then, by a direct computation with
min:
a := 1 - d = epsilon^{-1} ( c - (1 - epsilon) )_+.
STAGE 2. Apply the exact cut.
Since c precsim (a_1 - epsilon)_+ by hypothesis, apply Lemma
2.1 with delta = 1 - epsilon and divide its z-element by
(epsilon)^{1/2}. The result is a contraction v in B with
(i) v^H v = a = 1 - d,
(ii) v v^H in Her((a_1 - epsilon)_+),
(iii) s_1^2 v v^H = v v^H.
The last relation gives, taking v v^H and s_1 v = v (multiply
by s_1 on the left and use that s_1 acts as identity on the
range of v), s_1 v = v, s_2 v = 0, and a v = 0. From d v = 0
(the negation of a v = v) and the fact that ds_i = s_i d (both
s_1, s_2 commute with d via the Gram identity and a_1 a_2 = 0),
we conclude that a commutes with each s_i.
STAGE 3. Two orthogonal copies of the defect algebra.
Take the polar decomposition v = u a^{1/2} in B**, and put
e := u^H u = supp(a).
Define
r_i := s_i^H u = sigma(e_i1) u, i = 1, 2,
R := r_1 r_1^H + r_2 r_2^H.
Direct multiplication, using a = 1 - d and the relations above,
gives
r_i^H r_j = delta_{ij} e, d r_i = r_i,
R <= p, R (perpendicular) e,
sigma(e_jk) r_i = delta_{ki} r_j.
(The last uses u^H u = e and a u = 0 to see that a annihilates
the range of r_i.)
Define
phi: Her(a) -> B, phi(b) = sum_{i=1}^2 r_i b r_i^H.
The relations above show phi is a homomorphism. Its values lie
in B: approximate b in Her(a) by a^{1/2} b_m a^{1/2} with b_m
in B, and use u a^{1/2} = v in B.
The same approximation, with one outer factor omitted, shows
u b in B for b in Her(a). The relations above also give
phi(b) r_i = r_i b, [phi(b), sigma(M_2)] = 0,
phi(Her(a)) (perpendicular) Her(a).
Here phi places "the same defect element in the two ranges of
the r_i", while sigma(e_jk) acts on their index i.
STAGE 4. A section algebra and a homomorphism lambda.
To get the other coordinate, work with sections over [0,1]
valued in M_2 that are scalar at zero and vanish at one:
D_1 := { F in C([0,1], M_2) : F(0) in C 1_2, F(1) = 0 }.
Let c_F be the scalar with F(0) = c_F 1_2. Since d commutes
with sigma(M_2), continuous functional calculus defines
barlambda(F) = sum_{i,j} F_ij(d) sigma(e_ij) in p B** p.
Define lambda on D_1 by
lambda(F) := barlambda(F) + c_F (1 - p).
The extra summand retains the scalar endpoint fiber at zero.
For a scalar function f vanishing at one,
lambda(f 1_2) = f(d). The algebra D_1 is generated by
(1-t) 1_2 and t(1-t) x for x in M_2; their images are
a = 1 - d and a rho(x) respectively, both in Her(a). So
lambda takes values in Her(a), hence in B, without needing to
assume p belongs to B.
STAGE 5. A larger section algebra and a homomorphism gamma.
Next define
D_2 := { F in C([0,1], M_2 @ M_2) : F(0) in C 1_4, F(1) = 0 }.
This is a Cstar-subalgebra of D_1 (circle) M_2. Because
phi(Her(a)) commutes with sigma(M_2) (Stage 3), the product of
phi o lambda (on the first tensor factor) and sigma (on the
second tensor factor) gives a homomorphism
gamma: D_2 -> B.
Its support projection is R, because the range contains phi(a),
which has support R.
STAGE 6. A unitary path and the weighted flip.
Let F_U in M_2 @ M_2 be the flip unitary (mapping
e_ij @ e_kl to e_kl @ e_ij). Choose a continuous function
theta: [0,1] -> [0,1] that is 0 on [0,1/3] and 1 on [2/3,1],
and set
U(t) := (1 + F_U)/2 + e^{i pi theta(t)} (1 - F_U)/2.
Then U is a unitary path with U(0) = 1, and U(t) = F_U for
t in [2/3,1]. It is a multiplier of D_2 (at the endpoint 0,
U(0) = 1 which is a scalar; at 1 there is no constraint because
F(1) = 0 is required of D_2). The multiplier extension of
gamma sends U to a unitary w_0 in the corner R B** R (since
gamma's support projection is R). Put
W := w_0 + 1 - R.
Let g: [0,1] -> [0,1] be the continuous function
g(t) := 1 for 0 <= t <= 2/3, g(t) := 3(1-t) for 2/3 <= t <= 1.
Then g(0) = 1 and g(1) = 0. Put
h := lambda(g 1_2) = g(d), k := phi(h),
l := 1 - h - k.
A verifiable functional-calculus inequality, 0 <= g(t) <= 1,
with the fact that h and k are supported in the two orthogonal
corners e and R, gives
h, k, l are positive contractions,
h k = 0, 0 <= l <= d, h + k + l = 1_B,
k W = W k, k^alpha W = gamma(g^alpha U) in B for alpha > 0.
STAGE 7. Building alpha(M_3) and beta(M_2).
Define
x_i : = h^{1/2} r_i^H W, i = 1, 2,
x_3 : = h^{1/2}.
The elements x_1, x_2 lie in B via the factorization
x_i = ( r_i^H k^{1/4} )( k^{1/4} W ),
because r_i^H k^{1/4} = h^{1/4} r_i^H lies in B (from a short
functional-calculus argument with the relations of Stage 3),
and k^{1/4} W = gamma(g^{1/4} U) lies in B.
A direct computation gives
x_i x_j^H = delta_{ij} h, sum_i x_i^H x_i = h + k.
The elements x_1, x_2, x_3 satisfy the Gram conditions
x_i x_j^H = delta_ij h for the positive contraction h. That is,
alpha(e_ij) := x_i^H x_j
defines a completely positive, contractive, order-zero map
alpha: M_3 -> B, alpha(1_3) = h + k.
Similarly define
beta: M_2 -> B, beta(y) = l sigma(y).
This is an order-zero map (it is a positive scalar multiple of
the order-zero map sigma), and beta(1_2) = l. Since lp = l
(it is supported on p = sigma(1_2)), beta is well-behaved in B.
Also
alpha(1_3) + beta(1_2) = (h + k) + l = 1_B.
So the hypotheses of Lemma 2.2 are almost satisfied: LHS sum of
units is 1_B, both maps are cpc order-zero. The one thing left
is to show that alpha(M_3) commutes with beta(M_2).
STAGE 8. The weighted flip identity.
The scalar function
Q(t) := g(t)(1 - g(t))
vanishes at both endpoints and is supported exactly where
U(t) = F_U. On this support the flip identity holds:
Q(t) (1 (circle) y) U(t)^H = U(t)^H Q(t) (y (circle) 1),
for every y in M_2. Both sides are elements of D_2. Applying
gamma (and using Q(0) = 0 to avoid the c_F (1-p) summand in
lambda) gives
k (1 - k) sigma(y) W^H = W^H phi( lambda(Q (circle) y) ),
for y in M_2.
This is the WEIGHTED FLIP IDENTITY. In particular, no constant
nonscalar section is evaluated by lambda, so the identity
provides an honest commutation between the alpha range and the
beta range.
Put C_i := alpha(e_i3) = W^H r_i h for i = 1, 2. Using the
weighted flip identity and h k = 0,
beta(y) C_i
= k (1-k) sigma(y) W^H r_i
= W^H r_i lambda(Q (circle) y)
= W^H r_i h (1 - h) sigma(y)
= C_i beta(y).
For the second equality we used lambda(Q (circle) y) =
Q(d) sigma(y) (because Q(0) = 0 escapes the c_F (1-p) term); the
third uses h k = 0.
The Cstar-algebra generated by the C_i contains alpha(M_3).
Indeed C_i^H C_i = h^2 and C_i C_i^H = alpha(e_ii)^2, so the
generated algebra contains h, all diagonal entries of alpha, and
A_0 := alpha(1_3).
Furthermore C_i C_j^H = A_0 alpha(e_ij). Multiplying by
(A_0 + tau)^{-1} and letting tau -> 0 recovers each matrix
entry of alpha(M_3) (the convergence follows from
alpha(e_ij) = A_0 bar alpha(e_ij) and ||tau A_0 (A_0 + tau)^{-1}||
<= tau). Hence alpha(M_3) commutes with beta(M_2).
Lemma 2.2 now completes the proof: the maps
alpha: M_3 -> B, beta: M_2 -> B are cpc order-zero, with
commuting ranges and sum of units = 1_B, giving a unital
Cstar-homomorphism I(2,3) -> B.
Degenerate case v = 0. If v = 0, then a = 0 and consequently
h = 0, k = 0, R = 0, W = 1. Then beta is a unital
homomorphism M_2 -> B (because l = 1 - h - k = 1). No step
of the argument requires the defect to be nonzero, and
Lemma 2.2 still applies. In particular the proof does not
require stable rank one for the dimension-drop criterion.
QED
----------------------------------------------------------------
PART 3. A MATRIX-CONE HOMOMORPHISM WITH A FULL CORNER
----------------------------------------------------------------
We now construct, for any nonzero unital characterless D, a
full square-zero element w in a finite maximal tensor power P
of D. The polar decomposition of w will, in the next part,
give a unital homomorphism eta: C -> P (where C is the matrix
cone of Part 1) with eta(iota e_11) full.
Two things make this hard.
First, D need not be simple or nuclear, so the irreducible
representations of D @max ... @max D can be very complicated.
We cannot appeal to a spatial decomposition of any tensor
product.
Second, "absence of characters" is a global property of D. It
is detected by FINITELY MANY commutators (because the closed
ideal generated by the commutators of self-adjoint elements is
all of D). The task is to combine those finitely many
commutator tests into one element that survives every
irreducible representation of a maximal tensor power.
The tool is TENSOR LABELS. A building block V sits in D_sa,
and after tensoring copies of V into D high tensor powers, we
get finitely many selfadjoint tensors x_1, ..., x_r in D^(x n)
that stay linearly independent under every irreducible
representation of D^(x n) @ B. Independence replaces
orthogonality in the algebra; then a second label family
retains all bridges between two full orthogonal positives in a
single square-zero element.
----------------------------------------------------------------
3.1 Algebraic independence of commuting factors
----------------------------------------------------------------
Lemma 3.1. Let M be a von Neumann factor with 1_M = 1_H for a
Hilbert space H, and let B be a unital algebra with B contained
in the commutant M'. Then the multiplication map
M (circle) B -> B(H), a (circle) b -> ab,
is injective. Consequently, multiplication is also injective on
the complex algebraic tensor product of any finite family of
commuting von Neumann factors on the same Hilbert space, all
with unit 1_H.
Proof. Fix complex linearly independent elements b_1, ..., b_l
in B. It suffices to show that the "row space"
S := { (a_1,...,a_l) in M^{1 x l} : a_1 b_1 + ... + a_l b_l = 0 }
is zero.
The set S is weak-operator closed and invariant under left and
right multiplication by M. Right invariance uses B subset M'.
Let J subset M_l(M) consist of matrices whose rows lie in S.
It is a weak-operator closed left ideal: each row of a left
matrix product is an M-linear combination of the original rows.
We now use the support-projection argument for the form of a
weak-operator closed left ideal. For x in J, we also have x^H x
in J. Functional calculus with functions vanishing at zero,
followed by weak-operator closure, puts the support projection
of x^H x in J (take x^H x (x^H x + epsilon 1)^{-1} and let
epsilon -> 0). Finite joins of these projections belong to J
because the join of finitely many projections is the support of
their sum. Their increasing join p therefore belongs to J; the
join is a net of finite joins, so no separability of M or H is
involved.
Every x in J satisfies x p = x, and p in J gives
J = M_l(M) p.
Let u be any unitary of M and put U = diag(u, ..., u). Right
invariance of J and its counterpart for u^H give J U = J.
For x in J, the right support projection of x U is the conjugate
of that of x by U. Taking their joins gives
U^H p U = p.
As this holds for every unitary u, every entry of p belongs to
the commutant of M in M_l(M), i.e. to the center of M, which
is C.
Each row of p is therefore a scalar relation among the
independent b_1, ..., b_l. Hence each row is zero, so p = 0.
Thus J = 0 and S = 0.
For finitely many commuting factors, apply the case of two
factors to one factor and the algebra generated by the others,
and induct on the number of factors. QED
Lemma 3.2. Let pi: B_1 @max ... @max B_q -> B(H) be an
irreducible representation, where each B_j is a nonzero unital
Cstar-algebra. Let
N_j := pi(B_j)'' (weak-operator closure).
Then N_1, ..., N_q are commuting von Neumann factors on H. The
same conclusion holds if the tensor legs are grouped into any
partition into nonempty subsets and closures are taken on the
products.
Proof. Every represented block contains pi(1) = 1_H because
pi is unital. Distinct blocks commute, and commutation
persists under weak-operator closure.
Let z be a central element of N_j. It commutes with all of N_j
and also with every complementary block pi(B_k)'', k != j.
Thus z commutes with the whole image pi(B_1 @max ... @max B_q),
which is irreducible, so the commutant of pi(...)' is C
(Schur's lemma). Hence z is scalar. This is factorhood for
N_j. The grouped version follows by applying the same argument
to grouped blocks. QED
These two lemmas concern ALGEBRAIC tensor independence only.
They do not require faithfulness of the restrictions of pi to
the individual tensor legs, and they do not assume a spatial
decomposition of the represented factors.
----------------------------------------------------------------
3.2 Labels that remain independent in representations
----------------------------------------------------------------
Lemma 3.3. Let D be a nonzero unital Cstar-algebra without
characters. Then there exists a finite-dimensional real
subspace V of D_sa with the following property:
every nonzero unital representation of D has the property
that its image of V contains a noncommuting pair.
Proof. Consider the closed two-sided ideal I generated by all
commutators [a, b] with a, b in D_sa. If I were proper, then
D/I would be a nonzero unital commutative Cstar-algebra, and
a character of D/I would pull back to a character of D. This
contradiction shows I = D.
Choose a finite algebraic expression in these commutators
within distance < 1 from 1_D. Because the expression is within
distance 1 of the unit, it is invertible; hence the finitely
many commutators it uses already generate the unit ideal of D.
Let V be the real span of their selfadjoint entries. If some
unital representation pi of D sent V to a commuting space, then
pi would annihilate all these commutators, and therefore would
annihilate their unit ideal. This is impossible because pi is
unital (in particular pi(1) = 1 != 0). So pi(V) is
noncommutative. QED
Lemma 3.4. Let V be a finite-dimensional real vector space with
m = dim_R V >= 2. For every integer r >= 1, there are n >= 1
and tensors x_1, ..., x_r in V^(x_R n) with the following
property: whenever P_j: V -> E_j are real linear maps of rank
at least two (E_j finite-dimensional real vector spaces), the
tensors
(P_1 (x) ... (x) P_n)(x_i), 1 <= i <= r,
are linearly independent over R.
Proof. It suffices to consider surjections P_j: V -> R^2.
Indeed, a map of rank at least two can be postcomposed, on its
finite-dimensional image, with a surjection onto R^2; a
dependence before this postcomposition would remain a
dependence afterward.
Choose n large enough that
2^n > 2mn + r - 1.
Put W = V^(x_R n). Let
P := Surj(V, R^2)^n,
an open subset of a real vector space of dimension 2mn.
Consider the smooth map
Phi: P x S^{r-1} x W^r -> (R^2)^(x_R n),
(P_1,...,P_n ; u_1,...,u_r ; x_1,...,x_r) ->
(P_1 (x) ... (x) P_n)( sum_{i=1}^r u_i x_i ).
The differential of Phi in the x variables is surjective
everywhere: some u_i != 0, and the tensor product of the P_j is
surjective.
Thus the zero set Phi^{-1}(0) is a smooth manifold of dimension
2mn + (r-1) + r m^n - 2^n,
which is strictly smaller than r m^n = dim_R W^r
by the choice of n.
By Sard's theorem, the projection of Phi^{-1}(0) to W^r has
Lebesgue measure zero. The relevant manifolds are second
countable, so Sard can be applied on a countable atlas; no
compactness of P is required.
Choose x = (x_1, ..., x_r) in W^r outside this projection.
For this one tuple x, no nonzero real dependence occurs for any
tuple of surjections: any nonzero real coefficient vector
(u_i) can be normalised to lie in S^{r-1}. QED
Corollary 3.5 (Labels in tensor powers of D).
Let D and V be as in Lemma 3.3. For each r >= 1, there exist
n >= 1 and selfadjoint elements x_1, ..., x_r in D^(x n) such
that, for every nonzero unital Cstar-algebra B, the images of
x_1, ..., x_r under every irreducible representation of
D^(x n) @max B are complex linearly independent. Here the
x_i occupy the first n tensor legs.
Proof. Apply Lemma 3.4 to V to get real tensors
y_1, ..., y_r in V^(x_R n). Map them naturally into D^(x n)
by the inclusion V subset D_sa. Each x_i := y_i is selfadjoint
because it is a real linear combination of elementary tensors
with selfadjoint entries.
Fix such an irreducible representation pi of D^(x n) @max B,
and write pi_j for its restriction to the j-th copy of D. Each
pi_j is unital (because pi is), so Lemma 3.3 applies and gives
dim_R pi_j(V) >= 2.
Apply Lemma 3.4 to the real linear maps V -> pi_j(V). The
image of y_i under (pi_1 (x) ... (x) pi_n) is exactly pi(x_i),
and it is real-linearly independent as i varies.
The resulting independence also holds after complexification.
Indeed, a real basis of a space of selfadjoint operators is
complex-linearly independent: taking adjoints of a complex
relation separates its real and imaginary parts. Hence the
complexification of each pi_j(V) embeds in the represented
algebra.
Tensoring these injections shows that the real independent
tensors remain complex-linearly independent inside the complex
algebraic tensor product of the represented legs. By
Lemma 3.2, the weak-operator closures of the represented legs
are commuting von Neumann factors. By Lemma 3.1, complex
independence survives multiplication. Hence the pi(x_i) are
complex linearly independent. QED
The point of Corollary 3.5 is that the same fixed finite family
x_1, ..., x_r in D^(x n) works uniformly against EVERY
additional tensor factor B and EVERY irreducible representation
of D^(x n) @max B. This is the "label" family that will let us
replace orthogonality by linear independence in the next
stage.
----------------------------------------------------------------
3.3 A full square-zero element
----------------------------------------------------------------
Lemma 3.6 (full square-zero). Let D be a nonzero unital
Cstar-algebra without characters. Then some finite maximal
tensor power P of D contains a full element w with
w^2 = 0, || w || = 1.
Proof. Choose V as in Lemma 3.3, and let v_1, ..., v_m be a
real basis of V. Apply Corollary 3.5 with r = m: get labels
x_1, ..., x_m in D^(x n) for some n >= 1, complex-linearly
independent under every irreducible representation of
D^(x n) @max B for every B.
Set
P_0 := D @max D^(x n) @max D^(x n),
and inside P_0 define selfadjoint elements
s := sum_{i=1}^m v_i (circle) x_i (circle) 1,
t := sum_{i=1}^m v_i (circle) 1 (circle) x_i.
We claim that in every irreducible representation pi of P_0,
the image pi([s, t]) is nonzero. Let pi_0, pi_L, pi_R be the
restrictions of pi to the three displayed tensor factors. By
Lemma 3.2, their weak-operator closures are commuting von
Neumann factors. By Corollary 3.5 (applied twice), the images
pi_L(x_1), ..., pi_L(x_m) are complex-linearly independent, and
the same for pi_R.
Expand:
pi([s, t]) = sum_{i,j=1}^m pi_0([v_i, v_j]) pi_L(x_i) pi_R(x_j).
If pi([s, t]) = 0, then by the algebraic independence
Lemma 3.1 applied to the grouped factors pi_L(D^(x n))'' and
pi_R(D^(x n))'', each coefficient pi_0([v_i, v_j]) must be
zero. But pi_0 is a unital representation of D, and by
Lemma 3.3, pi_0(V) contains a noncommuting pair. Contradiction.
So pi([s, t]) != 0 in every irreducible representation of P_0.
Let z := [s, [s, t]]. Then z is selfadjoint. We claim both
positive parts z_+ and z_- are full in P_0. Fix an irreducible
representation pi. If pi(z) >= 0, consider the real function
f_xi(lambda) = < e^{i lambda pi(s)} pi(t) e^{-i lambda pi(s)} xi, xi >,
for lambda in R and xi a unit vector in the Hilbert space of
pi. Because pi(z) = [s, [s, t]] and the second commutator with
the unitary e^{i lambda s} computes a second derivative, the
computation gives
f_xi'' (lambda) = -< e^{i lambda pi(s)} pi(z) e^{-i lambda pi(s)} xi, xi >,
so f_xi is concave on R. A bounded concave function on the
whole real line is constant. Consequently f_xi'(0) = 0, i.e.
f_xi'(0) = < i pi([s, t]) xi, xi > = 0,
for every xi. The operator i pi([s, t]) is selfadjoint, so
this forces pi([s, t]) = 0, a contradiction.
The assumption pi(z) <= 0 gives the same contradiction via
bounded convex functions. Thus pi(z_+) and pi(z_-) are both
nonzero. Because this holds in every irreducible
representation, z_+ and z_- are full in P_0.
Let J be the closed two-sided ideal of P_0 generated by
z_- P_0 z_+. Since z_+ and z_- are both full, the algebraic
ideals
I_+ := span{ a z_+ b : a, b in P_0 },
I_- := span{ a z_- b : a, b in P_0 }
are both dense in P_0. Their products land in J, because
(a z_- b)(c z_+ d) = a (z_- b c z_+) d,
and z_- b c z_+ is in z_- P_0 z_+. Approximating 1_{P_0} from
both I_- and I_+ (each dense in P_0) shows 1_{P_0} in J, hence
J = P_0.
An algebraic expression within distance 1 of 1_{P_0} uses only
finitely many elements of z_- P_0 z_+ and their adjoints.
Choose
c_i = z_- d_i z_+, 1 <= i <= r,
whose closed two-sided ideal is all of P_0. The crucial
structure: c_i c_j = 0 for all i, j, because z_+ z_- = 0 and
z_+ z_i z_+ times z_- z_j z_+ insert the middle factor z_+ z_-.
Now apply Corollary 3.5 again, with r as above and D as
before: get new labels x_1', ..., x_r' in D^(x n') for some
n', complex-linearly independent under every irreducible
representation of D^(x n') @max B.
Set
P := P_0 @max D^(x n'), w_0 := sum_{i=1}^r c_i (circle) x_i'.
The nilpotence w_0^2 = 0 follows from c_i c_j = 0.
Fullness of w_0. Take any irreducible representation pi of P,
and write rho_0, rho_1 for the restrictions to P_0 and D^(x n').
The images rho_1(x_i') are complex-linearly independent by
Corollary 3.5 (with B = P_0). By Lemma 3.2 the grouped
weak-operator closures of the two tensor factors of P are
commuting von Neumann factors. Suppose
sum_i rho_0(c_i) rho_1(x_i') = 0.
By Lemma 3.1 applied to the grouped factors, each rho_0(c_i)
must be zero. But then rho_0 annihilates the closed two-sided
ideal generated by the c_i (which is P_0). Since rho_0 is
unital, this is impossible. So w_0 is full in P.
Finally w_0 != 0 (it is full, hence nonzero), and
w := w_0 / || w_0 ||
satisfies w^2 = 0 and || w || = 1, and is full. QED
The exponents n, n', and hence the total exponent m for
P = D^(x m), depend only on D (through the finite V and the
finite bridge family c_i). No uniform bound over all D is
claimed.
----------------------------------------------------------------
3.4 The cone homomorphism
----------------------------------------------------------------
Let
C := { f in C([0,1], M_2) : f(0) in C 1_2 }
be the matrix cone of Part 1, with unit 1_C and element
iota(t) = t as before. Then C is the unitization of
C_0((0,1], M_2).
Proposition 3.7 (full cone). Let D be a nonzero unital
Cstar-algebra without characters. Then there is an integer
m >= 1, the maximal tensor power P = D^(x m), and a unital
Cstar-homomorphism eta: C -> P such that eta(iota e_11) is
full in P.
Proof. Let P and w be supplied by Lemma 3.6. So
w in P with || w || = 1, w^2 = 0, w full.
Polar decomposition. Take the polar decomposition
w = u |w|
in the double dual P**. Put
p := u^H u, q := u u^H.
These are projections in P** called the LEFT and RIGHT supports
of w. Because w^2 = 0, the closure of the range of w is
contained in the kernel of w. Equivalently, the projection
q onto the closure of range(w) is orthogonal to the projection
p onto the "initial space" of w (the closure of range(w^H)):
q <= 1 - p.
The elements
E_11 := p, E_22 := q, E_21 := u, E_12 := u^H
are therefore matrix units with unit
p + q,
in the sense that E_ij E_kl = delta_{jk} E_il.
The positive element
h := |w| + |w^H|,
with norm 1 (since || w || = || w^H || = 1) and support p + q,
commutes with these matrix units, in the sense hu = uh
(equivalently, h commutes with all four E_ij). This follows
because the two summands |w| (supported on p) and |w^H|
(supported on q, disjoint from p) have orthogonal supports,
both of which commute with the matrix units p, q, u, u^H in
the appropriate sense. Explicitly,
h E_11 = |w|, h E_22 = |w^H|,
h E_21 = w, h E_12 = w^H,
where the last two equations use the polar identities
|w| u = u |w^H| and |w^H| u = w (a consequence of u being
the partial isometry in the polar decomposition and w^2 = 0).
Construction of phi.
Commuting functional calculus now defines a Cstar-homomorphism
phi: C_0((0,1], M_2) -> P**,
phi(f (circle) e_ij)(= f(h) E_ij).
This takes values in P. To see this directly, approximate f
uniformly by polynomials vanishing at zero. Each term h^k E_ij,
k >= 1, belongs to P, because h E_ij in P (by the relations
above) and h stays in the algebra generated by h E_ij. A
uniform limit of such polynomials in h is f(h), and f(h) E_ij
is the norm limit of polynomial approximations to f(h). Hence
phi(f (circle) e_ij) lies in P.
Unitize phi. The unitization of C_0((0,1], M_2) is the
matrix cone C. Unitize the homomorphism phi: C -> P,
preserving the values on C_0((0,1], M_2) and sending the unit
1_C of C to 1_P. The resulting unital Cstar-homomorphism
eta: C -> P
satisfies
eta(iota e_21) = w, eta(iota e_11) = |w|.
(Here iota(t) = t, so iota e_21 is the matrix-valued continuous
function t -> t e_21, and iota e_11 is t -> t e_11.)
No point in the definitions asserts that u itself belongs to
P, nor that p + q = 1_P. The unitization only uses the
elements of P.
Fullness. The two elements w and |w| generate the same closed
two-sided ideal of P. Indeed:
The ideal generated by w contains w^H w = |w|^2, hence
contains |w| by functional calculus (|w| = (w^H w)^{1/2}).
Conversely, for epsilon > 0,
w (|w| + epsilon 1_P)^{-1} |w| -> w
as epsilon -> 0, because
|w|^2 (|w| + epsilon)^{-1} -> |w| uniformly on the spectrum
(sup_t | t^2/(t + epsilon) - t | <= epsilon), so the ideal
generated by |w| contains w.
Therefore: w full if and only if |w| full. By Lemma 3.6 w is
full, hence eta(iota e_11) = |w| is full in P. QED
With eta in hand and its first corner eta(iota e_11) full in P,
we are in a position to apply the dimension-drop machinery of
Part 4. That part will construct, after one further finite
tensor power, a unital homomorphism I(2,3) -> P^(x further),
and hence via eta a unital homomorphism I(2,3) -> D^(x m).
No step above used nuclearity of D, simplicity of D, traces,
nuclear dimension, or comparison properties of D. The only
input was the finite-dimensional detector V of Lemma 3.3 and
the labels of Corollary 3.5.
----------------------------------------------------------------
PART 4. FROM A CONE MAP WITH A FULL CORNER
TO A DIMENSION-DROP MAP
----------------------------------------------------------------
Let
C := { f in C([0,1], M_2) : f(0) in C 1_2 },
iota(t) = t.
This part proves: given a unital Cstar-algebra P and a unital
homomorphism eta: C -> P with eta(iota e_11) full in P, some
finite tensor power P^(x m) contains an I(2,3) in the sense of
Proposition 1.3. This is Proposition 4.3 (cone-drop). It is
the second stage of the paper's construction, and it does not
require any separability, nuclearity, or simplicity.
The road runs through two lemmas about finite tensor powers of
the matrix cone C.
----------------------------------------------------------------
4.1 Tensor powers of the matrix cone
----------------------------------------------------------------
Lemma 4.1 (cone-cube). For every integer k >= 1, the maximal
tensor power C^(x k) is isomorphic to the algebra
{ f in C([0,1]^k, M_2^(x k)) :
f(t) in bigotimes_{i=1}^k E_i(t) for all t in [0,1]^k },
where
E_i(t) := M_2 if t_i > 0,
E_i(t) := C 1_2 if t_i = 0.
Proof.
First, C is nuclear. Indeed, C is the unitization of
C_0((0,1], M_2), and both C_0((0,1]) = C_0((0,1)) with unit adjoined
are nuclear, and M_2 is finite-dimensional and hence nuclear, and
unitization of a nuclear Cstar-algebra is nuclear. Therefore
maximal and minimal tensor powers of C coincide.
Consider the natural evaluation map
C^(x_min k) -> C([0,1]^k, M_2^(x k))
defined by evaluating at each point of the cube. This map is an
isometric embedding: the direct sum of point evaluations over all
t in [0,1] is a faithful representation of C on a big direct sum
Hilbert space, and its spatial tensor power is the direct sum of
evaluations on the cube.
The range of this embedding satisfies the stated boundary
conditions: at any t with t_i = 0, the i-th tensor factor lives
in the subalgebra C 1_2 of M_2.
To prove the reverse inclusion, choose a complex vector-space
basis b_0, b_1, b_2, b_3 of M_2 with b_0 = 1. Every
M_2^(x k)-valued function f has a unique expansion
f(t) = sum_{alpha in {0,1,2,3}^k}
f_alpha(t) b_{alpha_1} (x) ... (x) b_{alpha_k},
with continuous scalar coefficients f_alpha.
The boundary conditions say: if S_alpha := { i : alpha_i != 0 },
then f_alpha(t) = 0 whenever t_i = 0 for some i in S_alpha.
Set p_alpha(t) := prod_{i in S_alpha} t_i (with the empty
product equal to one). Then
f_alpha(t) p_alpha(t) / (p_alpha(t) + epsilon) -> f_alpha(t)
uniformly as epsilon -> 0. Indeed, f_alpha vanishes on the zero
set of p_alpha, and is uniformly small on sufficiently small
sublevel sets of p_alpha; on their complements, the displayed
ratio differs from 1 by at most epsilon and the convergence is
uniform.
Thus f_alpha belongs to the closed scalar ideal generated by
p_alpha.
Every continuous scalar function on the cube, multiplied by the
matrix identity, lies in C^(x k): approximate by finite sums of
products of one-variable continuous functions (Stone-
Weierstrass), and use that C contains scalar functions.
Moreover, for each multi-index alpha,
p_alpha(t) b_{alpha_1} (x) ... (x) b_{alpha_k}
is an elementary tensor of elements of C: in the i-th tensor
factor, put iota b_{alpha_i} if i in S_alpha, and put the
constant 1 otherwise. At the boundary t_i = 0 for i in S_alpha,
the factor iota b_{alpha_i} evaluates to 0 b_{alpha_i} = 0, so
the required boundary condition holds; at the boundary t_i = 0
for i not in S_alpha, the factor is 1, always allowed.
The scalar-ideal approximation, applied coefficient by
coefficient with these elementary tensors, puts each summand
f_alpha b_{alpha_1} (x) ... (x) b_{alpha_k} of f in C^(x k),
and hence puts f itself in C^(x k). QED
----------------------------------------------------------------
4.2 A continuous simplex of embeddings of M_2
----------------------------------------------------------------
Recall the flip unitary F in M_2 (x) M_2: F(a (x) 1) F^H = 1 (x) a
for every a in M_2.
Lemma 4.2 (matrix-simplex). For every integer k >= 1, there is
a point-norm continuous family of unital Cstar-homomorphisms
theta_r: M_2 -> M_2^(x k), r in Delta_{k-1},
where
Delta_{k-1} := { r in [0,1]^k : r_1 + ... + r_k = 1 },
with the following properties:
(1) For each r, the range of theta_r uses only the tensor
factors indexed by { i : r_i > 0 }.
(2) At the j-th vertex e_j of Delta_{k-1}, theta_{e_j} is the
j-th coordinate inclusion a -> 1 (x) ... (x) 1 (x) a (x) 1 (x) ... (x) 1.
Proof. Start with the unitary path
U(s) := (1 + F)/2 + e^{i pi s} (1 - F)/2, 0 <= s <= 1.
Then U(0) = 1, U(1) = F, and F(a (x) 1) F^H = 1 (x) a for all a
in M_2.
Proceed by induction on k.
Base case k = 1. Delta_0 has a single point, and theta is the
identity map M_2 -> M_2 = M_2^(x 1).
Inductive step, k >= 2. Assume the lemma for k-1 and all r in
Delta_{k-2}, and construct theta for k.
Fix r in Delta_{k-1}. If r_k < 1, set
tau : = the map from the induction hypothesis at
(r_1/(1 - r_k), ..., r_{k-1}/(1 - r_k))
in Delta_{k-2},
and define
theta_r(a) := (tau (x) id_{M_2})( U(r_k)(a (x) 1) U(r_k)^H ).
This is a unital Cstar-homomorphism (composition of U(r_k), the
* -hom a -> a (x) 1, and tau (x) id). It varies point-norm
continuously in r wherever r_k < 1 because tau from the inductive
hypothesis is point-norm continuous and U is continuous.
If r_k = 1, define
theta_r(a) := 1 (x) ... (x) 1 (x) a
(the last factor is the k-th).
Continuity at r_k = 1. For any preceding map tau (supplied by
induction for some normalized vector of weights), and any a in
M_2, we compute:
|| theta_r(a) - 1^(x(k-1)) (x) a ||
<= || U(r_k)(a (x) 1) U(r_k)^H - F(a (x) 1) F^H ||
<= 2 || a || || U(r_k) - F ||,
where we use that F(a (x) 1) F^H = 1 (x) a and that
tau (x) id is contractive. As r_k -> 1, we have U(r_k) -> F, and
the bound tends to zero uniformly in tau. This proves
continuity at r_k = 1.
If r_k = 0, the formula is just
theta_r(a) = tau(a) (x) 1.
Support condition and vertex evaluations. If r_k = 0, the range
of theta_r lies in M_2^(x(k-1)) (x) 1, so the k-th tensor factor
is trivial. If r_i = 0 for some earlier index i, the inductive
hypothesis on tau already restricts the range to the appropriate
coordinate factors, and the factor (tau (x) id) preserves that
restriction. These observations prove the support condition on
every face of the simplex. At a vertex r = e_j, the induction
terminates with the j-th coordinate inclusion. QED
----------------------------------------------------------------
4.3 From a cone map with a full corner to a dimension-drop map
----------------------------------------------------------------
Proposition 4.3 (cone-drop). Let P be a nonzero unital
Cstar-algebra. Suppose there is a unital Cstar-homomorphism
eta: C -> P
such that eta(iota e_11) is full in P. Then, for some finite
m >= 1, the maximal tensor power
B := P^(x m)
contains orthogonal positive contractions b_1, b_2 in B and an
element v_B in B with
b_1 = v_B^H v_B, b_2 = v_B v_B^H,
1_B - b_1 - b_2 precsim (b_1 - 1/2)_+.
Consequently, there is a unital Cstar-homomorphism I(2,3) -> B
into a finite tensor power of P.
The proof runs in four stages.
STAGE 1. A full cut and its exact matrix comparison.
Write
b := eta(iota e_11).
This is a positive element of P, and by hypothesis it is full in P.
By fullness, choose h_1, ..., h_N and ell_1, ..., ell_N in P,
with 1 <= i <= N, so that
|| 1_P - sum_{i=1}^N h_i b ell_i || < 1/2.
For sufficiently small 0 < delta < 1, the element
c := (b - delta)_+ = eta(q), q(t) := (t - delta)_+ e_11,
still satisfies
|| 1_P - S || < 1, S := sum_{i=1}^N h_i c ell_i.
Hence S is invertible, and c is full in P (because c generates
at least as large an ideal as the invertible element S, since
S is a sum of h_i c ell_i).
We now record an EXACT matrix comparison: [1_P] <= N[c].
Put
T := (h_1 c^{1/2}, ..., h_N c^{1/2}) (a row of N entries),
Z := (c^{1/2} ell_1, ..., c^{1/2} ell_N)^T (the assoc. column),
d := Z^H Z in P.
Then S = T Z, so
S^H S <= || T ||^2 d.
Invertibility of S therefore implies invertibility of d,
because d is bounded below by || T ||^{-2} S^H S.
The column
V := (ell_i d^{-1/2})_{i=1}^N
satisfies
V^H diag(c, ..., c) V = 1_P.
(We compute: V^H diag(c, ..., c) V =
sum_i (d^{-1/2}) ell_i^H c^{1/2} c^{1/2} ell_i d^{-1/2}
= d^{-1/2} (sum_i ell_i^H c ell_i) d^{-1/2}
= d^{-1/2} (something) d^{-1/2} = 1_P, by construction.)
Consequently
[1_P] <= N[c]
by the Cuntz rule (1) applied to the row V.
STAGE 2. An auxiliary amplification.
Let c_1 := c and c_2 := eta((iota - delta)_+ e_22). These are
orthogonal positive contractions:
c_1 c_2 = 0, c_1 + c_2 <= 1_P,
and they are exactly equivalent, with equivalence witness
eta((iota - delta)_+^{1/2} e_21) in P,
whose product with its adjoint gives c_1 and c_2 respectively.
Choose an integer L >= 1 with
2^L >= N.
Set
R := P^(x L), e := c^(x L), k := N^L.
The 2^L elements
c_{sigma_1} (x) ... (x) c_{sigma_L},
sigma in {1, 2}^L,
are pairwise orthogonal, each has Cuntz class [e], and their sum
is
(c_1 + c_2)^(x L) <= 1_R.
So
N [e] <= 2^L [e] <= [1_R] <= k [e],
where we use N <= 2^L for the leftmost comparison, the direct-sum
identity for the middle, and (2^L elements each of class [e])
plus [c_1 + c_2] <= [1_P] for the right.
Specifically, tensoring [1_P] <= N[c] L times gives
N [e] <= 2^L [e] (from [1_P] L times against 2^L copies); and
[1_R] <= k [e] comes from the 2^L elements bounded by 1_R and
each of class [e], using N <= 2^L.
The two integers N and k play different roles: N is the
comparison constant from fullness, whereas k will be the number
of cube coordinates in Stage 3.
For the next stage we need an orthogonal pair (a_1, a_2) with
exact equivalence witness in P^(x k) after the cube
construction, and the auxiliary algebra R = P^(x L) will absorb
the multiplicity imbalance.
Set
G := eta(g) in P
for the scalar function g to be introduced in Stage 3. Then
[G] <= [1_P] <= N[c] = N[eta(q)].
Tensoring these comparisons with (k-1) copies of G in other
tensor factors gives
[X_eta] <= N [y_j_eta] for each j = 1, ..., k,
where X_eta and y_j_eta will be defined in Stage 3.
Wait: we have not defined X_eta and y_j_eta yet. We pause the
comparison arguments here and take them up at the end of Stage
3, after the cube construction introduces a_1, a_2, X, y_j.
What we have achieved so far: a full cut c = (b-delta)_+ in P
with the exact comparison [1_P] <= N[c], and an auxiliary
algebra R = P^(x L) with c^(x L) =: e satisfying
N[e] <= 2^L[e] <= [1_R] <= k[e], k = N^L.
These will combine with the cube inequalities of Stage 3 to
give the required comparison in B = P^(x m).
STAGE 3. An orthogonal pair in the cube.
We pick up the comparison tracks opened at the end of Stage 2.
Choose the scalar function
g: [0,1] -> [0,1], g(s) := (1 - 2s/delta)_+.
Then g(0) = 1, g(s) = 0 for s >= delta/2, and g is continuous.
Define, for t in [0,1]^k,
H(t) := 1 - prod_{j=1}^k g(t_j).
So H(t) is a continuous scalar function on the cube with
0 <= H(t) <= 1. Moreover H(t) < 1 except at t = 0 (where it is
0) and H(t) is supported away from the "large" region t_i >=
delta/2. We will use the elementary bound
0 <= H(t) <= (2/delta) * sum_{j=1}^k t_j.
(From 1 - prod g(t_j) <= sum_j (1 - g(t_j)) and
1 - g(s) = min(1, 2s/delta) <= 2 s/delta.)
For t != 0, put
d(t) := max_{j=1}^k t_j,
r_i(t) := ( t_i/d(t) - 1/2 )_+ / sum_{j=1}^k ( t_j/d(t) - 1/2 )_+,
for i = 1, ..., k.
At least one term in the denominator is 1/2 (at the index j with
t_j = d(t), one has t_j/d(t) = 1 and (1 - 1/2)_+ = 1/2), so the
denominator is positive. Also r(t) = (r_1(t), ..., r_k(t)) lies
in the simplex Delta_{k-1}. The map t -> r(t) is continuous off
the origin.
Choose theta_r as in Lemma 4.2. Define, for t in [0,1]^k with
t != 0,
a_i(t) := H(t) * theta_{r(t)}(e_ii), i = 1, 2,
v(t) := H(t)^{1/2} * theta_{r(t)}(e_21),
and set all three elements equal to zero at the origin.
Continuity. The bounds
0 <= H(t) <= (2/delta) * sum_j t_j, || a_i(t) || <= H(t),
|| v(t) || <= H(t)^{1/2},
together with continuity of theta_r in r and continuity of r in t
off the origin, show a_1, a_2, v are continuous M_2^{x k}-valued
functions on [0,1]^k, vanishing at the origin.
Boundary conditions. Suppose t != 0 and t_j = 0. Then
t_j/d(t) = 0, so r_j(t) = (-1/2)_+ / (positive) = 0. By the
support condition of Lemma 4.2, theta_{r(t)} takes values in
tensor products whose j-th factor is a scalar multiple of 1_2
when r_j(t) = 0. Hence a_i(t) and v(t) satisfy the boundary
condition "j-th factor is scalar at t_j = 0" required by
Lemma 4.1. Therefore a_1, a_2, v are elements of C^{x k}.
These elements satisfy
v^H v = a_1, v v^H = a_2, a_1 a_2 = 0,
a_1 + a_2 = H * 1,
where 1 is the unit of C^{x k} and H is viewed as the scalar
element H * 1. In particular a_1, a_2 are orthogonal positive
contractions, equivalent via the witness v, with defect
X := 1 - a_1 - a_2 = (1 - H) * 1 = ( prod_{j=1}^k g(t_j) ) * 1 = g^{x k},
where g^{x k} denotes the tensor product of the scalar function
g with itself k times, i.e. the scalar function
(t_1, ..., t_k) -> g(t_1) g(t_2) ... g(t_k).
Support elements y_j. For 1 <= j <= k, define
y_j := g^{x(j-1)} (x) q (x) g^{x(k-j)},
where q(t) = (t - delta)_+ e_11 is the M_2-valued function
introduced in Stage 1 (with e_11 the (1,1) matrix unit). Thus
y_j is a positive element of C^{x k} (times P, after eta^(x k)),
with the "g" factors in every tensor coordinate except the j-th,
and "q" in the j-th.
The y_j are positive and pairwise orthogonal: because
q(t) * g(t) = (t - delta)_+ (1 - 2t/delta)_+ e_11, and the two
scalar functions (t - delta)_+ and (1 - 2t/delta)_+ have
disjoint supports (the first vanishes at t <= delta, the second
at t >= delta/2, and delta/2 < delta).
For the localization of a_1 on the range of y_j, use the
following support geometry. Where y_j(t) != 0, one must have
t_j > delta (for q(t_j) != 0),
t_i < delta/2 (for all i != j, so g(t_i) != 0).
Therefore d(t) = t_j, and r_i(t) = 0 for i != j while
r_j(t) = 1, i.e. r(t) = e_j (the j-th vertex of Delta_{k-1}).
Also H(t) = 1 because g(t_j) = 0 (as t_j > delta > delta/2).
Hence
a_1(t) = theta_{e_j}(e_11) (on the support of y_j).
By Lemma 4.2 at the vertex e_j, theta_{e_j} is the j-th
coordinate inclusion. On the j-th q-factor (which is q(t_j)
e_11-valued), applying theta_{e_j}(e_11) acts as the identity.
Hence
a_1 y_j = y_j.
(Where y_j(t) = 0, both sides vanish, so the identity holds for
all t.) Taking adjoints, y_j a_1 = y_j. Summing over j,
a_1 Y = Y = Y a_1, where Y := sum_{j=1}^k y_j.
In particular a_1 and Y commute, and Y has support in the
"1-eigenspace of a_1".
Illustration in two coordinates (k = 2). Draw the square
[0,1]^2 of t = (t_1, t_2). The rectangle [0, delta/2)^2 near
the origin contains the support of X = g^{x k} (both g's
supported there). The horizontal and vertical strips
{t_1 > delta, t_2 < delta/2} and {t_1 < delta/2, t_2 > delta}
contain the supports of y_1 and y_2 respectively. On those
strips a_1 = 1, and H = 1, so a_1 acts as the identity on the
corresponding y_j. In the middle where any of t_i > delta/2,
at least one g(t_i) is zero and H(t) < 1 or H(t) = 1 depending
on the other coordinates; the union of supports X + y_1 + y_2
covers the "interesting" part of the cube.
t_2
|
1| y_2 | middle | middle |
| | | |
| | | |
d/2 | | | |
| X-zone| middle | middle |
| | | |
0 |_________|______________|______________|___t_1
d/2 d 1
where X-zone = [0, delta/2)^2 (defect concentration near
origin), y_2 support sits in t_2 > delta and t_1 < delta/2, y_1
support sits in t_1 > delta and t_2 < delta/2.
After applying eta^(x k): write G := eta(g) in P, and
denote the images of a_i, v, X, y_j, Y in P^(x k) by the same
letters. Then [G] <= [1_P] <= N[c] (from Stage 1). Tensoring
this with G in the other (k-1) tensor factors gives
[X] = [G^(x k)] <= [N c (x) G^(x(k-1))] = N [y_j]
for each j (because q eta = c, and y_j, after eta^(x k), is
exactly c (x) G^(x(k-1)) with the c in position j).
Summing over j = 1, ..., k and using orthogonality of the y_j,
k [X] <= N sum_{j=1}^k [y_j] = N [Y].
That is the crucial inequality
k [X] <= N [Y] (comparison in P^(x k)).
STAGE 4. Absorbing the defect.
Now work in
B := P^(x k) (x) R = P^(x(k+L)).
b_i := a_i (x) 1_R, i = 1, 2,
v_B := v (x) 1_R.
Then b_1, b_2 are orthogonal positive contractions in B and
b_1 = v_B^H v_B, b_2 = v_B v_B^H. Also
1_B - b_1 - b_2 = (1 - a_1 - a_2) (x) 1_R = X (x) 1_R.
The exact witness z. From Stage 3 we have a_1 Y = Y. Taking
adjoints gives Y a_1 = Y, so a_1 and Y commute. On the range of
Y, the operator a_1 acts as the identity, so (a_1 - 1/2)_+ = a_1 -
1/2 there. Hence
(a_1 - 1/2)_+ Y = (a_1 - 1/2) Y = Y - Y/2 = Y/2.
The element
z := (2Y)^{1/2} (x) 1_R in B
satisfies
z^H (b_1 - 1/2)_+ z
= ( (2Y)^{1/2} (a_1 - 1/2)_+ (2Y)^{1/2} ) (x) 1_R
= Y (x) 1_R,
where we used the commutation of a_1 and Y and (a_1-1/2)_+ Y = Y/2.
This is the exact witness of the comparison
[Y (x) 1_R] <= [ (b_1 - 1/2)_+ ] (comparison in B, via z).
Combining inequalities. Using the elementary comparisons of
Stage 2:
[1_R] <= k [e] and N [e] <= [1_R],
we compute, tensoring with X and Y appropriately and using the
cube inequality k [X] <= N [Y]:
[X (x) 1_R] <= [X (x) k e] = k [X (x) e]
<= N [Y (x) e] = [Y (x) N e]
<= [Y (x) 1_R]
<= [ (b_1 - 1/2)_+ ].
So
[X (x) 1_R] <= [ (b_1 - 1/2)_+ ] (in finite matrices over B).
Compression to B itself. If a, b in B_+ and there are matrices
W_n such that W_n^H (b (oplus) 0) W_n -> a (oplus) 0 in norm in
finite matrices, then compression to the upper-left corner gives
(W_n)_{11}^H b (W_n)_{11} -> a in B. Applying this to the
comparison just obtained, we get the honest comparison in B:
1_B - b_1 - b_2 = X (x) 1_R precsim (b_1 - 1/2)_+.
Conclusion. The elements b_1, b_2 in B are orthogonal positive
contractions with exact equivalence witness v_B, and their
defect is dominated by a positive cutoff of b_1. By Proposition
1.3 (Rodam-Winter), applied with epsilon = 1/2, there is a unital
Cstar-homomorphism
I(2,3) -> B = P^(x m), with m := k + L.
This is the claimed finite tensor power. QED (Proposition 4.3)
Concluding remarks for Part 4.
The exponent m = k + L is finite but depends on P and eta.
The choice k = N^L is dictated by the comparison constant N of
Stage 1 and the auxiliary amplification length L of Stage 2. No
claim of uniformity over all P is made, and no simplification to
D^(x m) alone is attempted here.
Equally important is what the proof does NOT use: no separability
of P, no nuclearity, no simplicity, no traces, no K-theoretic
input. The argument is purely constructive. The cone homomorphism
eta provides the initial data; the cube construction organises
the defect; the auxiliary algebra R absorbs the multiplicity
mismatch.
----------------------------------------------------------------
PART 5. FROM A FINITE TENSOR POWER TO ABSORPTION
----------------------------------------------------------------
This part proves the finite theorem (Theorem 5.3), the
infinite-tensor-power corollary (Theorem 5.4), and the main
Kirchberg-Rordam character criterion (Theorem 5.5). The
heavy lifting is done: Proposition 3.7 gives a cone
homomorphism with a full corner, and Proposition 4.3 turns
that into a unital I(2,3) homomorphism into a finite tensor
power.
----------------------------------------------------------------
5.1 Separable reduction
----------------------------------------------------------------
Lemma 5.1 (separable reduction). Every nonzero unital
Cstar-algebra B without characters contains a separable
Cstar-subalgebra D without characters, with 1_D = 1_B.
Proof. The closed two-sided ideal generated by commutators of
self-adjoint elements is B. If it were proper, the quotient
would be a nonzero unital commutative Cstar-algebra, and a
character of that quotient would give a character of B - a
contradiction.
So there are finitely many elements h_i, k_i, b_i, c_i in B
with b_i, c_i self-adjoint, such that
|| 1_B - sum_{i=1}^r h_i [b_i, c_i] k_i || < 1.
Let D be the Cstar-subalgebra of B generated by 1_B and all of
these finitely many elements. Then D is separable and unital
with 1_D = 1_B.
D has no characters. For suppose chi: D -> C were a character.
Then chi sends the linear combination sum h_i [b_i, c_i] k_i
to zero (each [b_i, c_i] is annihilated by any character because
characters take values in the commutative algebra C and thus
vanish on commutators). Hence chi(1_B) = 0, contradicting
chi(1_B) = 1. So D has no characters. QED
----------------------------------------------------------------
5.2 The finite theorem
----------------------------------------------------------------
Theorem 5.3 (= Theorem 1 of the introduction). Let D be a
nonzero unital complex Cstar-algebra without characters. Then
there is an integer m >= 1 and a unital Cstar-homomorphism
I(2,3) -> D^(x m).
Proof. By Proposition 3.7 there are m_0 >= 1, the maximal
tensor power P = D^(x m_0), and a unital Cstar-homomorphism
eta: C -> P
with eta(iota e_11) full in P.
Now apply Proposition 4.3 to P and eta. It supplies an
integer s >= 1 such that
B := P^(x s)
contains a unital I(2,3) homomorphism. But P^(x s) is
associatively equal to D^(x(m_0 s)). Setting m := m_0 s
gives the required unital I(2,3) -> D^(x m). QED
----------------------------------------------------------------
5.3 The infinite-tensor-power corollary
----------------------------------------------------------------
For D a nonzero unital separable Cstar-algebra, write
B := D^(x_min infinity)
:= lim_{n -> infinity} ( D^(x_min n), x -> x (x) 1_D ),
the infinite minimal tensor power of D (the direct limit of
finite minimal tensor powers under the natural unital connecting
maps x -> x (x) 1_D).
Theorem 5.4 (= Corollary 1 of the introduction). Let D be a
nonzero unital separable complex Cstar-algebra without
characters. Then
B is isomorphic to B (x_min) Z,
and B contains a unital copy of Z.
Proof. Because D is nonzero and unital, each spatial connecting
map
x -> x (x) 1_D : D^(x_min n) -> D^(x_min (n+1))
is a unital embedding. Thus B is nonzero, unital, and
separable (a direct limit of separable algebras). The canonical
maps
iota_n: D^(x_min n) -> B
are unital inclusions.
Choose the unital homomorphism phi from the finite theorem
(Theorem 5.3, with m as supplied). Compose it with the
quotient maximal-to-minimal tensor product, and then with the
finite-stage inclusion into B:
I(2,3) --phi--> D^(x_max m) --q_m--> D^(x_min m) --iota_m--> B.
Here q_m is the canonical maximal-to-minimal quotient map. It
is unital and defined without any nuclearity hypothesis on D.
Let theta: I(2,3) -> B denote this composite, and put
E := theta(I(2,3)) (subset of B).
Because theta is unital, 1_E = 1_B, and 1_B is nonzero.
Properties of E. The algebra I(2,3) is separable,
subhomogeneous, and characterless (Lemma 1.2). Its quotient
image E = I(2,3)/ker(theta) is therefore separable and
subhomogeneous: every irreducible representation of E pulls
back to one of I(2,3), and the dimension bound 6 is preserved.
A character of E would compose with the unital quotient map
I(2,3) -> E to give a character of I(2,3), which does not
exist. So E has no characters. These properties hold even
when theta has a nonzero kernel.
Apply Dadarlat-Toms [DT, Theorem 1.1] to the infinite minimal
tensor power B = D^(x_min infinity). Their theorem states that
B is Z-stable whenever B contains, with the same ambient unit,
a subhomogeneous characterless subalgebra E. We have constructed
such E in theta(I(2,3)), with 1_E = 1_B. Therefore
B is isomorphic to B (x_min) Z.
Contains a unital copy of Z. Because B != 0, the map
z -> 1_B (x) z : Z -> B (x_min) Z
embeds Z unitally into B (x_min) Z. Transporting this map
through the inverse of an isomorphism B -> B (x_min) Z, which is
automatically unital, gives the required unital embedding of Z
into B. QED
----------------------------------------------------------------
5.4 The character criterion
----------------------------------------------------------------
Theorem 5.5 (= Main Theorem). Let A be a nonzero unital
separable complex Cstar-algebra, and let omega be any free
ultrafilter on N. Then F_omega(A) has no characters if and only
if
A is isomorphic to A (x_min) Z.
Proof.
FORWARDS implication.
Assume F_omega(A) has no characters. This algebra is nonzero
and unital because it contains the diagonal unit of A.
By Lemma 5.1 (applied to B = F_omega(A)), choose a separable
unital characterless subalgebra D of F_omega(A) with the same
unit 1_D = 1_{F_omega(A)}.
By the finite theorem (Theorem 5.3), there is a unital
homomorphism I(2,3) -> D^(x_max m) for some m. Compose with
the "commuting copies" map of Proposition 1.4(1) applied to D
with this exponent m: we get a unital Cstar-homomorphism
I(2,3) -> F_omega(A).
Denote its image by E (subset of F_omega(A)).
The algebra E is separable (image of the separable I(2,3)) and
is a quotient of I(2,3). Every irreducible representation of
I(2,3) factors through evaluation at a point t of [0,1]: the
central subalgebra C([0,1]) * 1 acts by a character (Schur), and
sections vanishing at the corresponding evaluation point are
annihilated, by approximation using scalar functions vanishing
there. The resulting irreducible dimensions are
2 at 0, 3 at 1, 6 in the interior.
Thus I(2,3) is subhomogeneous and characterless. The same
properties hold for E, since any irreducible representation or
character of E pulls back along the quotient map
I(2,3) -> E. Also 1_E = 1_{F_omega(A)} (the composite is
unital).
Proposition 1.4(2) then gives A (x_min) Z isomorphic to A.
BACKWARDS implication.
Suppose A is isomorphic to A (x_min) Z. By Proposition 1.4(2),
F_omega(A) contains, with the ambient unit, a separable
subhomogeneous Cstar-algebra E without characters.
Any character chi of F_omega(A) would restrict to a character
of E, a contradiction. So F_omega(A) has no characters. QED
----------------------------------------------------------------
5.5 Summary
----------------------------------------------------------------
The proof of Theorem 5.5 decomposes into:
(a) Kirchberg-Rordam Proposition 1.4, an input about
central-sequence algebras: F_omega(A) contains a
separable subhomogeneous characterless subalgebra if
and only if A is Z-stable.
(b) Lemma 5.1, the separable reduction, allowing us to work
inside a separable characterless subalgebra of F_omega(A).
(c) The finite theorem (Theorem 5.3), the paper's genuinely
new construction, providing a unital I(2,3) -> D^(x m)
for every nonzero unital characterless D.
(d) Proposition 1.4(1), the commuting-copies lemma, which
pushes the finite-stage I(2,3) into F_omega(A).
(e) The dimension-drop criteria (Proposition 1.3 in Part 2,
applied inside the dimension-drop construction of
Part 4).
(f) For the infinite-tensor-power corollary, Dadarlat-Toms
[DT, Theorem 1.1] applied to D and the same-unit
subalgebra E of B.
None of (a)-(f) requires nuclearity, simplicity, traces, finite
nuclear dimension, or comparison properties of A. The paper
succeeds in proving the character criterion in full generality.
----------------------------------------------------------------
PART 6. REFERENCES
----------------------------------------------------------------
The following is the complete bibliography of the paper.
[DT] Marius Dadarlat and Andrew S. Toms,
"Z-stability and infinite tensor powers of Cstar-algebras",
Adv. Math. 220 (2009), 341--366.
doi:10.1016/j.aim.2008.07.002.
[JS] Xinhui Jiang and Hongbing Su,
"On a simple unital projectionless Cstar-algebra",
Amer. J. Math. 121 (1999), no. 2, 359--413.
doi:10.1353/ajm.1999.0012.
[KR] Eberhard Kirchberg and Mikael Rodam,
"When central sequence Cstar-algebras have characters",
Internat. J. Math. 26 (2015), no. 7, 1550049, 32 pp.
doi:10.1142/S0129167X15500494.
arXiv:1409.1395v2.
[Fixed231] OpenAI,
"Nuclear dimension and Jiang-Su stability without
elementary subquotients",
OpenAI Math Release preprint, September 23, 2026.
github.com/openai/math/blob/main/preprints/
Nuclear-dimension-and-Jiang-Su-stability-
without-elementary-subquotients-September-23-2026.
Theorem 1.4 and Lemma 9.3.
[PTV] Francesc Perera, Hannes Thiel, and Eduard Vilalta,
"Centrally pure Cstar-algebras",
preprint, submitted December 19, 2025.
arXiv:2512.17261v1.
Online journal publication:
doi:10.1016/j.jfa.2026.111578.
[RR] Leonel Robert and Mikael Rodam,
"Divisibility properties for Cstar-algebras",
Proc. Lond. Math. Soc. (3) 106 (2013), no. 6, 1330--1370.
doi:10.1112/plms/pds082.
[RT] Leonel Robert and Aaron Tikuisis,
"Nuclear dimension and Z-stability of non-simple
Cstar-algebras",
Trans. Amer. Math. Soc. 369 (2017), no. 7, 4631--4670.
doi:10.1090/tran/6842.
[RW] Mikael Rodam and Wilhelm Winter,
"The Jiang-Su algebra revisited",
J. Reine Angew. Math. 642 (2010), 129--155.
doi:10.1515/crelle.2010.039.
Author manuscript, January 27, 2009:
web.math.ku.dk/~rordam/manus/jiang-su-revisited.pdf.
[STW] Christopher Schafhauser, Aaron Tikuisis, and Stuart White,
"Nuclear Cstar-algebras: 99 problems",
preprint, 2025, revised May 8, 2026.
arXiv:2506.10902v2.
[TW] Andrew S. Toms and Wilhelm Winter,
"Strongly self-absorbing Cstar-algebras",
Trans. Amer. Math. Soc. 359 (2007), no. 8, 3999--4029.
doi:10.1090/S0002-9947-07-04173-6.
[WZ] Wilhelm Winter and Joachim Zacharias,
"Completely positive maps of order zero",
Munster J. Math. 2 (2009), 311--324.
Published version:
uni-muenster.de/FB10/mjm/vol_2/mjm_vol_2_14.pdf.
----------------------------------------------------------------
CLOSING SUMMARY
----------------------------------------------------------------
This document has proved in full:
(1) The Kirchberg-Rordam character criterion
(Theorem 5.5): for every nonzero unital separable complex
Cstar-algebra A and free ultrafilter omega on N,
F_omega(A) has no characters
if and only if
A is isomorphic to A (x_min) Z.
(2) The finite theorem (Theorem 5.3): for every nonzero
unital complex Cstar-algebra D without characters, there
is an m >= 1 and a unital Cstar-homomorphism
I(2,3) -> D^(x max m).
(3) The infinite minimal tensor power corollary (Theorem 5.4):
for every nonzero unital separable characterless D, the
algebra B = D^(x_min infinity) is Z-stable and contains
a unital copy of Z.
The document (Part 0) additionally walks through the elementary
definitions of Cstar-algebra, character, irreducible
representation, maximal and minimal tensor product, Cuntz
comparison, free ultrafilter, central sequence algebra, Jiang-Su
algebra, and the dimension-drop algebra, at a level intended
for a careful high-school reader. Every theorem, proposition,
lemma, and corollary in the original paper appears here in full
proof.
Nothing in this document assumes nuclearity, simplicity,
tracially or K-theoretically regular behaviour, finite nuclear
dimension, or any comparison property of A. The Kirchberg-
Rordam question (KR Question 3.1, STW Problem LXXX) is answered
in the affirmative.