NON-Z-STABLE SIMPLE AH ALGEBRAS
by
ALLAN HENDRICKSON
A DISSERTATION
Presented to the Department of Mathematics
and the Division of Graduate Studies of the University of Oregon
in partial fulllment of the requirements
for the degree of
Doctor of Philosophy
September 2023
DISSERTATION APPROVAL PAGE
Student: Allan Hendrickson
Title: Non-Z-Stable Simple AH Algebras
This dissertation has been accepted and approved in partial fulllment of the
requirements for the Doctor of Philosophy degree in the Department of
Mathematics by:
Dr. Huaxin Lin Chair
Dr. N. Christopher Phillips Core Member
Dr. Marcin Bownik Core Member
Dr. Victor Ostrik Core Member
Dr. Brittany Erickson Institutional Representative
and
Dr. Krista Chronister Vice Provost for Graduate Studies
Original approval signatures are on le with the University of Oregon Division of
Graduate Studies.
Degree awarded September 2023.
2
© 2023 Allan Hendrickson
This work is licensed under a Creative Commons
Attribution-ShareAlike (CC BY-SA) License
3
DISSERTATION ABSTRACT
Allan Hendrickson
Doctor of Philosophy
Department of Mathematics
September 2023
Title: Non-Z-Stable Simple AH Algebras
We consider the problem of dimension growth in AH algebras A dened as
inductive limits A = lim (MRn(C(Xn)), ϕn) over nite connected CW-complexes X→∞ n.n
We show that given∣∣ any sequence (Xn) of nite connected CW-complexes and matrixdim(Xn)sizes (Rn) with Rn Rn+1 satisfying the dimension growth condition lim = c
n→∞ Rn
with c ∈ (0,∞), there always exists an AH algebra with injective connecting homo-
morphisms over a subsequence which does not have Blackadar's strict comparison
of positive elements, and therefore does not absorb tensorially the Jiang-Su alge-
bra Z. This demonstrates that no regularity condition can be placed on the spaces
Xn in order to stabilize AH algebras over them - there always exists a pathological
construction.
4
Dedicated to my brothers.
5
TABLE OF CONTENTS
Chapter Page
1. PRELIMINARY NOTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2. AH Algebras and Dimension Growth . . . . . . . . . . . . . . . . . . 9
1.3. Stable and Real Rank . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4. The Cuntz Semigroup and Strict Comparison . . . . . . . . . . . . . 15
1.5. Perforation and the Jiang-Su Algebra Z . . . . . . . . . . . . . . . . 20
2. CONSTRUCTIONS OF GOODEARL, VILLADSEN AND TOMS . . . . . 22
2.1. Goodearl Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2. Villadsen Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3. Villadsen's Chern Class Obstruction . . . . . . . . . . . . . . . . . . 30
2.4. Tracial States in Villadsen Algebras . . . . . . . . . . . . . . . . . . 31
3. GENERALIZATION TO AH ALGEBRAS OVER CW-COMPLEXES . . 35
3.1. Embedded Spheres in the Cube . . . . . . . . . . . . . . . . . . . . . 35
3.2. Prescribed Dimension Growth over CW-Complexes . . . . . . . . . . 47
3.3. Injectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6
CHAPTER 1
PRELIMINARY NOTIONS
1.1 Introduction
At the end of the 1980's, George Elliott conjectured that a large class of
simple, nuclear C∗-algebras can be classied from their K-theory and tracial state
space, along with the pairing between them, known as the Elliott Invariant. For a
unital C∗-algebra A, this is
Ell(A) = ((K0(A), K0(A)+, [1A]), K1(A), T+(A), ρ),
where ρ : K0(A) → T (A) is the pairing by evaluation at a K0 class. One major exam-
ple of interest were the approximately homogeneous (AH) algebras: those obtained
from inductive limits of matrix algebras An = MRn(C(Xn)). Throughout the 1990's,
much progress was made on the overall classication of nuclear C∗-algebras. Among
these major advances included
ˆ Kirchberg-Phillips' classication of purely innite C∗-algebras satisfying the
Universal Coecient Theorem.
ˆ Elliott-Gong-Li's classication of AH algebras of very slow dimension growth.
ˆ Lin's classication of tracially AF algebras, and later in 2003 of real rank zero
algebras.
But, in 2002, Rørdam [Rør03] gave a stark counterexample to Elliott's con-
jecture, exhibiting a simple, nuclear C∗-algebra A which contained both a nite and
innite projection. It was shown that by tensoring with an algebra called the Jiang-
Su algebra, denoted Z and introduced in [JS99], that Ell(A⊗Z) = Ell(A), although
their real rank satised RR(A) ̸= 0, while RR(A⊗Z) = 0.
Meanwhile, the 1998 construction of J. Villadsen in [Vil98] exhibited topolog-
ical obstructions in the form of perforation of the K0 group of certain AH algebras.
These algebras failed to have slow dimension growth: the property that
dim(Xn)
lim = 0,
n→∞ Rn
7
eventually leading to more counterexamples. Results such as those furthered by
Gong-Jiang-Su in [GJS00], showed the connection between perforation and absorb-
ing Z tensorially, deemed Z-stability. It became clear that the Elliott invariant, as it
existed, was not sucient for classication of many seemingly tractible C∗-algebras,
even among AH algebras. In fact, it was later realized Ell(Z) = Ell(C) although
Z ≄ C.
Throughout the 2000's, Toms and Winter conjectured that for AH algebras,
this so-called Z-stability was related to the dimension growth - this quantity
dim(Xn)
lim . They also studied notions of divisibility, nuclear dimension, and com-
n→∞ Rn
parison in the Cuntz semigroup, which generalizes the notion of K-theory of a C∗-
algebra A to general positive elements in A ⊗ K. An important example are the
Villadsen algebras of the rst type, built from coordinate projections over increasing
sequences of spaces Xn = (Xn−1)
kn from some seed space X1 and sequence (kn) in N,
which they analyzed in detail in [TW09].
In this thesis, we investigate inductive limits over algebras An = MRn(C(Xn))
for some increasing sequence Rn and potentially quite general nite CW-complexes
Xn. Given any quasitrace τ on A = lim (An, ϕn), we associate its dimension function
n→∞
dτ as follows: given a ≤ 1 in A+, we dene
1
dτ (a) = lim τ(an ).
n→∞
Elliott-Robert-Santiago showed in [ERS11] that dτ is a so-called functional on the
Cuntz semigroup, and B. Blackadar introduced the important notion of strict com-
parison: when dτ (a) < dτ (b) for every quasitrace τ implies a ≾ b in the Cuntz
semigroup. For the AH algebras, Toms and Winter showed that strict comparison is
equivalent to Z-stability, along with other equivalent notions such as slow dimension
growth for Villadsen algebras of the rst type, throughout the thesis just referred to
as Villadsen algebras. In the initial stages of the project, it was hoped that a line
could be drawn for dierent values of c ∈ (0,∞), possibly getting strict comparison
for certain cases.
This turned out not to be possible. The main result of the thesis explains
how regardless of the choice of nite CW-complexes Xn, and regardless of the matrix
growth, i.e. the constant c ∈ (0,∞), there is always a subsequence which admits
8
a construction which is not Z-stable. Often, AH algebras are considered with di-
agonal connecting homomorphisms, requiring that the size of the matrices increases
multiplicatively.
Theorem 1.1.1. Let Xn be given CW-complexes with dimension dim(Xn) = dn,
and let (Rn)n∈N be ∣∣a sequence in N. Suppose that (dn) and (Rn) are monotonicallyincreasing, with Rn Rn+1 and dn → ∞, and suppose
dn
lim inf = c ∈ (0,∞).
n→∞ Rn
Then, there exists a subsequence (an) of N and connecting homomorphisms
ϕn,n+1 : MRa (C(Xan)) → Mn Ra (C(Xn+1 an+1))
such that A := lim (MRa (C(Xan)), ϕn,n+1) is a simple, unital AH algebra which is
n→∞ n
not Z-stable.
A small modication allows for injective connecting maps, i.e. the information of
each CW-complex is preserved across connecting homomorphisms. We also show a
few other facts surrounding these interesting algebras.
1.2 AH Algebras and Dimension Growth
Denition 1.2.1. Let Xn be nite CW-complexes, Pn ∈ C(Xn,MRn) be projections
for some sequence Rn ∈ N, and An = PnC(Xn,MRn)Pn An inductive limit
A = lim (An, ϕn)
n→∞
is known as an approximately homogeneous (AH) algebra.
Throughout this paper, without loss of generality, we will consider the case Pn = I,
and Xn connected and nontrivial of monotonically increasing dimension. We also
consider those algebras A which have diagonal connecting homomorphisms: those
which have connecting homomorphisms ϕn : An → An+1 of the form
ϕn(a) = diag(a ◦ f1, ..., a ◦ fNn,n+1)
for some continuo∣us functions fj : Xn+1 → Xn. Note that we require Nn,n+1 :=Rn+1 ∈ N, i.e. R ∣n Rn+1. We can consider the dimension growth of an AH algebra A
Rn
as
dim(Xn)
lim = c ∈ [0,∞].
n→∞ Rn
9
Denition 1.2.2. An AH algebra A is said to have slow dimension growth if it
admits an inductive decomposition (An, ϕn) such that
dim(Xn)
lim = 0.
n→∞ Rn
One may recall the next theorem from algebraic topology (c.f. Husemoller:
Chapter 8 [Hus94]).
Theorem 1.2.3. Let X be a compact metrizable Hausdor space, and let ω, ξ
be complex vector bundles⌊over X.⌋ If the ber-dimension of ω exceeds the ber-dim(X)
dimension of ξ by at least at every point of X, then ξ is isomorphic to a
2
sub-bundle of ω.
By the Serre-Swan theorem, complex vector bundles over such a spaceX are identied
with Murray-von Neumann equivalence classes of projections in M∞(C(X)). Note
that ∥p − q∥ < 1 implies p ∼ q, which implies Rank(p) = Rank(q). Thus, all
projections in AH algebras are equivalent to one in ϕn,∞An coming from a projection
in An having nite rank. When Xn are contractible, An = MRn(C(Xn)) has only
trivial projections, i.e. K0(A) ≃ Z coming from the rank of the projection in the
nite stage. In such a case, p, q ∈ An with Rank(p) < Rank(q) implies [p] ≤ [q] in
K0(A). This motivates a notion of comparison of projections.
Denition 1.2.4. Let A be an AH algebra. We say A has strict comparison of
projections if, for all projections p, q ∈ An, we have Rank(p) < Rank(q) implies
ϕn,∞p ≤ ϕn,∞q in A, i.e. [p] ≤ [q] in K0(A).
Certainly, not every AH algebra has strict comparison of projections.
Example 1.2.5. Let A = C(S2), then, A does not have strict comparison of projec-
tions. For, if ξ is the Hopf bration, and 1 0 0 0 0 0 0 0θ =  ∈ M4(C(S2 × S2))0 0 0 0
0 0 0 0
is a trivial rank 1 projection, then ξ × ξ ∈ M (C(S24 × S2)) dened by
10
[ ]
ξ(x) 0
ξ × ξ(x, y) =
0 ξ(y)
has Rank(θ) = 1 < 2 = Rank(ξ × ξ), but we have [θ] ̸≤ [ξ × ξ] in K0(C(S2 × S2)).
In fact, Villadsen was able to notice a key generalization of this fact in Lemma 1 of
[Vil98], which is stated in a further section as Theorem 2.3.1.
Note that if p ∈ C(X,Mn) ≃ Mn(C(X)) is a projection, and f : Y → X is
continuous, then p ◦ f ∈ Mn(C(Y )) is a projection, and Rank(p) = Rank(p ◦ f) since
rank of projections is constant on connected components. The next proposition is
well-known and straightforward to prove.
Proposition 1.2.6. Let A = lim (MRn(C(Xn)), ϕn) be an AH algebra with slow
n→∞
dimension growth and diagonal connecting homomorphisms. Then, A has strict
comparison of projections.
Proof. Clearly, Rn → ∞ as n → ∞ if dim(Xn) > 0. Let p, q ∈ MR (C(Xi)) bei
projections such that Rank(p) = n < Rank(q) = m. Then, for all j ≥ i,
Rj Rj
Rank(ϕi,jp) = n < m = Rank(ϕi,jq).
Ri Ri
By slow dimension growth, there exists N ∈ N such that for all j ≥ N we have
dim(Xj) Rj
< . Thus, for all j ≥ N
2 Ri
dim(Xj) Rj Rj Rj Rj Rj
Rank(ϕi,jp) + < n+ ≤ n+ (m− n) = m = Rank(ϕi,jq).
2 Ri Ri Ri Ri Ri
We conclude ϕi,jp ≤ ϕi,jq for all j ≥ N , thus ϕi,∞p ≤ ϕi,∞q, soA has strict comparison
of projections.
We will occasionally have need to talk about the quasitraces in such inductive
limits of algebras MRn(C(Xn)). Haagerup showed quite generally in Theorem 5.11
of [Haa14] that quasitraces on exact unital C∗-algebras are traces. This includes the
AH algebras analyzed in this thesis, so from now on we omit mention of quasitraces
and just deal with traces.
11
1.3 Stable and Real Rank
Stable and real rank are generalizations of dimension to potentially noncom-
mutative settings. In this section, we summarize the main properties and charac-
terizations. Recall the following classic characterizations of covering dimension (c.f.
Engelking [Eng78] and Brown-Pedersen [BP91]).
Theorem 1.3.1. Let X be a compact Hausdor space; then dim(X) is the least
integer n such that every continuous function from X into Rn+1 can be approximated
arbitrarily closely with never-vanishing functions.
This denition gives n-tuples (f1, ..., fn), which can't vanish simultaneously. So this
is the same as
Theorem 1.3.2. Let X be a compact Hausdor space. Then dim(X) is the least
integer n such that every (n+ 1)-tuple of elements in CR(X) (real-valued functions)
can be approximated arbitrarily closely by (n+ 1)-tuples of elements which generate
CR(X) as an ideal.
These motivate the following denitions:
Denition 1.3.3. Let A be a Banach algebra; then Lgn(A) and Rgn(A) are the set
of n-tuples in A which generate A as a left/right ideal. These are called unimodular
rows.
Denition 1.3.4. Let A be a unital Banach algebra; then ltsr(A) is the least integer
n such that Lgn(A) is dense in An in the product topology, called the left topological
stable rank. We dene rtsr(A) analogously. If A is not unital, we unitize rst.
Remark 1.3.5. If A is a Banach algebra with continuous involution (e.g. a C∗-
algebra), then ltsr(A) = rtsr(A) := tsr(A), called the stable rank of A.
The following is Proposition 1.7 of Rieel [Rie83], giving the connection to the
dimension of X:
Proposition 1.3.6. Suppose X is a compact Hausdor space; then
⌊dim(X)⌋
tsr(C(X)) = + 1.
2
12
Likewise, the following result, Proposition 3.1 of Rieel [Rie83], is of fundamental
importance to the case of stable rank one:
Theorem 1.3.7. Let A be a Banach algebra; the following are equivalent:
(i) ltsr(A) = 1;
(ii) rtsr(A) = 1;
(iii) The invertible elements of A are dense in A.
Proof. Lg1(A) and Rg1(A) consist of invertible elements, so (3) =⇒ (1), (2). Sup-
pose ltsr(A) = 1, and let a be a left-invertible element with left-inverse b. We have
b ≈ c for some left-invertible element c, so ca ≈ 1, which implies ca is invertible, or in
particular |ca− 1| < 1. Hence a is invertible. The other implication is analogous.
Moreover, it is shown in Theorem 6.1 of Rieel [Rie83]:
Theorem 1.3.8. Let A be a C∗-algebra⌊. Then for every m ∈ N
A tsr(A)− 1
⌋
tsr(Mm( ) = + 1.
m
In particular, tsr(A) = 1 if and only if tsr(Mn(A)) = 1 for every n.
Theorem 4.1 of Elliott-Ho-Toms [EHT09] gives the following characterization of sim-
ple, unital diagonal AH algebras:
Theorem 1.3.9. Let A = lim (An, ϕn) be a simple, unital diagonal AH algebra.
n→∞
Then, A has stable rank one.
We will need a notion of cancellation for projections for analyzing comparison
in K0. This motivates the following denition:
Denition 1.3.10. A C∗-algebra A has cancellation of projections if for all projec-
tions p, q, e, f ∈ A with pe = qf = 0, e ∼ f , and p+ e ∼ q+ f , then we get p ∼ q. A
is said to have cancellation if Mn(A) has cancellation of projections for every n.
Remark 1.3.11. Notice that A has cancellation of projections if and only if p ∼ q
implies (1 − p) ∼ (1 − q). That is, A has cancellation of projections if and only if
p ∼ q implies p and q are unitarily equivalent.
The following theorem and its proof can be found as Theorem 3.1.14 of H. Lin's book
[Lin01]:
13
Theorem 1.3.12. Every unital C∗-algebra A with tsr(A) = 1 has cancellation.
Proof. Let p, q ∈ Mn(A) be projections with p ∼ q; hence there exists a partial-
isometry v ∈ Mn such that v∗v = p, vv∗ = q. Since tsr(A) = 1, there exists
x ∈ GLn(A) (invertible) such that
∥x− v∥ ≤ 1 .
8
1
We have x = u(x∗x) 2 as given by the polar decomposition, for u ∈ Un(A). We
calculate
∥x∗x− p∥ ≤ ∥x∗ 1x− x∗v∥+ ∥x∗v − v∗v∥ < ,
4
which implies
upu∗ ≈ u(x∗1 x)u∗ = x∗x ≈ 1 q,
4 4
hence ∥upu∗ − q∥ < 1. Thus there exists a unitary w such that w∗upu∗w = q.
Real rank is another notion of noncommutative dimension dened as follows:
Denition 1.3.13. Let A be a unital C∗-algebra; then RR(A) is the smallest integer
such that for every n-tuple (x1, ..., xn) ∈ AnSA with n ≤ RR∑(A) + 1, and for everyn
ε > 0, there exists an n-tuple (y1, ..., yn) ∈ An 2
∑ SA
such that yk is invertible and
k=1
n
∥ (x 2k − yk) ∥ < ε.
k=1
The following, Proposition 1.1 in Brown-Pedersen [BP91], gives the connection
to the dimension of a space X:
Proposition 1.3.14. Let X be a compact Hausdor space; then
RR(C(X)) = dim(X).
Proof. The covering dimension of X is the smallest integer n such that every contin-
uous function f : X → Rn+1 is approximated by g such that g(x)∑≠ 0 for every x.n+1
Since g = (g1, ..., gn+1), we have g(x) ≠ 0 for every x if and only if gk(x)2 > 0 for
k=1
every x, i.e. this sum is invertible. In this context, real rank and covering dimension
are the same notion.
14
Proposition 1.2 in Brown-Pedersen [BP91] importantly relates real rank to stable
rank:
Proposition 1.3.15. Let A be a C∗-algebra. Then
RR(A) ≤ 2tsr(A)− 1.
Brown-Pedersen [BP91] obtain in Theorem 2.6 a characterization of real rank:
Theorem 1.3.16. Let A be a C∗-algebra. The following are equivalent:
(i): RR(A) = 0;
(ii): The set of elements of ASA with nite spectrum are dense in ASA
(mutually orthogonal projections);
(iii): For every hereditary subalgebra B ⊂ A, b1, ..., bn ∈ B, and ε > 0,
there exists a projection p ∈ B such that for all j ∈ {1, ..., n}
∥bjp− bj∥ < ε.
1.4 The Cuntz Semigroup and Strict Comparison
The Cuntz semigroup provides a generalization of Murray-von Neumann com-
parison for a C∗-algebra to positive elements.
Denition 1.4.1. Let A be a C∗-algebra, a, b ∈ A+. We write a ≾ b if there exists
a sequence (xn) in A such that
xnbx
∗
n → a.
We write a ∼ b if a ≾ b and b ≾ a. The Cuntz Semigroup of A, denoted Cu(A), is
the set of Cuntz equivalence classes in (A ⊗ K)+. It is an ordered semigroup under
≾ and with [a] + [b] = [a′ + b′], where a′, b′ are orthogonal with a′ ∼ a and b′ ∼ b.
Remark 1.4.2. Let p, q ∈ (A ⊗ K)+ be projections with p ≾ q. Then, due to
perturbation properties of projections, p ≤ q in Murray-von Neumann subequivalence.
In [CEI08], it is shown by Coward, Elliott, and Ivanescu that Cu is a covariant
functor from the category of C∗-algebras to a subcategory of the category of ordered
abelian groups, known as Cu. Particularly, if ϕ : A → B is a homomorphism, then
[ϕ] : Cu(A) → Cu(B) is a homomorphism with [ϕ][a] := [ϕ(a)].
15
Example 1.4.3. Let f, g ∈ C([0, 1]) be dened by
√
f(x) = x, g(x) = x.
Suppose there exists h ∈ C([0, 1]) such that f(x) = h(x)g(x)h(x). Then, necessarily
1
we have h(x) = ±√ for all x ̸= 0. Thus, h ̸∈ C([0, 1]). No element can satisfy the
4 x
condition of Murray-von Neumann subequivalence for f, g. However, there does exist
a sequence (hn) in C[0, 1] such that hn(x)g(x)hn(x) → f(x) for every x ∈ [0, 1], forexample  √14 if x ∈ [ 1 , 1]
hn(x) =  x n .5(n 4 )x if x ∈ [0, 1 ]
n
√
Thus, f(x) ≾ g(x). Clearly g(x) ≾ f(x), since kn(x) = 4 x satises g(x) =
kn(x)f(x)kn(x) for all n ∈ N.
There is another category C of non-cancellative compact Hausdor cones with
jointly continuous + and scaling in [0,∞], with morphisms being continuous linear
maps between cones, which is intimately related to the Cuntz semigroup and the
traces on A. This is analyzed in full detail by Elliott, Robert, and Santiago in
[ERS11].
In summary, let F (Cu(A)) be the set of additive, order preserving maps on
Cu(A) sending 0 → 0 and which preserve suprema of increasing sequences: the so-
called linear functionals on Cu(A). Let T (A) be the space of lower-semicontinuous
traces onA. Then, F : Cu → C and T : C∗-Alg → C are continuous (with respect to
sequential inductive limits) contravariant functors. Moreover, F (Cu(A)) ≃ QT2(A)
where QT2(A) is the cone of lower-semicontinuous 2-quasitraces on A. Lastly, there
is a dual cone L(F (Cu(A))) ∈ Cu to F (Cu(A)), yielding a covariant functor
L(F (Cu(·))) from C∗-Alg to Cu. Altogether, we have the functorial diagram
Cu Cu
C∗-Alg F L
T C
Remark 1.4.4. Following [CEI08], the Cuntz semigroup has addition given by
[a]⊕ [b] = [a′ + b′],
16
where a′, b′ ∈ (A⊗K)+ are orthogonal to each other and Cuntz equivalent to a and
b respectively. Let ψ : A → B be a homomorphism between C∗-algebras A and B.
Then, Cu(ψ) : Cu(A) → Cu(B) is a well-dened semigroup homomorphism given by
Cu(ψ)[a] = [ψ(a)].
In particular, Cu(ψ) is a so-called Cu-morphism, meaning among other properties
that it is order preserving : if [a] ≤ [b] in the Cuntz semigroup Cu(A), then
Cu(ψ)[a] = [ψ(a)] ≤ [ψ(b)] = Cu(ψ)[b].
Critically is the notion of a lower-semicontinuous dimension function dτ asso-
ciated to a trace τ , which is a linear functional on the Cuntz semigroup.
Denition 1.4.5. Let τ ∈ QT2(A) be a quasitrace. The dimension function dτ
associated to τ is given by acting on a ∈ A+ with a ≤ 1 by
1
dτ (a) = lim τ(a k ).
k→∞
B. Blackadar proposed the notion of strict comparison of general positive elements,
much in the spirit of strict comparison of projections.
Denition 1.4.6. Let A be a C∗-algebra. We say A has Blackadar's strict compari-
son of positive elements , or simply say A has strict comparison or strict comparison of
positive elements , if, for every a, b ∈ A+, we have dτ (a) < dτ (b) for every τ ∈ QT2(A)
implies a ≾ b.
Note that, trivially, strict comparison of positive elements implies strict com-
parison of projections.
Example 1.4.7. Mn(C) has strict comparison of positive elements. In fact, every
element A ∈ Mn(C)+ is Cuntz equivalent to a trivial projection P with Rank(P ) =
Rank(A).
Proof. There is a unique trace τ on Mn(C), which is the standard one τ = Tr, and
Rank(P )
dτ (A) = Tr(P ) = Rank(P ) (or for the normalized trace) when P ∈ Mn(C)
n
is a projection. Note that A ≾ B in Mn(C) implies Rank(A) ≤ Rank(B) from stan-
dard linear algebra, for there are Cn ∈ Mn(C) such that C ∗nBCn → A.
17
Let A ∈ Mn(C)+; then the functional calculus gives A = idσ(A) in C(σ(a)),
i.e. A(x) = x. Let B ∈ C(σ(a)) be dened by
√1 if x ∈ σ(A) \ {0}B(x) = x .0 if x = 0
Then, P = BAB∗ = 1 on σ(A) \ {0}, and BAB∗√ (0) = 0; in particular, P is a trivial
projection in C∗(A) with A ≾ P . Let C = A; then, CPC(x) = x on σ(A), thus
P ≾ A. Therefore, Rank(P ) = Rank(A) and [A] = [P ] in the Cuntz semigroup.
Since dτ agrees on Cuntz equivalence classes, we have
dτ (P ) = Rank(P ) = Rank(A) = dτ (A).
Let A,B ∈ Mn(C) satisfy dτ (A) < dτ (B). Then, [A] = [P ] and [B] = [Q] in the Cuntz
semigroup for some trivial projections P,Q ∈ Mn(C), with Rank(P ) < Rank(Q).
Since they are trivial projections, P ≤ Q in K0(Mn(C)) ≃ Z, thus A ∼ P ≾ Q ∼ B
in the Cuntz semigroup.
A similar example to the following appears in Toms [Tom08a]:
Example 1.4.8. M6(C([0, 1]
6)) fails to have strict comparison of positive elements.
Proof. The traces in Mn(C(X)) have extreme points given by evaluation maps at
each x ∈ X. In other words,
dτ (A) < dτ (B)
for A,B ∈ Mn(C(X))+ if and only if
Rank(A(x)) < Rank(B(x))
for every x ∈ Mn(C(X)). Let ξ′ ∈ C(S2,M2(C)) ≃ M2(C(S2)) be a nontrivial Rank
1 1 1
1 projection, e.g. the Hopf bration. Let c = ( , , ) ∈ [0, 1]3 be the center of the
2 2 2
cube, ∣
S = { ∈ 1x [0, 1]3 ∣ ∥x− c∥ = }
4
be the sphere in [0, 1]3 of radius 1 in with center c, and ζ : S → S2 a homeomorphism.
4
Thus, ξ = ξ′ ◦ ζ ∈ C(S,M2(C)) is a nontri∣vial projection. Let
U = {x ∈ [0, 1]3S ∣ ∥ 1 3x− c∥ ∈ ( , )
8 8
18
be an annular neighborhood of S in [0, 1]3. We have, ξ × ξ ∈ C(S × S,M6(C)) is a
projection, as given by  ξ(x) 02×2 02×2(ξ × ξ)(x, y) = 02×2 ξ(y) 02×2 .
02×2 02×2 02×2
Let ρ : [0, 1]3 \ {c} → S be projection along radial lines emanating from
1 1 1
c = ( , , ), f a positive function in C0(US) with f(x) = 1 for x ∈ S, and g a
2 2 2
positive function in C 60([0, 1] \ (S[× S)). De]ne
g(x) 0
G(x) = ∈ M 62([0, 1] ).
0 g(x)
Thus, for x, y ∈ [0, 1]3, 

04×4 04×2
  if (x, y) = (c, c)02×4 G(x) 
Φ(x, y) = 
f(x) · ξ(ρ(x)) 0 0 0 f(y) · ξ(ρ(y)) 0  if (x, y) ̸= (c, c)
0 0 ∣G(x)
denes a function Φ ∈ M6(C([0, 1]6)) such that Φ∣S×S = ξ × ξ and Rank(Φ(x)) ∈
{2, 4} for all x ∈ [0, 1]6. Put [ ]
1 01x5
θ1 = ,
05x1 05×5
which is a trivial Rank 1 projection in M6([0, 1]
6). Then, by∣∣ Villadsen [Vil98] Lemma1, stated in a further section as Theorem 2.3.1, we have θ1 S×S ̸≤ ξ × ξ in
K0(S × S) ≃ K 2 20(S × S ).
For the above elements, we have
Rank(θ1(x)) = 1 < 2 ≤ Rank(Φ(x))
for every x ∈ [0, 1]6, i.e. dτ (θ1) < dτ (∣∣Φ) for ev∣∣ery trace τ . Suppose θ1 ≾ Φ inCu(M6(C([0, 1]6))); this would imply θ1 S×S ≾ Φ S×S = ξ × ξ, which would imply
θ1 ≤ ξ × ξ in K0(S × S). That is a contradiction, therefore M6(C([0, 1]6)) does not
have strict comparison of positive elements.
19
1.5 Perforation and the Jiang-Su Algebra Z
In [JS99], Jiang and Su constructed a simple, separable, nuclear, innite-
dimensional algebra Z with the same Elliott invariant as C. It is an inductive limit
of sums of so-called dimension drop algebras. It was later shown that it is intimately
related to the notion of slow dimension growth and Blackadar's strict comparison of
positive elements.
Denition 1.5.1. Suppose A is a C∗-algebra. Then K0(A) is called weakly unper-
forated if nx ∈ K0(A)+ \ {0} implies x ∈ K0(A)+.
Example 1.5.2. Let A = Mn(C(X)), with X contractible. Then K0(A) is weakly
unperforated.
Example 1.5.3. Let A be a C∗-algebra with strict comparison of projections and
cancellation of projections. Then, K0(A) is weakly unperforated.
Remark 1.5.4. It is shown by Elliott-Ho-Toms [EHT09] that all simple AH algebras
have stable rank one, therefore have cancellation of projections by Theorem 1.3.12.
Since strict comparison in the Cuntz semigroup implies strict comparison of projec-
tions, all simple AH algebras with strict comparison of positive elements necessarily
have K0(A) is weakly unperforated. The converse is not true, as we will investigate
more later.
The following important notion is shown by Gong-Jiang-Su in [GJS00]:
Theorem 1.5.5. Let A be a simple, unital C∗-algebra with K0(A) weakly unperfo-
rated. Then Ell(A) ≃ Ell(A⊗Z).
Moreover, Rørdam [Rør04] was able to show that Z-stability, that A⊗Z ≃ A, gen-
erally implied strict comparison for simple, unital, exact, nite C∗-algebras.
Toms and Winter conjectured, and later proved for the AH algebras, a strong
equivalence known as the Toms-Winter Conjecture. A generalization of the next
theorem can be found in [Tom11].
Theorem 1.5.6. Let A be an AH algebra. Then A ⊗ Z ≃ A if and only if A has
Blackadar's strict comparison of positive elements.
20
Strict comparison has an important connection to almost unperforation.
Denition 1.5.7. Let A be a C∗-algebra. We say Cu(A) is almost unperforated if
(n+ 1)[x] ≤ n[y] in Cu(A) for some n implies [x] ≤ [y].
In Proposition 3.2.4 of [Bla+12], B. Blackadar, L. Robert, A. Tikuisis, A. Toms, and
W. Winter show an important connection to strict comparison:
Theorem 1.5.8. Let A be a simple, stably nite, unital C∗-algebra. Then, A has
strict comparison if and only if Cu(A) is almost unperforated.
Example 1.5.9. Cu(M6(C([0, 1]
6))) is not almost unperforated.
Proof. We know that this is true from Example 1.4.8. But, on the other hand,
let θ1,Φ be the elements constructed in this example. Then, 101[θ1] ≤ 100[Φ] in
Cu(M6(C([0, 1]
6))) from Toms' result in [Tom08b], because
1
Rank(101[θ1](x)) + dim([0, 1]
6) = 101 + 3 < 200 ≤ Rank(100[Φ](x))
2
for every x ∈ [0, 1]6. But, as we know based on the example, [θ1] ̸≤ [Φ]. Therefore,
Cu(M6(C([0, 1]
6))) is not almost unperforated.
21
CHAPTER 2
CONSTRUCTIONS OF GOODEARL, VILLADSEN AND TOMS
2.1 Goodearl Algebras
In [Goo92], K. R. Goodearl constructed an AH algebra with a curious property,
later expanded on by J. Villadsen.
Denition 2.1.1. A Goodearl algebra over seed space X is an inductive limit A =
lim (MRn(C(X)), ϕn) where
n→∞
ϕn(a) = diag(a, a, ..., a, δn(a), ...., δn(a)
for δn(a) := xn for some xn ∈ X are (constant) evaluation maps, and at least one
identity and evaluation map occur.
Remark 2.1.2. We have A is simple if and only if the evaluation points are dense
in X. Thus, when A is simple and dim(X) < ∞, Rn → ∞, therefore A has slow
dimension growth.
A. Toms (c.f [Tom08b]) has shown the analogous result to Theorem 1.2.3
for Cuntz subequivalence for the commutative C∗-algebra C(X), and the following
generalization for Mn(C(X)):
Theorem 2.1.3. Let X be a compact metric space with dim(X) being the covering
dimension. Let a, b ∈ Mn(C(X)) be positive, and suppose for all x ∈ X
1
Rank(a(x)) + dim(X) < Rank(b(x)).
2
Then we have a ≾ b.
The next proposition and its proof illustrates the Toms-Winter conjecture for the
simple Goodearl algebras over a nite-dimensional space X.
Proposition 2.1.4. Let X satisfy dim(X) = d, and let A be a simple Goodearl
algebra constructed from X. Then A has strict comparison of positive elements.
Proof. Suppose a, b ∈ A+, and dτ (b)− dτ (a) = δ > 0 for every 2-quasitrace τ . Since
dτ is lower-semicontinuous, we have
U = {c ∈ A ∣∣ δ+ dτ (c)− dτ (a) > }
2
22
is an open set containing b; thus, there exists r > 0 such that
∥b− c∥ δ< r implies dτ (c) > dτ (a) + .
2
Let ε ∈ (0, r) be arbitrary. A result of Kirchberg and Rørdam states that
∥a − b∥ < η implies (a − η)+ ≾ b. Moreover, Cuntz inequality is preserved under ∗-
homomorphisms, and there are increasing sequences (a′n), (b
′
n) such that ϕ a
′
n,∞ n → a
and ϕn,∞b
′
n → b, with a′n, b′n ∈ An for each n ∈ N. That is, there exists n ∈ N and
a′, b′ ∈ An such that
(i) ∥ϕ ′n,∞a − a∥ < ε;
(ii) ∥b− ϕ ′n,∞b ∥ < ε < r;
which satisfy
(iii) (a− ε)+ ≾ ϕn,∞a′ ≾ a;
(iv) ϕ b′n,∞ ≾ b.
So, we have
δ δ
dτ (b) ≥ dτ (ϕ b′n,∞ ) > dτ (a) + ≥ d (a′τ ) + > dτ (a− ε)+.
2 2
Let τ̃ be any 2-quasitrace, with τn = τ̃ ◦ ϕn,∞ its pullback to An, and let
∥a∥ = 1 with a ∈ A+. Write τ := τn−1 for brevity. For a typical element a ∈ An−1,
let us compute a trace of
ϕn−1(a) = diag(αn.a, βn.(δn(a))) ∈ An,
where αn.a denotes a repeated on the diagonal αn times, and analogously for βn.(δn(a)).
We nd
1
τn(ϕn−1,na) = [αnτ(a) + βnτ(δn(a))] ,
ν(n)
1
τn+1(ϕn−1,n+1a) = · [αnαn+1τ(a)]
ν(n+ 1)
1 1
+ · βn · τ(δn(a)) + · βn+1αn · τ(δn+1(a)),
ν(n) ν(n+ 1)
. . . ∑ ( )s k1 βk 1 ∏
τs(ϕn−1,s(a)) = [ωs+1,n · τ(a)] + · αj · τ(δk(a)).
ν(n) αk ν(k)
k=n j=n
23
In particular, we have that ∑s
τs(ϕn−1,s(a)) = c0τ(a) + ckτ(δk(a))
k=n
for coecients ck ∈∑(0,∞) where ck → 0 as k → ∞ and c0 is decreasing; indeed fors
all s we have c0 + ck = 1.
k=n
The function dn := dτn = dτ̃ ◦ ϕn,∞ is a lower-semicontinuous dimension func-
δ
tion on An with dn(b′)− dn(a′) > > 0. Therefore, there exists an open set V ⊂ X
4
such that for every x ∈ V we have Rank(b′(x)∑) > Rank(a′(x)). Because the sequence
(xk)k≥M is dense in X for every M ∈ N, and ck < ∞, with ck > 0 for all k, there
k
exists a subsequence (xj)j∈I for some nite index set I ⊂ N such that
(i) x∑j ∈ V for every j ∈ I;δ
(ii) cj < ;
8
j∈I
1
(iii) |I| > ⊕d.2 ⊕
Let a = δ (a′γ k ) and likewise bγ = δk(b
′) as block diagonals (along with
k∈I k∈I
zero blocks) such that, for suciently large s,
aγ ≤ ϕn,sa′ and bγ ≤ ϕ ′n,sb .
We have ∑s
τs(ϕ
′
n,sb − bγ) = c0τ(b′) + ckτ(δk(b′))
k=∑n, k∉ Is
> c τ(a′
δ
0 ) + ckτ(δk(a
′)) +
8
k=n, k∉ I
δ
= τs(ϕ
′
n,sa − aγ) + ;
8
therefore, for all x ∈ X, Rank((ϕn,sb′ − bγ)(x)) ≥ Rank((ϕ ′n,sa − aγ)(x)).
1
On the other hand, we have selected more than d elements xj which are all
2
constant matrices satisfying Rank(δ (b′j )) ≥ Rank(δj(a′)) + 1. Whence,
1
Rank(bγ(x)) > Rank(aγ(x)) + d;
2
24
we conclude, since a ′γ ⊥ ϕn,sa and bγ ⊥ ϕn,sb′, that for all x ∈ X
1
Rank(ϕ ′n,sb (x)) > Rank(ϕn,sa
′(x)) + d.
2
By Toms' result, since ε > 0 is arbitrary, we have ϕ a′n,∞ ≾ ϕ ′n,∞b . Putting it all
together, we have found
(a− ε) ≾ ϕ a′ ≾ ϕ ′+ n,∞ n,∞b ≾ b.
Since ε > 0 is arbitrary, we conclude a ≾ b.
It is not an obvious fact that even Goodearl algebras over an innite-dimensional
seed space X have strict comparison. It follows from sucient divisibility properties
of these algebras, as investigated by Fu, Li, and Lin in [FLL22].
Denition 2.1.5. Let A be a simple C∗-algebra. A is said to have property (TAD)
if given any ε > 0, s ∈ A+ with s ̸= 0, n ∈ N, and any nite subset F ⊂ A, there
exists θ ∈ A+ and a C∗ subalgebra D ⊗Mn ⊂ A such that
(i) θx ≈ε xθ for all x ∈ F ;
(ii) (1− θ)x ∈ε D ⊗ 1n for all x ∈ F ;
(iii) θ ≾ s.
Denition 2.1.6. Let A be a simple C∗-algebra. A is said to be tracially approxi-
mately divisible if for any ε > 0, nite F ⊂ A, element e 1F ∈ A+ with
eFx ≈ε/4 x ≈ε/4 xeF for all x ∈ F , every nonzero s ∈ A+ and n ∈ N, there exists
θ ∈ A1+ and a C∗ algebra D ⊗Mn ⊂ A, and a c.p.c. map β : A → A such that
(i) x ≈ε x1 + β(x) for all x ∈ F , with ∥x1∥ ≤ ∥x∥, x1 ∈Her(θ);
(ii) β(x) ∈ε D ⊗ 1n, and eFβ(x) ≈ε β(x) ≈ε β(x)eF for all x ∈ F ;
(iii) θ ≾ s.
IfA has property (TAD) it is shown to be tracially approximately divisible in [FLL22].
In this work, simple C∗ algebras with property (TAD) are shown to have strict com-
parison.
25
Proposition 2.1.7. Let A be a real rank one Goodearl algebra over an innite-
dimensional space X. Then A has property (TAD). Therefore, it has strict compari-
son of positive elements.
Proof. Let ε > 0, F = {x1, ..., xℓ} ⊂ A, s ∈ A+ nonzero, and n ∈ N. There exists
some large N ∈ N and s 1 ℓN , xN , ..., xN ∈ AN such that
(i) bNxN ≈ ε xNbN for every bN ∈ AN . Such an N exists because A is
3
real rank one and so the connecting homomorphisms are approximately
id⊗1N as N → ∞;
(ii) ϕ jN,∞x εN ≈ x for all j = 1, ..., ℓ;3
(iii) ϕN,∞sN ≈ ε s with sN ≾ s.
3
Put θN =
1
∥ ∥ss N , where and D = A ≃ A⊗Mn. Then, with θ = ϕN,∞θN , we haveN
∥θx− xθ∥ ≤ ∥θx− θϕn,∞xN∥+ ∥ϕn,∞xNθ − xθ∥+ ∥ϕn,∞xNθ − θϕn,∞xN∥
≤ 2∥θ∥∥x− ϕN,∞xN∥+ ∥ϕn,∞(xNθN − θNxN)∥
≤ · · ε ε2 1 + = ε.
3 3
Corollary 2.1.8. Let A be a simple Goodearl algebra; then K0(A) is weakly unper-
forated.
Proof. A has cancellation of projections, since it has stable rank one, and so we have
τ(p) < τ(q) implies p ≾ q for any projections p, q.
2.2 Villadsen Algebras
The Villadsen algebras employ an illustrative construction rst investigated
by J. Villadsen in [Vil98].
Denition 2.2.1. A map ϕj : C(Xj) → MR (C(Xj+1)) is called diagonal if it hasj
the form
 f ◦ λ1 0 · · · 0 0 f ◦ λ2 0 · · · 0
ϕj(f) = diag(f ◦ λ1, . . . f ◦ λR ) :=j 
.. . . . . . ..  ,
0 · · · 0 f ◦ λ Rj−1 0 
0 · · · 0 f ◦ λR )j
26
where each λj : Xj+1 → Xj is a continuous map, called the eigenvalue maps of ϕj.
Remark 2.2.2. Let X be a locally compact Hausdor space with dim(X) ≥ 1; put
X1 = X,
n
Xj+1 = X
j
j (n > 1)
∏j−1
for some sequence (nj) in N. For each i, j ∈ N with i ≤ j, put Di,j = nk. Notice
k=i
dim(Xn) = dim(X1) ·D1,n and Di,j ·Dj,ℓ = Di,ℓ.
Thus, Di,j is the number of distinct coordinate projections from Xj → Xi, for we
have
dim(Xj)
Di,j = and X = (X )
Di,j
j i .
dim(Xi) ∣
Denition 2.2.3. Let (kj) be an increasing sequence in N with k ∣j kj+1 for all j; a
unital diagonal homomorphism
ϕj : M ⊗ C(XkjR ) → MR (C(Xkj+1))j j+1
is called a Villadsen map of the rst type if each eigenvalue map is either a point
evaluation or a coordinate projection. An inductive limit A = lim (An, ϕ→∞ n) overn
A kjj = MR (C(X )), where ϕn are Villadsen maps of the rst type, is called a Villadsenj
algebra (of the rst type).
Throughout this paper, we will refer to a Villadsen algebra to mean a Villadsen alge-
bra of the rst type where dim(X1) ≥ 1 and X1 is connected. As with the Goodearl
algebras, a Villadsen algebra A is simple when the evaluation maps are suciently
dense when included in XN.
Rj
PutMi,j = and denoteDi,j as the number of distinct coordinate projections
Ri
appearing in the connecting homomorphisms from stage i to j. ThusMi,j ·Mj,ℓ = Mi,ℓ,
so
Di,j · Dj,j+1 Di,j+1=
Mi,j Mj,j+1 Mi,j+1
and generally
Di,j Di,i+1
= · · Dj−1,j. . .
Mi,j Mi,i+1 Mj−1,j
27
is a product of terms which have innitely many terms strictly less than(1 whe)n A
Di,j
is simple, as there are innitely many point-evaluation maps. Therefore is
Mi,j
a decreasing positive sequence in j, so this sequence converges. We will mostly be
interested, without loss of generality, in the case Di,j = Ni,j, i.e. all the distinct
coordinate projections are included. In Toms-Winter [TW09], it is shown that when
Di,j
lim = ε > 0, then A doesn't have strict comparison.
j→∞ Mi,j
Similarly, let αn denote the number of total projections in the denition of ϕn,
including multiplicity. Put
Ri
ωi,j = αi · ... · αj−1 .
Rj
Analogously, we have
ωi,j · ωj,ℓ = ωi,ℓ,
so ωi,j is a decreasing positive sequence in j which also converges. In Goodearl's paper
[Goo92], it is shown that in the case Xn = X for all n, i.e. the Goodearl algebras,
when lim ω1,j = ε > 0, then A has real rank one. It turns out that this phenomenon
j→∞
with real rank is typical of the Villadsen algebras as well. Note that many of the
following results are shown in one form or another throughout the literature. The
next lemma's proof is inspired by a similar proof in Goodearl's paper.
Lemma 2.2.4. Suppose that A = lim (MRn(C(Xn)), ϕn) is a simple Villadsen alge-
n→∞
bra of the rst type. Let αn be the number of projections appearing in the connecting
homomorphism ϕn and put
Ri
ωi,j = αi · ... · αj−1 .
Rj
If
lim ω1,j = ε > 0,
j→∞
then A has real rank one.
Proof. Since A has stable rank one, it has real rank zero or one by Proposition 1.3.15.
There exists x, y in X and f : X → C continuous such that f(x) = 1 and f(y) = 0;
put a = diag(f, ..., f) ∈ A1. Appealing to Theorem 1.3.16, suppose that there exists
b ∈ A εsuch that b is a linear combination of projections and ∥b−ϕ1,∞(a)∥ < . Then
4
there exists s ∈ N and c ∈ As such that
28
(i) c is a linear combination of projections;
ε
(ii) ∥c− ϕ1,s(a)∥ < .
2
There is a particular x̃ ∈ Xns such that πr1πr2 ◦ ... ◦ πrs : Xns1 2 s → X has
πr1πr21 2 ◦ ... ◦ πrss (x̃) = x
for all choices of rj. In particular, that element is x̃ = (x, x, . . . , x). Likewise,
ỹ = (y, y, . . . , y) has the analogous statement. Thus, ϕ1,s(x̃) − ϕ1,s(ỹ) is a diagonal
matrix which has at least ε · ν(s) entries of 1 and the rest are zero; i.e.
|Tr(ϕ1,s(x̃)− ϕ1,s(ỹ)| > εν(s).
On the other hand, since Tr is a continuous map on linear combinations of projections
into Z, one has Tr(c(x̃)) = Tr(c(ỹ)). Therefore
| εTr(c(z))− Tr(ϕ1,s(a)(z))| ≤ ∥c(z)− ϕ1,s(a)(z)∥ · ν(s) < ν(s)
2
for all z ∈ Xns . But since Tr(c(x̃)) = Tr(c(ỹ)) we have
|Tr(ϕ1,sa(x̃))− (ϕ1,sa(ỹ))| < ε · ν(s).
This is a contradiction; therefore such an element a cannot have been approximated
by a linear combination of projections. Therefore, A cannot be real rank zero by
Proposition 1.3.16.
Note that Di,j ≤ αi,j for all i, j, since we are just discounting multiplicity. The
next statement is proved in Toms, Winter [TW09]; their proof includes elements of
both the above and below proofs.
Corollary 2.2.5. Suppose A = lim (A , ϕ
→∞ n n
) is a simple Villadsen algebra with real
n
rank zero. Then, A has strict comparison of positive elements.
Proof. Suppose A does not have strict comparison. Let D1,j denote the number of
Rj
distinct projections from stage 1 to j, and M1,j = denote the relative matrix size.
R1
By Toms [TW09] Lemma 5.1,
D1,j
lim → ε > 0.
j→∞ M1,j
But, D1,j ≤ αi,j since we are just ignoring the distinction; therefore
lim ω1,j ≥ ε > 0.
j→∞
By Lemma 2.2.4, A has real rank one.
29
The converse is essentially also proved in Christensen [Chr18], though not
stated explicitly. He cites A. Toms as well:
Lemma 2.2.6. Suppose that A = lim (M
→∞ Rn
(C(Xn)), ϕn) is a simple Villadsen alge-
n
bra of the rst type. Let αn be the number of projections appearing in the connecting
homomorphism ϕn, and put
Ri
ωi,j = αi · ... · αj−1 .
Rj
If
lim ω1,j = 0
j→∞
then A has real rank zero.
Proof. We have D1,j ≤ α1,j. Therefore if ω1,j → 0, A has slow dimension growth.
Since A is simple and has slow dimension growth, it has real rank zero if and only if
the projections separate the traces (see Denition 2.4.1 later on) of A (c.f. Blackadar,
Dardalat, Rørdam [BDR91]). But, when the product is zero, then A has a unique
trace, as illustrated in Example 2.1.4 (c.f. also Christensen [Chr18] - Theorem 3.6).
Therefore, A has real rank zero. One can also surely use practically the same direct
proof as appears in Goodearl's paper [Goo92].
These lemmas witness the following generalization of the phenomenon of real
rank one appearing in the Goodearl algebras to the Villadsen algebras:
Theorem 2.2.7. Suppose that A = lim (MRn(C(Xn)), ϕn) is a simple Villadsen
n→∞
algebra of the rst type. Let αn be the number of projections appearing in the
connecting homomorphism ϕn, and put
Ri
ωi,j = αi · ... · αj−1 .
Rj
We have lim ω1,j = 0 if and only if A has real rank zero.
j→∞
2.3 Villadsen's Chern Class Obstruction
Toms exhibited in [Tom08a] a simple Villadsen algebra with weakly unper-
forated K0 group, in particular over a contractible seed space X1, but which failed
to have strict comparison of positive elements. It showed the necessity of the Cuntz
semigroup for classication, but also suggested the diculty in explicitly nding the
30
Cuntz semigroup of even modest algebras. This was largely based on the work of
Villadsen, who showed the following in Lemma 1 of [Vil98].
Theorem 2.3.1. Let ζ be a complex line bundle over a nite CW-complex B, and
let n ∈ N. Let θ1 denote the trivial line bundle. If [ζ×n]− [θ1] ∈ K0(Bn)+, then the
n-th tensor power of the Euler class of ζ is zero.
The lack of strict comparison illustrated in Example 1.4.8 can be preserved
across the Villadsen algebras if the number of projections is suciently large. This
phenomenon is known as Villadsen's Chern class obstruction. LetA = lim (C (X ))
n→∞ Rn n
be a simple Villadsen algebra, and let Ni,j and Mi,j denote the number of distinct
Rj
projections and the relative matrix size Mi,j = , respectively, in the connecting
Ri
homomorphisms ϕi,j from stages i to j. In [TW09], Toms and Winter were able to
prove the dichotomy that A has strict comparison if and only if
Ni,j
lim = 0
j→∞ Mi,j
for all i, and A doesn't have strict comparison if and only if
Ni,j
lim lim = 1.
i→∞ j→∞ Mi,j
In particular, in this case
Ni,j
( ) lim = εi > 0j→∞ Mi,j∞
Ni,j
for all i; this sequence is a decreasing sequence in j, and (ε )∞i
M i=1
is an
i,j j=1
increasing sequence in i with lim εi = 1 from the multiplicative properties. This
i→∞
number εi is the limit of the proportion of projection maps from Ai compared to the
size of the matrices. When it is positive, as we will see later, the projection block
manages to preserve enough information to prevent strict comparison.
2.4 Tracial States in Villadsen Algebras
The extremal tracial states in Mn(C(X)) come from point evaluations at x:
given A ∈ Mn(C(X)), x ∈ X, the function
τx(A) = Tr(A(x))
31
denes an extreme tracial state on Mn(C(X)). Moreover, for the Villadsen algebras,
we nd for a ∈ An = MRn(C(Xn))+, and x ∈ Xn,
τx′(ϕn,∞a) := lim Tr(ϕn,ka(x, x, . . . , x)) = lim τ(x,x,...,x)(ϕn,ka)
k→∞ k→∞
∞
dim(Xk) ⋃
where (x, x, . . . , x) has Dn,k := copies, denes a tracial state on Ai which
dim(Xn) i=n
extends to A. Thus the extreme traces from point evaluations on An extend naturally
to traces on A. In this section, we make a summary of some results informing the
Villadsen algebras.
Denition 2.4.1. Let A be a C∗ algebra. We say the projections in A separate the
traces if τ1(p) = τ2(p) for every projection p ∈ A implies τ1 = τ2.
The following statement is concluded by Theorem 1.3 in Blackadar, Bratteli,
Elliott, and Kumjian's paper [Bla+92]:
Theorem 2.4.2. Let A be a simple Villadsen algebra. Consider the following state-
ments:
(i) The projections in A separate the traces;
(ii) For every a ∈ Ai,SA, ε > 0, there is j ≥ i such that TV (ϕi,j(a)) < ε,
where TV is the variation of the normalized trac∣e, denoted for b ∈ Aj
TV (b) = sup{|Tr(b(x))− Tr(b(y))| ∣ x, y ∈ Xj},
(iii) A has real rank zero.
We have (iii) =⇒ (ii) ⇐⇒ (i).
Along similar lines, the following was concluded in T. Ho's Ph.D Thesis as
applied to the Villadsen algebras:
Theorem 2.4.3. Let A be a simple Villadsen algebra. The following are equivalent:
(i) A has real rank zero;
(ii) ω1,∞ := lim ω1,j = 0;
j→∞
(iii) TR(A) = 0, where TR is the tracial topological rank ;
(iv)A has slow dimension growth and projections inA separate the traces.
32
We can strengthen these results for the Villadsen algebras.
Lemma 2.4.4. Let A be a simple Villadsen algebra with real rank one. Then the
projections in A do not separate the traces.
Proof. Let
Ri
ωi,∞ := lim ωi,j = lim αi · ... · αj−1 ;
j→∞ j→∞ Rj
We have ωi,∞ > 0 for all i since A is real rank one, by Theorem 2.2.7. For x ∈ Xn
for some n ∈ N, let τx′ be the tracial states on A coming from evaluation at (x, x, . . .).
Let p ∈ A be a projection. Then, there exists a projection pn ∈ An for some
large n such that p ∼ pn. For projections, Tr is a continuous map from A → Z. In
fact,
Tr(pn(x)) = Tr(ϕn,kpn(x, x, . . . , x)) = Tr(ϕn,kpn(y, y, . . . , y)) = Tr(pn(y))
for every k ≥ n and x, y ∈ Xn. Thus,
τx′(p) = Tr(pn(x)) = Tr(pn(y)) = τy′(p)
for every projection p ∈ A and x, y ∈ Xn. But, generally τx′ ̸= τy′ for given x, y. For
example, if x, y ∈ X1 and f ∈ C(X1) is a function which has f(x) = 0 and f(y) = 1,
putting a = diag(f, f, ..., f) ∈ A1 we can see
|τx′(ϕ1,∞a)− τy′(ϕ1,∞a) = lim |Tr(ϕ1,ka(x, x, . . . , x))− Tr(ϕ1,ka(y, y, . . . , y)|
k→∞
≥ ω1,∞ > 0.
Thus, the projections in A do not separate the traces.
Corollary 2.4.5. For simple Villadsen algebras, in Theorem 2.4.2, (i), (ii), and (iii)
are equivalent.
Proof. If A does not have real rank zero, it has real rank one, since it has stable rank
one, by Theorem 1.3.15. Thus, the negation of (iii), i.e. real rank one, implies the
negation of (ii) and (i).
Below we summarize a general statement about Villadsen algebras, which gen-
eralizes in some cases to suitable diagonal AH algebras (c.f Theorem 3.4 of [TW09]).
Equivalence of (vi)-(xv) is well established by Toms, Winter, Niu, Lin, and others.
33
Corollary 2.4.6. Let A be a simple Villadsen algebra. The following are equivalent:
(i) A has real rank zero;
(ii) ω1,∞ = 0;
(iii) TR(A) = 0
(iv) For every a ∈ Ai,SA, ε > 0, there is j ≥ i such that TV (ϕi,j(a)) < ε;
(v) The projections in A separate the traces.
Any of the above implies the following, which are equivalent:
(vi) A has Blackadar's strict comparison of positive elements;
(vii) A has slow dimension growth;
(viii) A tensorially absorbs the Jiang-Su algebra Z;
(ix) A has nite nuclear dimension;
(x) A has nite decomposition rank;
(xi) A is tracially approximately divisible;
(xii) A is tracially Z-absorbing;
(xiii) A has Niu's mean dimension zero;
(xiv) A has Lin's tracial approximate oscillation zero;
(xv) A has almost unperforated Cuntz semigroup.
These each imply
(xvi) K0(A) is weakly unperforated.
Lastly, there are examples witnessing:
(vi)-(xv) ̸ =⇒ (i)-(v): Goodearl [Goo92];
(xvi) ̸ =⇒ (vi)-(xv): Toms [Tom08a].
34
CHAPTER 3
GENERALIZATION TO AH ALGEBRAS OVER CW-COMPLEXES
3.1 Embedded Spheres in the Cube
In this section, we establish some basics about how some sets homeomorphic
to spheres embed naturally into sets homeomorphic to cubes.
Lemma 3.1.1. For each i ∈ N, let Xi be a ∣CW-complex and Qi ⊂ Xi be subsets
such that Qi ≃ [0, 1]Ni for some Ni with N ∣i Ni+1. Let ηi : Q → [0, 1]Nii be these
Nj
homeomorphisms, for each j > i put Ni,j = , and let
( Ni)
πi,j
Ni,j
k : [0, 1]
Nj ≃ [0, 1]Ni → [0, 1]Ni
for k ∈ {1, ..., Ni,j} be the coordinate projections. Let γj : Xj → Qj be a retract, and
for each k ∈ {1, ..., Ni,j}, put
λi,j := η−1k i ◦ π
i,j
k ◦ ηj ◦ γj : Xj → Qi ⊂ Xi.
Then,
(i) There exist injections ιi,jk : Qi→ Qj ⊂ Xj such that for all i, j andx ∈ Qi, 
(λi,j ◦ ιi,jk1 k )(x) =2 x if k1 = k2 .η−1i (0) if k1 ̸= k2
(ii) Given i ∈ N and D ∈ N such that D ≤ Ni − 1, there exists a subset
S̃D ⊂ Qi such that SD ≃ S̃D, a relatively open neighborhood Ui ⊂ Qi of
S̃D, and a retract τ : U → S̃Di .
(iii) Given i ∈ N, j > i, k ∈ {1, ..., Ni,j}, and D ∈ N with D ≤ Ni − 1, let
S̃D ⊂ Qi be the subset in (ii), and ιi,jk be the inj∏ections in (i). There existsNi,j
a subset Sj ⊂ Qj and a homeomorphism ζ : ιi,ji,j k (Qi) → Qj such that
k=1
S ≃ (SD)×Ni,j , and for all ℓ ∈ {1, ..., N D Ni,ji,j} and (x1, ..., xN ) ∈ (S̃ ) ,i,j
λi,jk (S
D i,j i,j
j) = S̃ and λk ◦ ζi,j(ι1 (x1), ..., ι
i,j
N (xi,j N ) = xi,j k.
We will call ζi,j the rectifying homeomorphism of Sj.
35
( )
Proof of (i). For each k ∈ {1, ..., Ni,j}, let (ιi,jk )′ : [0, 1]Ni → [0, 1]Nj ≃ [0, 1]N
N
i i,j be
inclusion in the k slot with 0 in the other coordinates. Let ιi,j1 , ..., ι
i,j
N : Qi → Qj bei,j
dened by
ιi,jk = η
−1
j ◦ (ι
i,j ′
k ) ◦ ηi. ∣
Note that, since γj : Xj → Qj is a retract, we have γ ∣j = idQ , so γ ◦ ιi,j = ιi,jj k k forQj j
each k. Thus, for each x ∈ Qi, we have
(λi,j ◦ ιi,jk k )(x) = (η
−1
i ◦ π
i,j
k ◦ η ◦ η
−1 ◦ (ιi,j)′j j k ◦ ηi)(x)1 2 1 2
= η
−1 ◦ (πi,j ◦ (ιi,j)′i k k )(η1 2 i(x))
x if k1 = k2= .η−1i (0) if k1 ≠ k2
Proof of (ii). Let
∣∣ ( ) 1D∑+1 2U ′ 1i = {(x1, ..., xN ) ∈ [0, 1]Ni (xk − )2 ∈ 1 3( , )},i 2 8 8
k=1
∣ (D1 1 ∑ ) 1+1 21 1
S ′ = {(x1, ..., x Ni ∣ 2D+1, , ..., ) ∈ [0, 1] (xk − ) = }
2 2 2 4
k=1
′ ′ 1 1 1and let τ : Ui → S ′ be retraction along radial lines emanating from(( , , ..., ) in the
D∑2 2 2 ) 1+1 21
rst D+1 coordinates, and projection to 1 in the rest, i.e. if d = (x − )2 ,
2 k 2
k=1
then
′ 1 1 1 1τ (x1, ..., xN ) = ( xi 1, ..., xD+1, , ..., ).4d 4d 2 2
Put
S̃D = η−1S ′i , , Ui = η
−1
i (U
′
i) τ = η
−1
i ◦ τ ′ ◦ ηi.
Then, since η is a homeomorphism and (τ ′)2 = τ as a retract, we have
SD ≃ S ′ ≃ η−1 ′i S , that Ui is an open neighborhood, and τ 2 = τ is a retract.
36
N∏i,j
Proof of (iii). Let ζi,j : ι
i,j
k (Qi) → Qj be the map
k=1 ( )
ζi,j(ι
i,j
1 (x1), ..., ι
i,j −1 i,j ′ −1 i,j ′
N (xi,j N ) = ζi,j ηj ◦ (ι1 ) ) ◦ ηi(x1), ..., ηj (◦ (ιN ) ) ◦ ηi,j i,j i(xN )i,j )
:= η−1j ηi(x1), ..., ηi(xN ) ,i,j
which is well-dened since ηi, η
i,j
j and each ιk are injections for each i, j, k. Thus, for
all k ∈ {1, ..., Ni,j}, ( )
λi,jk ◦ ζ(ι
i,j
1 (x1), ..., ι
i,j (x ) = λi,j η−1N N k j (ηi(x1)(, ..., ηi(x ))i,j i,j Ni,j
= η−
)
1
i ◦ πk ◦ η ◦ η−1j j (ηi(x1), ..., ηi(xN ))i,j
= η−1i (ηi(xℓ))
= xℓ.
The continuous map ζi,j has inv(erse )
ζ−1 i,j i,j i,j i,ji,j (x) = ι1 (λ1 (x)), ..., ιN (λi,j N (x) ,i,j
for we have
ζ−1i,j ◦ ζ (ι
i,j
i,j 1 (x1), ..., ι
i,j
N (x
−1 −1
N ) = ζi,j (ηj (ηi(x1), ..., ηi,j i,j i(xN )))i,j
= (ιi,j1 (λ
i,j(η−11 j (ηi(x1), ..., ηi(xN )), ..., ι
i,j
N (λ
i,j −1
N (ηj (ηi(x1), ..., ηi(xi,j i,j i,j N ))i,j
= (ιi,j1 (x1), ..., ι
i,j
N (x )).i,j Ni,j
Since Qi, Qj are compact Hausdor spaces, a∏nd ζ is a continuous bijection, then it isNi,j
indeed a homeomorphism. Letting S = ζ ( ιi,j(S̃Di,j k )) ⊂ Qj, we have
k=1
N∏i,j
S ≃ ιi,j Dk (S̃ ) ≃ (S
D)×Ni,j
k=1
and λi,jk (Sj) = S̃
D, specically the sphere S̃D coming from the k-th coordinate.
Porism 3.1.2. For each i ∈ N, let Xi b∣∣e a CW-complex and Qi ⊂ Xi be subsets suchthat Q ≃ [0, 1]Ni for some N i,j i,ji i with Ni Ni+1. Let λk : Xj → Qi and ιk : Qi → Xj
be as described in Lemma 3.1.1 for each k ∈ {1, ..., Ni,j}. Then, for each k, ιk ◦ λk is
a local retract from Xj → ιk(Qi) ⊂ Qj.
37
Proof. We have λk ◦ ιk = id, so (ιk ◦ λk) ◦ (ιk ◦ λk) = ιk ◦ λk.
In the next theorem, we establish how projections over a sphere carry across
connecting homomorphisms in Villadsen algebras. This allows us to see the failure of
strict comparison, similar to Example 1.4.8.
Theorem 3.1.3. Suppose the following setup: ∣
(i) For k ∈ N, put Xk = [0, 1]Dk , where D ∣k Dk+1, with D1 ≥ 3, and put
Ak = MR (C(Xk)). Suppose A = lim (An, ϕn,n+1) is a simple Villadsenk n→∞
algebra with seed space X1.
(ii) Let j ∈ N, and suppose N1,j is the number of projection maps from
A Rj1 to Aj in the connecting homomorphisms ϕ1,j, and M1,j = is the
R1
Dj
total number of eigenvalue maps. Suppose for simplicity N1,j = , i.e.
D1
all projection maps appear from X1 → Xj.
(iii) Let U ⊂ X1 be an open neighborhood of some set S ⊂ X1, such
that S ≃ S2 and there exists a retract τ : U → S ≃ S2. Without loss of
generality, we just refer to S as S2.
(iv) Let f ∈ C 20(X1) have support U and equal 1 on S .
(v) Let η ∈ MR1(C(S2)) be a projection.
N
(vi) Let ρj : MR (C(X )) = M (C(X
1,j)) → M (C((S2)N1,j)) be re-
j j Rj 1 Rj
striction, and let τf : M
2
R1(C(S )) → MR1(C(X1)) be the function
f(x)ξ(τ(x)) if x ∈ Uτf (ξ)(x) = .0 if x ̸∈ U
Then, up to rearrangement of the blocks,
ρj ◦ ϕ1,j ◦ τf (η) = diag((η)×N1,j , Kj(η)),
where Kj(η) is a constant block of rank at most Rank(η) · (M −N ), and (η)×N1,j1,j 1,j
is the projection in M (C((S2)N1,jR1·N )) given by1,j
(η)×N1,j(y1, ..., yN ) := diag(η(y1), . . . , η(y1,j N )).1,j
38
Proof. Put ψ1,j = ρj ◦ ϕ1,j ◦ τf . We follow the diagram
ϕ1,j
MR1(C(X1)) MR (C(Xj))j
τ∗f ρj
ψ1,j
MR1(C(S
2)) MR (C((S
2)×N1,j))
j
N
When (x1, ..., x
1,j
N ) ∈ Xj = X1 , up to rearrangement of the blocks,1,j
(ϕ ∗1,j ◦ ι ◦ τf (η))(x1, ..., xN ) = diag(f(π1(x1, ..., xN )) · η(τ(π1(x1, ..., xN ))), . . . ,1,j 1,j 1,j
. . . , f(πN (x1, ..., xN )) · η(τ(π1,j 1,j N (x , ..., x ))), K (η))1,j 1 N1,j j
= diag(f(x1)η(τ(x1)), ..., f(xN )η(τ(xN )), K (η))1,j 1,j j
where
Kj(η) = diag(τ
∗ ∗
f ◦ δ1, ..., τf ◦ δβ )j
is a constant block of βj := M1,j − N1,j point evaluations of τf (η) in the connecting
homomorphisms from A1 to Aj. In particular, Rank(τf (η)(x)) is either Rank(η) when
x ∈ U or 0 when x ̸∈ U , so the rank of this constant point evaluation block Kj(η) is
at most
Rank(Kj(η)) ≤ Rank(η) · (M1,j −N1,j) ≤ Rj − Rank(η) ·N1,j.
Given (x1, ..., xN ) ∈ Xj for x1, ..., xN∣ ∈ X1, then (x1, ..., x ) ∈ (S2)N1,jN if and1,j 1,j 1,j
only if f(xk) = 1 for all k. Likewise, τ ∣ 2 = idS2 as a retract, thusS
(x1, ..., xN ) ∈ (S2)N1,j implies1,j
diag(f(x1)η(τ(x1)), ..., f(xN )η(τ(x1,j N )), K (η)) = diag(η(x ), . . . , η(x ), K (η))1,j j 1 N1,j j
= diag((η)×N1,j , Kj(η))(x1, ..., xN1,).
Remark 3.1.4. Note that, when x ∈ S2, we have τ, f are trivial, so if all the point
evaluations in the connecting homomorphisms δ1, ..., δβ from A1 to A are in S2j orj
outside U , then the block
Kj(η) = diag(τ
∗
f ◦ δ1, ..., τ ∗f ◦ δβ )j
is actually a constant (trivial) projection, and so for x ∈ (S2)N1,j ,
39
ψ1,j(ζ · ξ)(x) = ϕ1,j(ζ · ξ)(x)
= ϕ1,j(ζ)(x) · ϕ1,j(ξ)(x)
= ψ1,jζ(x) · ψ1,jξ(x).
So, ψ1,j is a homomorphism in this case. Since there are nitely many point eval-
uations, such a particular neighborhood U could always be chosen for any given j.
Generally, we note that the constant block Kj(η) is Cuntz equivalent to a constant
projection by Example 1.4.7.
Remark 3.1.5. Suppose (i) and (ii) in the theorem above. Let
ι , ..., ι : [0, 1]Di1 N → [0, 1]Dji,j
be inclusion satisfying
πi,jk ◦ ιk = id .
Dene
U ⊂ Xi, f ∈ C0(U), ξ ∈ MR (C((SD))), and τf : MR (C(SD)) → MR (C(Xi))i i i
in (iii) - (vi) analogously for SD ⊂ Xi = [0, 1]Di , for some D ∈ N. Since f = 1 on
SD with support U , for (x1, ..., xN ) ∈ UNi,j the restriction to (SD)Ni,j ⊂ [0, 1]Dj ofi,j
ϕi,j ◦ τf (ξ∣) is∣ ( )ϕ ◦τ ξ (x , . . . , x ) = diag τ ξ ◦ πi,j, . . . , τ ξ ◦ πi,ji,j f Ni,j 1 N f 1 f N , Kj(ξ)) (x1, . . . , x )U i,j i,j Ni,j
= diag(τfξ ◦ πi,j1 (x1, . . . , xN ), . . . , τ ξ ◦ π
i,j
f N (x1, . . . , xN ), Kj(ξ))i,j i,j i,j
= diag(τ i,jfξ ◦ π1 (x1, , 0, 0, . . .), . . . , τ ξ ◦ π
i,j
f N (0, . . . , 0, xN ), Kj(ξ))i,j i,j
= diag(τ ξ ◦ πi,j i,jf 1 ◦ ι1(x1), . . . , τfξ ◦ πN ◦ ιN (xN ), Ki,j i,j i,j j(ξ)).
This formulation is useful for generalizing this calculation to sets Qi homeomorphic
to cubes.
40
Thanks to H. Lin for his particularly pointed advice on honing these next
formulations.
Remark 3.1.6. For each∣∣ i ∈ N, letXi be a nite connected CW-complex, and (Rn) bea sequence in N with Rn Rn+1 for all n; put An = MRn(C(Xn)). Let Cn : [0, 1] → Xn
and Fn : Xn+1 → [0, 1] be continuous maps, and δ′n := Cn ◦ F ′n : Xn+1 → Xn, i.e. δn
is a map which factors through an interval. Assuming dim(Xj) ≥ 1, one can choose
Fn to be surjective (as a retraction) and Cn : [0, 1] → Xn to be an embedding. In
this case, we obtain a unital homomorphism ψ′I,n : An → MRn(C([0, 1])) which is
surjective and a homomorphism ψ′n : MRn(C([0, 1])) → An+1 which is injective, so δ′n
induces ψ′ ◦ ψ′n I,n : An → An+1.
Let δn : Xn → Xn−1 be a constant map, i.e. δn(y) = xn for some xn ∈ Xn−1
and for all y ∈ Xn. Suppose ϕn,n+1 : An → An+1 is of the form
ϕ (a) = diag(a ◦ λn,n+1, . . . , a ◦ λn,n+1 , a ◦∆n,n+1, ..., a ◦∆n,n+1n,n+1 1 α )n,n+1 1 βn
Rn+1
for each n, where β = − α and ∆n,n+1n n,n+1 k = δn or ∆
n,n+1
k = δ
′
n for eachRn
k ∈ {1, ..., βn} and each λn,n+1i : Xn+1 → Xn is a continuous map. Then with
An = MRn(C(Xn)), we may form the AH algebra A = lim (A , ϕ→∞ n n,n+1).n
Denition 3.1.7. Let Xn be nite connected CW-complexes for each n ∈ N with
dim(Xn) ≥ 1 and suppose A = lim (An, ϕn,n+1), where ϕn,n+1 is of the form
n→∞
ϕ (a) = diag(a ◦ λn,n+1, . . . , a ◦ λn,n+1 , a ◦∆n,n+1 n,n+1n,n+1 1 αn,n+1 1 , ..., a ◦∆β ).n
We will say A is a generalized interval Villadsen algebra if each λn,n+1i : Xn+1 → Xn is
a continuous map and every ∆n,n+1j is either a point-evaluation, or induces ψ
′
n ◦ ψ′I,n,
where ψ′I,n : An → MRn(C([0, 1])) is a surjective unital homomorphism and a homo-
morphism ψ′n : MRn(C([0, 1])) → An+1 which is injective.
Note that one may write
ϕn,n+1(f) = diag(Ψn(f), ψn ◦ ψI,n(f)) for allf ∈ An,
where Ψn : An → Mαn,n+1Rn(C(Xn+1)) is a unital homomorphism and ψI,n : An → Cn
is a unital surjective homomorphism and ψn : Cn → An+1 is an injective homomor-
phism, and where Cn is a nite direct sum of full matrix algebras and C
∗-algebras of
the form Mr(C([0, 1])).
41
Lemma 3.1.8. Let X, Y be CW-complexes, A1 = MR(C(X)), A2 = MR(C(Y )),
ω : Y → [0, 1] and P : [0, 1] → X be continuous maps. Then, for every a ∈ A1,
a ◦ P ◦ ω ≾ IR×R
where IR×R ∈ A2 is the R×R identity matrix.
Proof. By Theorem 1 of L. Robert [Rob13], Cu(Mn(C([0, 1]))) ≃ LSC(X,N), thus
Mn(C([0, 1])) has strict comparison. Given a ∈ Mn(C[0, 1]), the value of
[a] ∈ LSC(X,N) is [a](x) = Rank(a(x)) ≤ R, in particular. Let a ∈ MR(C(X));
then,
a ◦ P ∈ MR(C([0, 1])) ≾ IR×R ∈ MR(C([0, 1])).
Let Ω : MR(C([0, 1])) be the homomorphism Ω(A)(x) = A(ω(x)). Thus, as a Cuntz-
morphism, Cu(Ω) preserves Cuntz inequality, and certainly Ω(IR×R) = IR×R as a
constant matrix. Hence
Ω(a ◦ P ) = a ◦ P ◦ ω ≾ IR×R.
Recall that, by Theorem 2.2.7, when A is a simple Villadsen algebra of the
αi,j
rst type with → 0, where αi,j is the total number of projection maps (including
Mi,j
multiplicity) from Ai to Aj, then A has real rank zero. Therefore, it is Tracially AF
by Theorem 2.1 of H. Lin [Lin03]. An analogous result can be shown for the general-
ized interval Villadsen algebras. Special thanks to H. Lin who assisted greatly in the
following proof.
Denote by I the class of C∗-algebras which are nite direct sums of C∗-algebras
of the form Mr or Mk(C(X)) for some compact subset X ⊂ [0, 1].
Lemma 3.1.9. Let A be a unital simple generalized interval Villadsen algebra as in
Denition 3.1.7, and a ∈ A+\{0}
Rn+1
. Suppose that for all n, we have αn,n+1 < , i.e.
Rn
at least one interval or evaluation map occurs. Then, there is a nonzero projection
e ∈ An for some n such that ϕn,∞(e) ≾ a and e(x) = e(x′) for all x, x′ ∈ Xn.
42
Proof. First, we show that A has the following property: given any ε > 0, and nite
subset F ⊂ A which contains nonzero b, there exists a nonzero projection q ∈ A such
that for all f ∈ F ,
(i) ∥qf − fq∥ < ε;
(ii) qfq ∈ε C for some C ∈ I;
(iii) ∥qbq∥ ≥ (1− ε)∥b∥.
To prove this, we may assume without loss of generality b ∈ A+ with ∥b∥ = 1. Let
ε > 0, and dene f ∈ C([0, 1])+ such that
0 ≤ f ≤ 1, f(t) = 1 for t ∈ [1− ε , 1], and f(t) = 0 for t ∈ [0, 1− ε ].
4 2
Put bε = f(b); since A is simple, there exist x1, . . . xℓ ∈ A such that
∑ℓ
x∗i bεxi = 1.
i=1
We may assume F = ϕm,∞(G) for some nite set G ⊂ Am and for some m ∈ N. In
particular, there exists b′ ∈ G ∩ (Am)+ such that bε = ϕm,∞(b′). We may further
assume that there exists x′i ∈ Am for i ∈ {1, . . . , ℓ} such that
∑ℓ
∥ 1(x′)∗b′x′i i − 1Am∥ < .4
i=1
We may then nd y ∈ Am such that
∑ℓ
y∗(x′ ∗ ′ ′i) b xiy = 1Am (∗).
i=1
Choose G1 = G ∪ {y, y∗
Rm+1
, xi, x
′ ∗ ′
i, y (xi)
∗, xiy}. Since βm,m+1 = − αm,m+1 ≥ 1, we
Rm
may write
ϕm(f) = diag(Ψm(f), ψm ◦ ψI,m(f))
for all f ∈ Am, where Ψm : Am → MRm·αm,m+1(C(Xm+1)) is a unital homomorphism
and ψI,m : Am → C ′ and ψ ′m : C → Mβm(C(Xm+1)) unital homomorphisms which are
surjective and injective respectively, for some C ′ ∈ I.
43
Put C = ϕ ′m+1,∞(ψm(C )) and q0 = ψm◦ψI,m(1Am). Thus, C ∈ I and q0 ∈ Am+1
is a projection. Putting q = ϕm+1,∞(q0) we have qg = gq for all g ∈ G1. In particular,
qb = bq, so we have s∑hown (i) and (ii) hold. From (∗) we haveℓ
ϕ ∗ ′m,∞(y (xi)
∗)qϕm,∞(b
′)qϕ ′m,∞(xiy) = q,
i=1
which implies f(qbq) = qbεq ̸= 0. Hence,
∥qbq∥ ≥ (1− ε) = (1− ε)∥b∥,
which shows (iii). Since A is a simple C∗-algebra, by (i)-(iii) and by the proof of
Theorem 3.2 of [Lin07], A has property (SP): that every non-zero hereditary C∗-
subalgebra of A contains a nonzero projection. We may apply property (SP) to get
the desired conclusion.
Let a ∈ A+ \ {0}, and assume without loss of generality ∥a∥ = 1. Since A has
property (SP), there is a nonzero projection q ∈ aAa, so q ≾ a. By Lemma 2.7.2 in
[Lin01], there is i ∈ N and a projection q′ ∈ Ak such that ϕ ′k,∞(q ) ∼ q. As before,
we may write
ϕm(f) = diag(Ψm(f), ψm ◦ ψI,m(f))
for all f ∈ Am, where Ψm : Am → MRm·αm,m+1(C(Xm+1)) is a unital homomorphism
and ψ ′′I,m : Am → C and ψm : C ′′ → Mβm(C(Xm+1)) unital homomorphisms which
are surjective and injective respectively, for some C ′′ ∈ I.
Put p1 = ψk ◦ ψ (q′I,k ) ∈ ψ ′′k(C ). Since ψk ◦ ψI,k is injective, p1 ̸= 0. As-
sume Rank(q′) = r ∈ (0, βkRk) and choose a constant projection q ′′0 ∈ C such that
Rank(ψk(q0)) = r. Put C1 = ψ (C ′′k ); then, in C1, we have q1 := ψk(q0) ∼ p1. Thus, e
must also be constant in Mβ (C(Xk m+1)). With n = k + 1, p = ϕk+1,∞(q0), we then
have
p ∼ ϕ ′k+1,∞(p1) ≾ ϕk+1,∞(q ) ∼ q ≾ a.
Theorem 3.1.10. Let A be a unital simple generalized interval Villadsen algebra as
in Denition 3.1.7. Suppose that for all j we have
Rj
lim α
→∞ j,n
· = 0.
n Rn
Then, A has tracial rank at most one.
44
Proof. Following Denition 5.3 of [Lin07], we show the following: given any ε > 0,
element a ∈ A+ \ {0}, and nite subset F ⊂ A, there exists a projection p ∈ A and
a C∗-algebra C ∈ I such that
(i) ∥pf − fp∥ < ε for all f ∈ F ,
(ii) pfp ∈ C for all f ∈ F , and
(iii) 1− p ≾ a.
To show this, we may assume without loss of generality that F = ϕm,∞(F1) for
some m ∈ N and some nite subset F1 ⊂ Am. By Lemma 3.1.9, choose a nonzero
projection e ∈ An such that e is constant in MRm(C(Xm)) and q := ϕm,∞(e) ≾ a.
Since A is simple, we have ∣
σ = inf{τ(q) ∣ τ ∈ T (A)} > 0.
We claim there exists n0 ∈ N such that
σ
τ(ϕm,n(e)) >
2
for all τ ∈ T (An) and n ≥ n0. Otherwise, there would be a subsequence {nk} such
that nk → ∞ and τn ∈ T (An ) withk k
σ
τn (ϕm,n (e)) <k k 2
for all k.
Since each An is nuclear, by Theorem 2.3.13 of [Lin01], there is a sequence of
completely-positive contractive (cpc) maps Lk : A → ϕk,∞(Ak) ≃ Ak such that for
any m ∈ N we have
lim ∥Lk(ϕm,∞(a)− a∥ = 0 (∗)
k→∞
for all a ∈ Am. Dene tk : A → C by
tk(b) = τn ◦ Lk(b)k
for each b ∈ A, k ∈ N. Let t be a weak-∗ limit of {tk}. Note that each tk is a state,
and A is unital, and so is t, so by (∗), we have t is a tracial state on A. However, we
have
σ
t(ϕm,∞(e)) ≤
2
45
∣
from the above estimates, contradicting that σ = inf{τ(q) ∣ τ ∈ T (A)} > 0. Whence,
the claim is proved.
Now, choose n ≥ m such that
Rm σ
αm,n · <
Rn 2
and
σ
t(ϕm,n+1(e)) ≥
2
for all t ∈ T (An+1). We can write
ϕm,n+1(f) = diag(Ψ(f), ψ ◦ ψI(f))
for all f ∈ Am, where Ψ : Am → An+1 is a homomorphism, Ψ(1Am) is a constant
projection of rank Rm ·αm,n+1, ψI : Am → C0 and ψ : C0 → An+1 are homomorphisms
which are surjective and injective, respectively, for some C0 ∈ I. Put
C = ϕn+1,∞ ◦ ψ(C0); then, C ∈ I. Put q0 = Ψ(1Am) and
p = ϕn+1,∞(ψ ◦ ψI(1Am)) = 1− ϕn+1,∞(q0).
Thus, 1− p = ϕn+1,∞(q0), and for any f ∈ F1 ⊂ Am we have
(a) (1− q0)ϕm,n+1(f) = ϕm,n+1(f)(1− q0) and
(b) (1− q0)ϕm,n+1(f)(1− q0) ∈ ψ(C0).
Hence, we have (i) and (ii) above. Since Ψ(1Am) has rank αm,n+1 · Rm, we have
Rm σ σ
t(q0) = αm,n+1 ·Rm. Thus, since αm,n · < and t(ϕm,n+1(e)) ≥ , we nd
Rn 2 2
t(q0) < t(ϕm,n+1(e))
for all t ∈ T (An+1). Since both q0 and e are constant projections in MRn+1(C(Xn+1)),
we conclude
q0 ≾ ϕm,n+1(e).
Thus, (iii) follows from
1− p = ϕn+1,∞(q0) ≾ ϕm,∞(e) ≾ a.
Therefore, A has tracial rank at most one.
46
3.2 Prescribed Dimension Growth over CW-Complexes
In this section we pose and answer two questions:
Question 3.2.1. Given any number c ∈ (0,∞), does there always exist an AH
dim(Xn)
algebra A = lim (M (C(X ))) such that lim = c and A does not have
n→∞ Rn n n→∞ Rn
strict comparison?
Question 3.2.2. Given a sequence (Xn) of CW-complexes, and a sequence Rn such
dim(Xn)
that → c ∈ (0,∞) and dim(Xn) → ∞, does there always exist an AH
Rn
construction A = lim (MR (C(Xn )), ϕn ), at least over some subsequence of (Xn)
i→∞ ni i i
and (Rn), which fails to have strict comparison?
We will answer both questions in the armative by generalizing the Villadsen
construction to the setting of CW-complexes by factoring Xj through a set Qj ⊂ Xj,
a set which is homeomorphic to a cube. First, we exhibit how to get to a particular
constant c ∈ (0,∞) with a simple Villadsen algebra.
Theorem 3.2.3. Let c ∈ (0,∞). There exists a simple Villadsen algebra
A = lim (MRn(C(Xn)), ϕn) such that
n→∞
dim(Xn)
lim = c
n→∞ Rn
and which does not have strict comparison of positive elements.
Proof. Suppose c ∈ (0, 1). Let us dene a sequence qn in Q. There exists a rational
c 1
number q1 ∈ (c, 1). Let q2 satisfy < q2 < 1 with q1q2 − c < . Likewise, there
q1 2
c 1
exists q3 such that < q3 < 1 and q1q2q3 − c < . Continuing in this fashion
q1q2 3
recursively, we have
1 c
qn : q1...q2qn − c < with ∏n−1 < qn < 1.n
j=1 qj
∏n
Then, c < qj < 1 is a decreasing sequence with limit c. We necessarily have
j=1
qk → 1 as k → ∞.
47
Put
nk
qk = ,
∏mkk
Rk = mj, and
j=1
Nj = nj,
X1 = I := [0, 1],
Xk = (X
×Nk
k−1) .
Thus, ∏k
dim(Xk) = dim(X1) · Nj = dim(X1) ·N1,k.
j=1 ∏k
Let (x ∞ j=1 Njj) be a dense sequence in X1 with xk ∈ I := Xk without loss of
generality. Put ∏k
Ak = MR (C(Xk)) = M∏k m C(I j=1 nj)k j=1 j
and dene ϕk−1,k : Ak−1 → Ak by
ϕk−1,k(a) = diag(a ◦ πk1 , ..., a ◦ πkN , δ (a), ..., δ (a))k k k
where πkj are the projection maps from Xk → Xk−1 for each j ∈ {1, ..., Nk} and
δk(a) = a(xk) with multiplicity mk − nk ≥ 1. Then, A ≃ (An, ϕn) is a simple
Villadsen algebra with ∏k ∞N ∏1,k nj dim(Xk)
lim = lim = = dim(I) · qk = c.
k→∞ M1,k k→∞ m Rj=1 j k k=1
Since c > 0, we have A does not have strict comparison by Toms-Winter [TW09]
Lemma 5.1.
c
Suppose c ≥ 1; there exists an integer M > c; thus ∈ (0, 1). Let X1 be
M
any compact Hausdor space of dimension M ; obtain (xj), nj,mj as before. Then
the analogue of the abo∏ve givesk ∏∞N1,k nj dim(Xk) c
lim = lim = dim(X1) · = M · qk = M · = c
k→∞ M1,k k→∞ mj Rk Mj=1 k=1
which, likewise, fails to have strict comparison by Toms-Winter [TW09] Lemma 5.1.
48
Next, we present a particular construction by factoring through cubes embed-
ded in given CW-complexes.
Construction 3.2.4. For each i ∈ N, let Xi be a nite connected∣∣ CW-complex with1 ≤ dim(Xi) ≤ dim(Xi+1), and (Rn) be a sequence in N with Rn Rn+1 for all n; put
An = MRn(C(Xn)). ∣∣As in Lemma 3.1.1, let Qi ⊂ Xi be subsets such that Qi ≃ [0, 1]Nifor some Ni with Ni Ni+1. Further, let ηi : Qi → [0, 1]Ni be these homeomorphisms,
Nj
for each j > i put Ni,j = and let
Ni ( )
i,j Nj ≃ N Ni i,jπk : [0, 1] [0, 1] → [0, 1]
Ni
for k ∈ {1, ..., Ni,j} be the coordinate projections. Let γj : Xj → Qj be a retract, and
for each k ∈ {1, ..., Ni,j}, put
λi,j := η−1 ◦ πi,jk i k ◦ ηj ◦ γj : Xj → Qi ⊂ Xi
and
ϕ n,n+1n,n+1(a) = diag(a ◦ λ1 , . . . , a ◦ λ
n,n+1 , a ◦∆n,n+1, ..., a ◦∆n,n+1Nn,n+1 1 β ),n
where every ∆n,n+1j is either a point-evaluation, or induces the composition ψ
′ ◦ψ′n I,n,
where ψ′I,n : An → MRn(C([0, 1])) is a surjective unital homomorphism and
ψ′n : MRn(C([0, 1])) → An+1 is an injective homomorphism. Therefore, we have
A = lim (An, ϕn,n+1) is a generalized interval Villadsen algebra as in Denition 3.1.7.
n→∞
Remark 3.2.5. Every Villadsen algebra over Xi = [0, 1]
Ni for some D ∈ N is trivially
a generalized interval Villadsen algebra produced by Construction 3.2.4, with Qi =
[0, 1]Ni and ηi = id. Of course the maps ∆
n,n+1
k are all just constant maps in this
case. We allow for factoring through an interval because it both trivialize vector
bundles and allows us to make an argument later to produce injectivity in connecting
homomorphisms between potentially quite dierent CW-complexes.
49
Theorem 3.2.6. Let Xn be given CW-complexes with dimension dim(Xn) = dn,
and let (Rn)n∈N be ∣∣a sequence in N. Suppose that (dn) and (Rn) are monotonicallyincreasing, with Rn Rn+1 and dn → ∞, and suppose
dn
lim inf = c ∈ (0,∞).
n→∞ Rn
Then, there exists a subsequence (an) of N and connecting homomorphisms
ϕn,n+1 : MRa (C(Xan)) → MRa (C(Xn n+1 an+1))
such that A := lim (MRa (C(Xan)), ϕn,n+1) is a simple, unital AH algebra which is
n→∞ n
not Z-stable. ( )
3
Proof. Using the fact that Rn, dn → ∞, let (qn) be a sequence in Q∩ , 1 , writing
4
Nn
qn = for Nn,Mn ∈ Z, satisfying the following properties:
Mn∏∞ 2
(i) qj > ;
3
j=1
Rn
(ii) Mk =
k+1 > 2k for some increasing subsequence (nk) of N;
Rnk
1
(iii) qk · c ·Rn < dn − 2 < dn < · c ·Rk k k n .q kk
Restricting (Rk) and (dk) to the subsequence (nk), we regard Rn as Rj and dj n as dj j
from here on. Write An := MRn(C(Xn)). Without loss of generality, we may assume
assume 1d
2 1
> 6 and M1 ≥ 4.
1
Put D1 = 6 ≤ d1 and
2
Dk = D1 ·N1 · ... ·Nk−1;
note that
Rk = M1 ·M2 · ... ·Mk−1 ·R1.
50
Therefore,
D2 = D1 ·N1
1
< d1 · q1 ·M1
2
1 · 1< · c ·R1 · q1 ·M1
2 q1
1 · 1= · c ·R2 · q1
2 q1
1
< · 1 · d2.
2 q1
In general,
k−1
1 ∏· 1 3Dk < dk < dk
∣ 2 qj=1 k 4 1
with 3∣Dk. Note that there exists L > 1 such that D1 > d1, so we have for all k
L
Dk D1
= · q1 ·
2 D1 2 1 d1 2 1 c
... · qk−1 > · > · · > · · q1 · c > > 0 (∗).
Rk R1 3 R1 3 L R1 3 L 2L
Since Xn is a CW-complex with Dn ≤ dim(Xn) = dn, there exists a compact
subset Qn ⊂ Xn for each n such that
Q ≃ IDn = (IDn−1)×Nn−1n .
Let ηn : Q → IDnn be such a homeomorphism. For k ∈ {1, ..., Nn−1} let
πnk : I
Dn → IDn−1
be projection to the k-IDn factor for each k ∈ {1, . . . , Nn−1}. Since Xn is a CW-
complex, there exists a retract
γn :∏Xn → Qnj−1Dj
for each n. For j ≥ i, put Di,j = = Nk. When j > i we have IDj = (IDi)×Di,j ;
Di
k=i
let πi,j Dj Di Dik : I → I be projection onto the k-I factor, for each k ∈ {1, ...Di,j}.
For each i ∈ N, j > i, k ∈ {1, . . . , Di,j}, put
λi,j := η−1 ◦ πi,jk i k ◦ ηj ◦ γj : Xj → Qi ⊂ Xi.
51
Let (xn) be a dense sequence in I
∞ such that x ∈ IDnn ⊂ I∞ for each n. Dene
δn : Xn+1 → Xn to be the constant map
δn(x) = xn;
thus, a ◦ δn(x) = a(xn) is a constant matrix. Put An = MRn(C(Xn)) and dene
ϕn,n+1 : An → An+1 by
ϕn,n+1(a) = diag(a ◦ λn,n+1 n,n+11 , ..., a ◦ λN , a ◦ δn n, . . . , a ◦ δn)
with A = lim (An, ϕn,n+1). Thus, A has been produced via Construction 3.2.4. But,
n→∞
in fact, we can show that A itself is isomorphic to a simple Villadsen algebra over
the cube, noting that the above maps ϕn,n+1 are not injective.
Consider the maps
ιn : MRn(C(I
Dn)) → MRn(C(Xn)) and ρ Dnn : MRn(C(Xn)) → MRn(C(I ))
dened by ∣
ιn(a)(x) = a(ηn ◦ γn(x)), ρn(a)(x) = a∣ (η−1n (x)),Qn
and ϕ̃ Dn Dn+1n,n+1 : MRn(C(I )) → MRn+1(C(I )) dened by
ϕ̃n,n+1(a) = diag(a ◦ πn,n+11 , ..., a ◦ π
n,n+1
D , a ◦ δ
′
n, ..., a ◦ δ′n,n+1 n),
where a ◦ δ′n(x) = a(xn) is again point evaluation. Put Bn = M DnRn(C(I )); then
by density of the sequence (xn), we have B := lim (Bn, ϕ ˜→∞ n,n+1) is a simple Villadsenn
algebra of the rst type. By construction, in particular (∗) above, Di,j is the number
Rj
of projections appearing in eigenvalue maps from Bj to Bi and Mi,j := is the total
Ri
number of eigenvalue maps, then for every i ∈ N, with
Ni,j c
lim > > 0.
j→∞ Mi,j 2L
Therefore, by Toms-Winter [TW09] Lemma 5.1, B does not have strict comparison.
52
Consider the following diagram:
ϕ1,2 ϕ2,3 ϕ3,4
MR1(C(X1)) MR2(C(X2)) MR3(C(X3)) ...
ι1 ρ1 ι2 ρ2 ι3 ρ3
D ϕ̃1,2 D ϕ̃2,3 D ϕ̃3,4MR1(C(I
1)) M 2 3R2(C(I )) MR3(C(I )) ...
For a ∈ MRn(C(Xn)), x ∈ Xn+1, the map ϕn := ϕn,n+1 is given by the diagram:
(ιn,n+1 ◦ ϕ̃n ◦ ρ∣∣n)(a)(x) = (ϕ̃n ◦ ρn)(a)(ηn+1 ◦ γn+1(x))= ϕ̃n(a ∣∣◦ η
−1
n )(ηn+1 ◦ γn+1(x))Qn ∣ ∣
= diag((a ◦ η−1) ◦ πn∣,n+1, . . . , a∣ ◦ η−1) ◦ πn,n+1 , δ′ ((a∣ ◦ η−1)), . . . ,Qn n 1∣ Qn n Dn,n+1 n Qn n, . . . , δ′ −1n((a ◦ η )))(ηQ n n+1 ◦ γn+1(x))n
= diag(a ◦ η−1 ◦ πn,n+1n 1 ◦ ηn+1 ◦ γn+1(x), . . . ,
, . . . , a ◦ η−1n ◦ π
n,n+1
1 ◦ ηn+1 ◦ γn+1(x), δn(a), . . . , δn(a))
= ϕn(a)(x).
Note that we have ηn ◦γn ◦η−1n = idIDn for each n, since γn = idQn as a retract. Thus,
for a ∈ M (C(IDnRn )) and x ∈ IDn+1 ,
∣
(ρn+1 ◦ ϕn ◦ ιn)(a)(x) = (ϕn ◦ ιn)(a)∣ (η−1Qn+1 n+1(x))
= diag(ιn(a) ◦ η−1 n,n+1n ◦ π1 ◦ ηn+1 ◦ γn+1, . . . , ∣
, . . . , ιn(a) ◦ η−1 ◦ πn,n+1n D ◦ ηn+1 ◦ γn+1, δn(ιn(a)), . . . , δ (ι (a)))∣ (η−1n n n Qn+1 n+1(x))
= diag(a ◦ (η −1 −1n ◦ γn ◦ ηn ) ◦ π1 ◦ (ηn+1 ◦ γn+1 ◦ ηn+1)(x), . . . ,
, . . . , a ◦ (η −1 −1n ◦ γn ◦ ηn ) ◦ π1 ◦ (ηn+1 ◦ γn+1 ◦ ηn+1)(x),
a(ηn ◦ γ ◦ η−1n n (xn)), . . . , a(ηn ◦ γn ◦ η−1n (xn)))
= diag(a ◦ πn,n+1(x), . . . , a ◦ πn,n+11 D (x), a(xn), . . . , a(xn n))
= diag(a ◦ πn,n+1, . . . , a ◦ πn,n+1 ′1 D , δn(a), ..., δ
′
n(a))n
= ϕ̃n(a)(x).
Indeed, the diagram is fully intertwining; whence A ≃ B and A does not have strict
comparison.
53
3.3 Injectivity
In the nale, we answer a followup questions about modifying such a con-
struction to preserve information between the connecting homomorphisms, answering
again in the armative:
Question 3.3.1. Can the connecting homomorphisms ϕn be made injective without
losing signicatnt information about a lack of Z-stability?
Lemma 3∣∣.3.2. For each i ∈ N, let Xi be a CW-complex, and (Rn) be a sequence inN with Rn Rn+1 for all n; put An = MRn(C(Xn)). Suppose A = lim (A , ϕ→∞ n n,n+1) is an
generalized interval Villadsen algebra as in Denition 3.1.7 which is produced by Con-
struction 3.2.4, assuming this construction's notation along with that of Lemma 3.1.1.
Thus, the connecting homomorphisms ϕn,n+1 are obtained from factoring through an
interval, or projections over a cube. Let
(i) η ∈ MR (C(S̃D)) be a projection, for some i,D ∈ N where S̃D ⊂ Qi i ⊂
Xi has S̃
D ≃ SD as given by Lemma 3.1.1 (ii),
(ii) Ui ⊂ Xi be an open neighborhood of S̃D with retract τ : U → S̃Di , as
given by Lemma 3.1.1 (ii),
(iii) j > i, and Sj ⊂ Qj be the subset given in Lemma 3.1.1 (iii) with
rectifying homeomorphism ζi,j,
(iv) ζ ′ : (S̃D)Ni,j → Sj be the homeomorphism
ζ ′(x1, ..., xN ) = ζi,j(ι
i,j
1 (x1), . . . ι
i,j
i,j N (xi,j N ),i,j
where ιi,j, ..., ιi,j1 N are the injections given in Lemma 3.1.1 (i) with respecti,j
to which the rectifying homeomorphism ζi,j satises, for all k,
λi,j ′ i,j i,j i,jk ◦ ζ (x1, . . . , xk) = λk ◦ ζi,j(ι1 (x1), ..., ιN (xi,j N ) = x .i,j k
(v) f ∈ C0(Ui) be equal to 1 on S̃D and 0 o Ui.
(vi) τ ∗f η ∈ MR (C0(Ui)) ⊂ MR (C(Xi)) be the functioni i
τ ∗ 0 if x ̸∈ Uif η(x) = .f(x)η(τ(x)) if x ∈ Ui
54
(
Then, ϕ τ ∗
) ∣
η∣ ∣( i,j f) S ∈ MR (C(Sj)) ≃ C(Sj,MR ) satisesj j j
ϕ ∗ ∣ ′i,jτf η S ◦ ζ (x1, ..., xN ) = diag(η(x1), . . . η((xN ), Ki,j)(η)(x1, ..., xN )),j i,j i,j i,j
Rj
where Ki,j(η) is a block of rank at most Rank(η) − ·Ni,j comprised of either
Ri
maps which are factored through an interval or which are constant.
Rj
Proof. Put βi,j = − Ni,j, which is the total number of maps which are constant
Ri
or factored through an interval from stage i to stage j. Therefore, the diagonal
block Ki,j(η)
′ ∈ Mβ (C(Xj)) appearing in ϕ τ ∗i,j f η of such maps has rank at mosti,j
Rank(η) · βi,j. Specically, K ′i,j(η) is
K (η)′ = diag(τ ∗i,j f η ◦∆
i,j
1 , . . . τ
∗η ◦∆i,jf β ),i,j
where ∆i,j1 , . . .∆
i,j
β : Xj → Xi are the continuous maps which are constant o∣r factori,j
through an interval from Construction 3.2.4. Thus, puttingK ′∣ ′i,j(η) := Ki,j(η) S ◦ζ ∈j
Mβ (C((S̃
D)Ni,j), we have Ki,j(η) has at most the same rank. Since i, j are xed,i,j
for the main calculation we may suppress the symbol i, j" and put
λk := λ
i,j
k , ιk := ι
i,j
k , ζ := ζi,j, K(η) := Ki,j(η), K(η)
′ := Ki,j(η)
′ and N := Ni,j.
Let (x1, ..., xN) ∈ (S̃D)N be arbitrary; put
y = ζ ′(x1, ..., xN) = ζ(ι1(x1), ..., ιN(xN)),
w(hich is u)n∣ique in Sj since ζ
′ is a ho∣meomorphism. Analogously to Remark 3.1.5:
ϕ ∗ ∣ ′i,jτf η ◦ζ (x1, ...(, xN) = ϕ ∗i,jτf η∣S (y)Sj j ∣ )
= diag(τ ∗η ◦ λ1, . . . , τ ∗η ◦ λ ,K(η)′∣N S ) (∣y)j )
= diag τ ∗η ◦ λ1(y), . . . , τ ∗η ◦ λ ′∣N(y), K(η) S (y))j
= diag(τ ∗η ◦ λ1 ◦ ζ(ι1(x1), . . . , ιN(xN)), . . . ∣
. . . , τ ∗η ◦ λN ◦ ζ(ι1(x1), . . . , ιN(xN)), K(η)′∣S ◦ ζ ′(x1, ..., xN)).j
= diag(τ ∗η(y1), . . . , τ
∗η(yN), K(η)(x1, ..., xN)).
= diag(η(x1), . . . , η(xN), K(η)(x1, ..., xN))
55
Lemma 3.3.3. Suppose A = lim (An, ϕn,n+1) is a generalized interval Villadsen
n→∞
algebra as in Denition 3.1.7 which is produced by Construction 3.2.4, assuming
this construction's notation along with that of Lemma 3.1.1. Thus, the connecting
homomorphisms ϕn,n+1 are obtained from factoring through an interval, or projections
over a cube. Let η ∈ M∣∣ R (C(S̃D)) be the projection in (i), for some i,D. SupposeiB ∈ MR (C(Xi)) has B D = 0, B is orthogonal to τ ∗f η, and A ∈ MR (C(X )) is giveni S̃ i i
by
A = τ ∗f η +B.
Let ψ : (SD)Ni,j → (S̃D)Ni,j be the natural coordinate-wise homeomorphism.
With t∣∣his setup, if for some Q ∈ N, k ≥ i, we have ΘQ ∈ MR (C(Xk))ksatisfying [ΘQ S ] = [θQ], a trivial rank Q projection, andk
(ΘQ) ≾ (ϕi,kA) ,
then, for j ≥ k, in K0(MR (C((SD)Ni,j)))j
[θ ]×Nk,jQ = [θQ×N ] ≤ [η ◦ ψ]×Ni,j ⊕ θk,j ℓ ,i,j
where [σ]Ni,j denotes the equivalence class of σ×Ni,j ∈ M D N· C((S ) i,jR N )) given byi i,j
σ×Ni,j(x1, ..., xN ) := diag(σ(x1), . . . , σ(xi,j N )),i,j
and θℓ is a trivial projection of rank ℓi,j withi,j
ℓi,j := 2(Rj −Ri ·Ni,j)
Proof. Let Ψ : M (C(S̃D)Ni,j) → M (C((SD)Ni,j)) ≃ C((SD)Ni,jR R ,MR ) be the iso-j j j
morphism
∣ ΨA ∣= A ◦ ψ.
Note that since B∣ 2 = 0, we have ϕ ∣i,jB 2 is a matrix factored through [0, 1] of rankS̃ S̃
1
no more than ℓi,j = Ri(Rj −Ni,j), for all j ≥ i. Likewise, so is Ki,j(η). By Lemma
2
3.1.8, these elements are comparable to trivial ones after having factored through
Cu(Mn(C[0, 1])): we have ∣
ϕi,jB∣ D ≾ θ 1 ℓ and Ki,j(η) ≾ θ 1 .S̃ 2 i,j ℓ2 i,j
56
∣
Likewise, we have [ϕ Θ ∣ ×Nk,jk,j Q ] ≥ [θQ] = [θQ·N ] by cutting down by the trivialSj k,j
block.
By functoriality of Cu(·) (c.f. Remark 1.4.4), we have (ΘQ) ≾ (ϕi,kA) implies
in the Cuntz se[migroup C∣ u](M (C((S[
D)Ni,jR )))j ∣ ]
Cu(Ψ) (ϕ ∣k,jΘQ) S ≤ Cu(Ψ) [((ϕi,jA) ∣j Sj ) ∣ ]
= Cu(Ψ)( ϕ τ ∗ ∣i,j f η + ϕ[i,jB) Sj∣ ])
= Cu(Ψ)([(ϕi,jη)])⊕ [(ϕ ∣i,jB)) Sj [ ])
= Cu(Ψ) [η]×
∣
Ni,j ⊕ K ∣i,j(η)]⊕ (ϕi,jB)) Sj
≤ [η ◦ ψ]×Ni,j ⊕ [θ 1 ℓ ]⊕ [θ 1 ]
2 i,j
ℓ
2 i,j
= [η ◦ ψ]×Ni,j ⊕ [θℓ ].i,j
As noted, we have [ ∣ ]
Cu(Ψ) (ϕk,jΘ ) ∣Q S ≥ [θQ·N ].j k,j
So, putting these inequalities together, we have that if (ΘQ) ≾ (ϕi,kA), then
[θ ]×Nk,j = [θ ] ≤ [η ◦ ψ]×Ni,jQ Q×N ⊕ θk,j ℓ ,i,j
in Cu(M (C((SD)Ni,jR ))). But, for projections Cuntz inequality is the same asj
Murray-von Neumann inequality, therefore we conclude in K0((C((S
D)QNi,j)))
[θ ]×Nk,jQ = [θQ·N ] ≤ [η ◦ ψ]×Ni,j ⊕ θ .k,j ℓi,j
In the nal theorem, we establish that the construction in Theorem 3.2.6 can
be modied to allow for injective connecting homomorphisms, thereby preserving the
information of each CW-complex.
57
Theorem 3.3.4. Let Xn be given nite connected CW-complexes and let (Rn)n∈N
be ∣∣a sequence in N. Suppose that (dn) and (Rn) are monotonically increasing, withRn Rn+1 and dn → ∞, and suppose
dim(Xn)
lim inf = c ∈ (0,∞).
n→∞ Rn
Then, there exists a subsequence (an) of N and injective connecting homomorphisms
ϕn,n+1 : MRa (C(Xn an)) → MRa (C(X ))n+1 an+1
such that A := lim (MRa (C(Xan)), ϕn,n+1) is a simple, unital AH algebra which is
n→∞ n
not Z-stable.
Proof. Construct the subsequence and algebras Ak = Ra (C(Xa )) with continuousk k
maps λn,n+1k : Xa → Xa factoring through a cube and δn evaluation on the densek+1 k
sequence (xn) in I
∞ as in the proof of Theorem 3.2.6. Relabeling as before, we may
replace each Aan with just An. Without loss of generality, we may choose the se-
Nn
quence qn such that qn = has Mn −Nn ≥ 2.
Mn
Since Xn is a nite connected CW-complex, it is a Peano space; thus, there
exists a continuous surjective map ωn : [0, 1] → Xn, a Peano curve, by the Hahn-
Mazurkiewicz Theorem. Let Pn : Xn+1 → [0, 1] be a surjective map. Replace one
of the constant maps δn in each connecting homomorphism ϕn,n+1 with the map
δ′n := ωn ◦ Pn : Xn → Xn−1.
Thus, A = lim (A
→∞ n
, ϕn,n+1) is a simple, unital generalized interval Villadsen
n
algebra as in Denition 3.1.7 which is produced by Construction 3.2.4. The con-
necting homomorphisms ϕn,n+1 are obtained from factoring through an interval, or
projections over a cube, and are injective thanks to surjectivity of the maps δ′n.
Let η′ ∈ M2(C(S̃2)) be the Hopf bration, a nontrivial line bundle for which
no tensor power of the Euler class is zero. Thus, by Villadsen's Lemma, Theorem
2.3.1, we have
[θ ] ≰ [η′ ◦ ψ]×n1 (∗)
in K0(C((S
2)n)), for any n ∈ N.
58
Without loss of generality, we may assume i is large enough that Ri ≥ 3,
dim(Qi) ≥
dim(Qi+1)
3, and Di,i+1 := ≥ 2.
dim(Qi)
By construction of A, the number Ni,j of projections, and ℓi,j, which is twice
the number of constant maps along with those factored through an interval, satises
Ri c
Ni,j >> ℓi,j, since Ni,j > > 0 for some L > 0, and for all i and j ≥ i. Without
Rj 2L
1
loss of generality we may assume i is also large enough that 2 ≤ ℓi,j < Ni,j for every
2
j ≥ i.
Assume all the notation of Lemma 3.3.2, with S2 ≃ S̃2 ⊂ Qi ⊂ Xi. Put
η = diag(η′, 0, . . . , 0) ∈ MR (C(S̃2∣)). So, by Lemma 3.3.2, in K (C((S
2)Ni,i+10 )),i
[ϕ ∗ ∣ ×Ni,i+1i,i+1τf η S ] ≤ [η] ⊕ [θℓ ],i+1 i,i+1
where ℓi,i+1 is twice the number of trivial maps in ϕi,i+1 : Ai → Ai+1.
Appealing to Lemma 3.3.3, let θQ′ ∈ MR (C(Si+1)) be a trivial bundle ofi+1
rank Q′ = N > ℓ . For each j ≥ i, let ψ : (S2)Ni,j 2 Ni,ji,i+1 i,i+1 → (S̃ ) be the natural
coordinate-wise homeomorphism and put A,B ∈ MR (C(Xi)) asi
B = diag(02×2, h, 0, . . .), A = τ
∗
f η +∣ B,
where h ∈ C (∣∣X \ S̃2) is a strictly positive element. Thus, B∣0 i 2 = 0, B is orthogonalS̃to τ ∗f η, and A 2 = η. Let g ∈ C0(Ui+1) be a strictly positive function, where US̃ i+1
is a ne∏ighborhood of S ≃ (S2)Ni,i+1i+1 obtained from the rectifying homeomorphism
ζi,i+1( ιi,i+1(Ui)), i.∣∣e. one such that the support agrees with the nontrivial part ofϕi,i+1τ ∗f η, and with g S = 1. Thus, wehave in MR (C(Xi+1 i+1))i+1
0 if x ∉ Ui+1Rank(gθQ′(x)) = Ni,i+1 if x ∈ Ui+1
and 1 if x ̸∈ Ui+1
Rank(ϕi,i+1A(x)) =  ,Ni,i+1 + ℓi,i+1 if x ∈ Ui+1
59
so we nd
Rank(ϕi+1,jgθQ′(x)) < Rank(ϕi,jA(x))
for all x ∈ Xj, j ≥ i, i.e. dt ′x(ϕi,jgθQ ) < dtx(ϕi,jA) for all j ≥ i and all extreme traces
tx ∈ T (Aj). Hence,
dt(ϕi+1,∞gθQ′) < dt(ϕi,∞A)
holds for all t ∈ T (A).
Suppose that ϕi+1,jgθQ′ ≾ ϕi,jA for suciently large j. P∣ut ΘQ = ϕi+∣1,jgθQ′
in Lemma 3.3.3, where Q = Q′ ·N ∣ ∣i+1,j = Ni,j. Then, (ϕi+1,jgθQ) S ≾ (ϕi,jA) S ) forj j
suciently large j, where S ≃ (S2)Ni,jj . By Lemma 3.3.3, we have
[θ ×Ni+1,jN ] = [θN ] ≤ [η ◦ ψ]×Ni,j ⊕ θi,i+1 i,j ℓi,j
in K ((C(S2)Ni,j0 )), where
Rank(θℓ ) = ℓi,j = 2Ri(Rj −Ni,j) < Ni,j i,j.
By cancellation of projections, since ℓi,j < Ni,j, we reduce to
[θV ] ≤ [η ◦ ψ]×Ni,j
inK0((S
2)Ni,j , where V ≥ 1. But, this contradicts (∗) above - the phenomenon known
as Villadsen's Chern class obstruction. Therefore, A does not have strict comparison
of positive elements.
Remark 3.3.5. One could dispense with the need for connectedness by adjusting
the construction to include maps ζn to each component.
60
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