Regularity properties in the classification program for separable
  amenable C*-algebras
REGULARITY PROPERTIES IN THE CLASSIFICATION
PROGRAM FOR SEPARABLE AMENABLE C∗-ALGEBRAS
GEORGE A. ELLIOTT AND ANDREW S. TOMS
Abstract. We report on recent progress in the program to classify separable
amenable C∗-algebras. Our emphasis is on the newly apparent role of regular-
ity properties such as finite decomposition rank, strict comparison of positive
elements, and Z-stability, and on the importance of the Cuntz semigroup. We
include a brief history of the program’s successes since 1989, a more detailed
look at the Villadsen-type algebras which have so dramatically changed the
landscape, and a collection of announcements on the structure and properties
of the Cuntz semigroup.
1. Introduction
Rings of bounded operators on Hilbert space were first studied by Murray and
von Neumann in the 1930s. These rings, later called von Neumann algebras, came
to be viewed as a subcategory of a more general category, namely, C∗-algebras.
(The C∗-algebra of compact operators appeared for perhaps the first time when
von Neumann proved the uniqueness of the canonical commutation relations.) A
C∗-algebra is a Banach algebra A with involution x 7→ x∗ satisfying the C∗-algebra
identity:
||xx∗|| = ||x||2, ∀x ∈ A.
Every C∗-algebra is isometrically ∗-isomorphic to a norm-closed sub-∗-algebra of
the ∗-algebra of bounded linear operators on some Hilbert space, and so may still
be viewed as a ring of operators on a Hilbert space.
In 1990, the first named author initiated a program to classify amenable norm-
separable C∗-algebras via K-theoretic invariants. The graded and (pre-)ordered
group K0 ⊕ K1 was suggested as a first approximation to the correct invariant,
as it had already proved to be complete for both approximately finite-dimensional
(AF) algebras and approximately circle (AT) algebras of real rank zero ([15], [17]).
It was quickly realised, however, that more sensitive invariants would be required
if the algebras considered were not sufficiently rich in projections. The program
was refined, and became concentrated on proving that Banach algebra K-theory
and positive traces formed a complete invariant for simple separable amenable C∗-
algebras. Formulated as such, it enjoyed tremendous success throughout the 1990s
and early 2000s.
Date: November 1, 2021.
2000 Mathematics Subject Classification. Primary 46L35, Secondary 46L80.
Key words and phrases. C∗-algebras, classification.
This work was partially supported by NSERC.
http://arxiv.org/abs/0704.1803v3
2 GEORGE A. ELLIOTT AND ANDREW S. TOMS
Recent examples based on the pioneering work of Villadsen have shown that
the classification program must be further revised. Two things are now appar-
ent: the presence of a dichotomy among separable amenable C∗-algebras dividing
those algebras which are classifiable via K-theory and traces from those which will
require finer invariants; and the possibility—the reality, in some cases—that this di-
chotomy is characterised by one of three potentially equivalent regularity properties
for amenable C∗-algebras. (Happily, the vast majority of our stock-in-trade simple
separable amenable C∗-algebras have one or more of these properties, including, for
instance, those arising from directed graphs or minimal C∗-dynamical systems.)
Our plan in this article is to give a brief account of the activity in the classifica-
tion program over the past decade, with particular emphasis on the now apparent
role of regularity properties. After reviewing the successes of the program so far,
we will cover the work of Villadsen on rapid dimension growth AH algebras, the
examples of Rørdam and the second named author which have necessitated the
present re-evaluation of the classification program, and some recent and sweeping
classification results of Winter obtained in the presence of the aforementioned regu-
larity properties. We will also discuss the possible consequences to the classification
program of including the Cuntz semigroup as part of the invariant (as a refinement
of the K0 and tracial invariants).
2. Preliminaries
Throughout the sequel K will denote the C∗-algebra of compact operators on a
separable infinite-dimensional Hilbert space H. For a C∗-algebra A, we let Mn(A)
denote the algebra of n × n matrices with entries from A. The cone of positive
elements of A will be denoted by A+.
2.1. The Elliott invariant and the original conjecture. The Elliott invariant
of a C∗-algebra A is the 4-tuple
(1) Ell(A) :=
(K0A,K0A
+,ΣA),K1A,T
+A, ρA
where the K-groups are the topological ones, K0A
+ is the image of the Murray-
von Neumann semigroup V(A) under the Grothendieck map, ΣA is the subset
of K0A corresponding to projections in A, T
+A is the space of positive tracial
linear functionals on A, and ρA is the natural pairing of T
+A and K0A given by
evaluating a trace at a K0-class. The reader is referred to Rørdam’s monograph
[45] for a detailed treatment of this invariant. In the case of a unital C∗-algebra
the invariant becomes
(K0A,K0A
+, [1A]),K1A,TA, ρA
where [1A] is the K0-class of the unit, and TA is the (compact convex) space of
tracial states. We will concentrate on unital C∗-algebras in the sequel in order to
limit technicalities.
The original statement of the classification conjecture for simple unital separable
amenable C∗-algebras read as follows:
2.1. Let A and B be simple unital separable amenable C∗-algebras, and suppose
that there exists an isomorphism
φ : Ell(A) → Ell(B).
It follows that there is a ∗-isomorphsim Φ : A→ B which induces φ.
REGULARITY PROPERTIES FOR AMENABLE C
-ALGEBRAS 3
It will be convenient to have an abbreviation for the statement above. Let us call
it (EC).
2.2. Amenability. We will take the following deep theorem, which combines re-
sults of Choi and Effros ([7]), Connes ([9]), Haagerup ([27]), and Kirchberg ([32]),
to be our definition of amenability.
Theorem 2.2. A C∗-algebra A is amenable if and only if it has the following
property: for each finite subset G of A and ǫ > 0 there are a finite-dimensional
C∗-algebra F and completely positive contractions φ, ψ such that the diagram
idA //
commutes up to ǫ on G.
The property characterising amenability in Theorem 2.2 is known as the completely
positive approximation property.
Why do we only consider separable and amenable C∗-algebras in the classification
program? It stands to reason that if one has no good classification of the weak
closures of the GNS representations for a class of C∗-algebras, then one can hardly
expect to classify the C∗-algebras themselves. These weak closures have separable
predual if the C∗-algebra is separable. Connes and Haagerup gave a classification
of injective von Neumann algebras with separable predual (see [10] and [28]), while
Choi and Effros established that a C∗-algebra is amenable if and only if the weak
closure in each GNS representation is injective ([8]). Separability and amenability
are thus natural conditions which guarantee the existence of a good classification
theory for the weak closures of all GNS representations of a given C∗-algebra. The
assumption of amenability has been shown to be necessary by Dădărlat ([14]).
The reader new to the classification program who desires a fuller introduction is
referred to Rørdam’s excellent monograph [45].
2.3. The Cuntz semigroup. One of the three regularity properties alluded to
in the introduction is defined in terms of the Cuntz semigroup, an analogue for
positive elements of the Murray-von Neumann semigroup V(A). It is known that
this semigroup will be a vital part of any complete invariant for separable amenable
C∗-algebras ([51]). Given its importance, we present both its original definition,
and a modern version which makes the connection with classical K-theory more
transparent.
Definition 2.3 (Cuntz-Rørdam—see [12] and [49]). Let M∞(A) denote the alge-
braic limit of the direct system (Mn(A), φn), where φn : Mn(A) → Mn+1(A) is
given by
Let M∞(A)+ (resp. Mn(A)+) denote the positive elements in M∞(A) (resp. Mn(A)).
Given a, b ∈ M∞(A)+, we say that a is Cuntz subequivalent to b (written a - b) if
there is a sequence (vn)
n=1 of elements in some Mk(A) such that
||vnbv
n − a||
−→ 0.
4 GEORGE A. ELLIOTT AND ANDREW S. TOMS
We say that a and b are Cuntz equivalent (written a ∼ b) if a - b and b - a. This
relation is an equivalence relation, and we write 〈a〉 for the equivalence class of a.
The set
W(A) := M∞(A)+/ ∼
becomes a positively ordered Abelian semigroup when equipped with the operation
〈a〉+ 〈b〉 = 〈a⊕ b〉
and the partial order
〈a〉 ≤ 〈b〉 ⇔ a - b.
Definition 2.3 is slightly unnatural, as it fails to consider positive elements in
A ⊗ K. This defect is the result of mimicking the construction of the Murray-
von Neumann semigroup in letter rather than in spirit. Each projection in A ⊗ K
is equivalent to a projection in some Mn(A), whence M∞(A) is large enough to
encompass all possible equivalence classes of projections. The same is not true,
however, of positive elements and Cuntz equivalence. The definition below amounts
essentially to replacing M∞(A) with A⊗K in the definition above (this is a theorem),
and also gives a new and very useful characterisation of Cuntz subequivalence. We
refer the reader to [35] and [39] for background material on Hilbert C∗-modules.
Consider A as a (right) Hilbert C∗-module over itself, and let HA denote the
countably infinite direct sum of copies of this module. There is a one-to-one corre-
spondence between closed countably generated submodules of HA and hereditary
subalgebras of A ⊗ K: the hereditary subalgebra B corresponds to the closure of
the span of BHA. Since A is separable, B is singly hereditarily generated, and it is
fairly routine to prove that any two generators are Cuntz equivalent in the sense of
Definition 2.3. Thus, passing from positive elements to Cuntz equivalence classes
factors through the passage from positive elements to the hereditary subalgebras
they generate.
Let X and Y be closed countably generated submodules of HA. Recall that
the compact operators on HA form a C
∗-algebra isomorphic to A⊗ K. Let us say
that X is compactly contained in Y if X is contained in Y and there is compact
self-adjoint endomorphism of Y which fixes X pointwise. Such an endomorphism
extends naturally to a compact self-adjoint endomorphism of HA, and so may be
viewed as a self-adjoint element of A⊗K. We write X - Y if each closed countably
generated compactly contained submodule of X is isomorphic to such a submodule
of Y .
Theorem 2.4 (Coward-Elliott-Ivanescu, [11]). The relation - on Hilbert C∗ mod-
ules defined above, when viewed as a relation on positive elements in M∞(A), is
precisely the relation - of Definition 2.3.
Let [X ] denote the Cuntz equivalence class of the module X . One may con-
struct a positive partially ordered Abelian semigroup Cu(A) by endowing the set
of countably generated Hilbert C∗-modules over A with the operation
[X ] + [Y ] := [X ⊕ Y ]
and the partial order
[X ] ≤ [Y ] ⇔ X - Y.
The semigroup Cu(A) coincides with W(A) whenever A is stable, i.e., A ⊗ K ∼=
A, and has some advantages over W(A) in general. First, suprema of increasing
REGULARITY PROPERTIES FOR AMENABLE C
-ALGEBRAS 5
sequences always exist in Cu(A). This leads to the definition of a category including
this structure in which Cu(A) sits as an object, and as a functor into which it
is continuous with respect to inductive limits. (Definition 2.3 casts W(A) as a
functor into just the category of partially ordered Abelian semigroups with zero.
This functor fails to be continuous with respect to inductive limits.) Second, it
allows one to prove that if A has stable rank one, then Cuntz equivalence of positive
elements simply amounts to isomorphism of the corresponding Hilbert C∗-modules.
This has led, via recent work of Brown, Perera, and the second named author, to
the complete classification of all countably generated Hilbert C∗-modules over A
via K0 and traces, and to the classification of unitary orbits of positive operators
in A⊗K through recent work of Ciuperca and the first named author ([4], [5], [6]).
Essentially, W(A) and Cu(A) contain the same information, but we have chosen
to maintain separate notation both to avoid confusion and because many results in
the literature are stated only for W(A).
Cuntz equivalence is often described roughly as the Murray-von Neumann equiv-
alence of the support projections of positive elements. This heuristic is, modulo
accounting for projections, precise in C∗-algebras for which the Elliott invariant is
known to be complete ([42]). In the stably finite case, one recovers both K0, the
tracial simplex, and the pairing ρ (see (1)) from the Cuntz semigroup, whence the
invariant
(Cu(A),K1A)
is finer than Ell(A) in general. Remarkably, these two invariants determine each
other in a natural way for the largest class of simple separable amenable C∗-algebras
in which (EC) can be expected to hold ([4], [5]).
3. Three regularity properties
Let us now describe three agreeable properties which a C*-algebra may enjoy.
We will see later how virtually all classification theorems for separable amenable
C∗-algebras via the Elliott invariant assume, either explicitly or implicitly, one of
these properties.
3.1. Strict comparison. Our first regularity property—strict comparison—is one
that guarantees, in simple C∗-algebras, that the heuristic view of Cuntz equivalence
described at the end of Section 2 is in fact accurate for positive elements which
are not Cuntz equivalent to projections (see [42]). The property is K-theoretic in
character.
Let A be a unital C∗-algebra, and denote by QT(A) the space of normalised 2-
quasitraces on A (v.[2, Definition II.1.1]). Let S(W(A)) denote the set of additive
and order preserving maps d from W(A) to R+ having the property that d(〈1A〉) =
1. Such maps are called states. Given τ ∈ QT(A), one may define a map dτ :
M∞(A)+ → R
(2) dτ (a) = lim
τ(a1/n).
This map is lower semicontinuous, and depends only on the Cuntz equivalence class
of a. It moreover has the following properties:
(i) if a - b, then dτ (a) ≤ dτ (b);
(ii) if a and b are orthogonal, then dτ (a+ b) = dτ (a) + dτ (b).
6 GEORGE A. ELLIOTT AND ANDREW S. TOMS
Thus, dτ defines a state on W(A). Such states are called lower semicontinuous
dimension functions, and the set of them is denoted by LDF(A). If A has the
property that a - b whenever d(a) < d(b) for every d ∈ LDF(A), then let us say
that A has strict comparison of positive elements or simply strict comparison.
A theorem of Haagerup asserts that every element of QT(A) is in fact a trace
if A is exact ([29]). All amenable C∗-algebras are exact, so we dispense with the
consideration of quasi-traces in the sequel.
3.2. Finite decomposition rank. Our second regularity property, introduced by
Kirchberg and Winter, is topological in flavour. It is based on a noncommutative
version of covering dimension called decomposition rank.
Definition 3.1 ([34], Definitions 2.2 and 3.1). Let A be a separable C∗-algebra.
(i) A completely positive map ϕ :
i=1Mri → A is n-decomposable, if there is
a decomposition {1, . . . , s} =
j=0 Ij such that the restriction of ϕ to
preserves orthogonality for each j ∈ {0, . . . , n}.
(ii) A has decomposition rank n, drA = n, if n is the least integer such that
the following holds: Given {b1, . . . , bm} ⊂ A and ǫ > 0, there is a completely
positive approximation (F, ψ, ϕ) for b1, . . . , bm within ǫ (i.e., ψ : A → F and ϕ :
F → A are completely positive contractions and ‖ϕψ(bi) − bi‖ < ǫ) such that ϕ is
n-decomposable. If no such n exists, we write drA = ∞.
Decomposition rank has good permanence properties. It behaves well with re-
spect to quotients, inductive limits, hereditary subalgebras, unitization and sta-
bilization. Its topological flavour comes from the fact that it generalises covering
dimension in the commutative case: if X is a locally compact second countable
space, then drC0(X) = dimX . We refer the reader to [34] for details.
The regularity property that we are interested in is finite decomposition rank,
expressed by the inequality dr < ∞. This can only occur in a stably finite C∗-
algebra.
3.3. Z-stability. The Jiang-Su algebra Z is a simple separable amenable and
infinite-dimensional C∗-algebra with the same Elliott invariant as C ([30]). We
say that a second algebra A is Z-stable if A ⊗ Z ∼= A. Z-stability is our third
regularity property. It is very robust with respect to common constructions (see
[56]).
The next theorem shows Z-stability to be highly relevant to the classification
program. Recall that a pre-ordered Abelian group (G,G+) is said to be weakly
unperforated if nx ∈ G+\{0} implies x ∈ G+ for any x ∈ G and n ∈ N.
Theorem 3.2 (Gong-Jiang-Su, [26]). Let A be a simple unital C∗-algebra with
weakly unperforated K0-group. Then,
Ell(A) ∼= Ell(A⊗Z).
Thus, modulo a mild restriction on K0, the completeness of Ell(•) in the simple
unital case of the classification program would imply Z-stability. Remarkably, there
exist algebras satisfying the hypotheses of the above theorem which are not Z-stable
([46], [51], [52]).
REGULARITY PROPERTIES FOR AMENABLE C
-ALGEBRAS 7
3.4. Relationships. In general, no two of the regularity properties above are
equivalent. The most important general result connecting them is the following
theorem of M. Rørdam ([47]):
Theorem 3.3. Let A be a simple, unital, exact, finite, and Z-stable C∗-algebra.
Then, A has strict comparison of positive elements.
We shall see in the sequel that for a substantial class of simple, separable, amenable,
and stably finite C∗-algebras, all three of our regularity properties are equivalent.
Moreover, the algebras in this class which do satisfy these three properties also
satisfy (EC). There is good reason to believe that the equivalence of these three
properties will hold in much greater generality, at least in the stably finite case; in
the general case, strict comparison and Z-stability may well prove to be equiva-
lent characterisations of those simple, unital, separable, and amenable C∗-algebras
which satisfy (EC).
4. A brief history
We will now take a short tour of the classification program’s biggest successes,
and also the fascinating algebras of Villadsen. We have two goals in mind: to edify
the reader unfamiliar with the classification program, and to demonstrate that the
regularity properties of Section 3 pervade the known confirmations of (EC). This
is a new point of view, for when these results were originally proved, there was no
reason to think that anything more than simplicity, separability, and amenability
would be required to complete the classification program.
We have divided our review of known classification results into three broad cat-
egories according to the types of algebras covered: purely infinite algebras, and
two formally different types of stably finite algebras. It is beyond the scope of this
article to provide and exhaustive list of of known classification results, much less
demonstrate their connections to our regularity properties. We will thus choose,
from each of the three categories above, the classification theorem with the broadest
scope, and indicate how the algebras it covers satisfy at least one of our regularity
properties.
4.1. Purely infinite simple algebras. We first consider a case where the theory
is summarised with one beautiful result. Recall that a simple separable amenable
C∗-algebra is purely infinite if every hereditary subalgebra contains an infinite pro-
jection (a projection is infinite if it is equivalent, in the sense of Murray and von
Neumann, to a proper subprojection of itself—otherwise the projection is finite).
Theorem 4.1 (Kirchberg-Phillips, 1995, [31] and [43]). Let A and B be separable
amenable purely infinite simple C∗-algebras which satisfy the Universal Coefficient
Theorem. If there is an isomorphism
φ : Ell(A) → Ell(B),
then there is a ∗-isomorphism Φ : A→ B.
In the theorem above, the Elliott invariant is somewhat simplified. The hypothe-
ses on A and B guarantee that they are traceless, and that the order structure on
K0 is irrelevant. Thus, the invariant is simply the graded group K0⊕K1, along with
the K0-class of the unit if it exists. The assumption of the Universal Coefficient
Theorem (UCT) is required in order to deduce the theorem from a result which is
8 GEORGE A. ELLIOTT AND ANDREW S. TOMS
formally more general: A and B as in the theorem are ∗-isomorphic if and only
if they are KK-equivalent. The question of whether every amenable C∗-algebra
satisfies the UCT is open.
Which of our three regularity properties are present here? As noted earlier,
finite decomposition rank is out of the question. The algebras we are considering
are traceless, and so the definition of strict comparison reduces to the following
statement: for any two non-zero positive elements a, b ∈ A, we have a - b. This,
in turn, is often taken as the very definition of pure infiniteness, and can be shown
to be equivalent to the definition preceding Theorem 4.1 without much difficulty.
Strict comparison is thus satisfied in a slightly vacuous way. As it turns out, A and
B are also Z-stable, although this is less obvious. One first proves that A and B
are approximately divisible (again, this does not require Theorem 4.1), and then
uses the fact, due to Winter and the second named author, that any separable and
approximately divisible C∗-algebra is Z-stable ([57]).
4.2. The stably finite case, I: inductive limits. We now move on to the case
of stably finite C∗-algebras, i.e., those algebras A such that that every projection in
the (unitization of) each matrix algebra Mn(A) is finite. (The question of whether
a simple amenable C∗-algebra must always be purely infinite or stably finite was
recently settled negatively by Rørdam. We will address his example again later.)
Many of the classification results in this setting apply to classes of C∗-algebras
which can be realised as inductive limits of certain building block algebras. The
original classification result for stably finite algebras is due to Glimm. Recall that
a C∗-algebra A is uniformly hyperfinite (UHF) if it is the limit of an inductive
sequence
−→ Mn2
−→ Mn3
−→ · · · ,
where each φi is a unital ∗-homomorphism. We will state his result here as a con-
firmation of the Elliott conjecture, but note that it predates both the classification
program and the realisation that K-theory is the essential invariant.
Theorem 4.2 (Glimm, 1960, [24]). Let A and B be UHF algebras, and suppose
that there is an isomorphsim
φ : Ell(A) → Ell(B).
It follows that there is a ∗-isomorphism Φ : A→ B which induces φ.
Again, the invariant is dramatically simplified here. Only the ordered K0-group
is non-trivial. The strategy of Glimm’s proof (which did not use K-theory explicitly)
was to “intertwine” two inductive sequences (Mni , φi) and (Mmi , ψi), i.e., to find
sequences of ∗-homomorphisms ηi and γi making the diagram
φ1 //
φ2 //
φ3 //
· · ·
ψ1 //
;;xxxxxxxx
ψ2 //
;;xxxxxxxx
ψ3 //
==zzzzzzzzz
· · ·
commute. One then gets an isomorphism between the limit algebras by extending
the obvious morphism between the inductive sequences by continuity.
The intertwining argument above can be pushed surprisingly far. One replaces
the inductive sequences above with more general inductive sequences (Ai, φi) and
REGULARITY PROPERTIES FOR AMENABLE C
-ALGEBRAS 9
(Bi, ψi), where the Ai and Bi are drawn from a specified class (matrix algebras over
circles, for instance), and seeks maps ηi and γi as before. Usually, it is not possible
to find ηi and γi making the diagram commute, but approximate commutativity
on ever larger finite sets can be arranged for, and this suffices for the existence of
an isomorphism between the limit algebras. This generalised intertwining is known
as the Elliott Intertwining Argument.
The most important classification theorem for inductive limits covers the so-
called approximately homogeneous (AH) algebras. An AH algebra A is the limit of
an inductive sequence (Ai, φi), where each Ai is semi-homogeneous:
pi,j(C(Xi,j)⊗K)pi,j
for some natural number ni, compact metric spaces Xi,j , and projections pi,j ∈
C(Xi,j) ⊗ K. We refer to the sequence (Ai, φi) as a decomposition for A; such
decompositions are not unique. All AH algebras are separable and amenable.
Let A be a simple unital AH algebra. Let us say that A has slow dimension
growth if it has a decomposition (Ai, φi) satisfying
lim sup
dim(Xi,1)
rank(pi,1)
, . . . ,
dim(Xi,ni)
rank(pi,ni)
Let us say that A has very slow dimension growth if it has a decomposition satisfying
the (formally) stronger condition that
lim sup
dim(Xi,1)
rank(pi,1)
, . . . ,
dim(Xi,ni)
rank(pi,ni)
Finally, let us say that A has bounded dimension if there is a constant M > 0 and
a decomposition of A satisfying
{dim(Xi,l)} ≤M.
Theorem 4.3 (Elliott-Gong and Dădărlat, [20] and [13]). (EC) holds among simple
unital AH algebras with slow dimension growth and real rank zero.
Theorem 4.4 (Elliott-Gong-Li and Gong, [22] and [25]). (EC) holds among simple
unital AH algebras with very slow dimension growth.
All three of our regularity properties hold for the algebras of Theorems 4.3 and
4.4, but some are easier to establish than others. Let us first point out that an
algebra from either class has stable rank one and weakly unperforated K0-group
(cf.[1]), and that these facts predate Theorems 4.3 and 4.4. A simple unital C∗-
algebra of real rank zero and stable rank one has strict comparison if and only if its
K0-group is weakly unperforated (cf.[41]), whence strict comparison holds for the
algebras covered by Theorem 4.3. A recent result of the second named author shows
that strict comparison holds for any simple unital AH algebra with slow dimension
growth ([55]), and this result is independent of the classification theorems above.
Thus, strict comparison holds for the algebras of Theorems 4.3 and 4.4, and the
proof of this fact, while not easy, is at least much less complicated than the proofs
of the classification theorems themselves. Establishing finite decomposition rank
requires the full force of the classification theorems: a consequence of both theorems
is that the algebras they cover are all in fact simple unital AH algebras of bounded
10 GEORGE A. ELLIOTT AND ANDREW S. TOMS
dimension, and such algebras have finite decomposition rank by [34, Corollary 3.12
and 3.3 (ii)]. Proving Z-stability is also an application of Theorems 4.3 and 4.4: one
may use the said theorems to prove that the algebras in question are approximately
divisible ([21]), and this entails Z-stability for separable C∗-algebras ([57]).
Why all the interest in inductive limits? Initially at least, it was surprising to
find that any classification of C∗-algebras by K-theory was possible, and the earliest
theorems to this effect covered inductive limits (see, for instance, the first named
author’s classification of AF algebras and AT-algebras of real rank zero []). But it
was the realisation by Evans and the first named author that a very natural class
of C∗-algebras arising from dynamical systems—the irrational rotation algebras—
were in fact inductive limits of elementary building blocks that began the drive to
classify inductive limits of all stripes ([19]). This theorem of Elliott and Evans has
recently been generalised in sweeping fashion by Lin and Phillips, who prove that
virtually every C∗-dynamical system giving rise to a simple algebra is an inductive
limit of fairly tractable building blocks. This result continues to provide strong
motivation for the study of inductive limit algebras.
4.3. The stably finite case, II: tracial approximation. Natural examples of
separable amenable C∗-algebras are rarely equipped with obvious and useful induc-
tive limit decompositions. Even the aforementioned theorem of Lin and Phillips,
which gives an inductive limit decomposition for each minimal C∗-dynamical sys-
tem, does not produce inductive sequences covered by existing classification theo-
rems. It is thus desirable to have theorems confirming the Elliott conjecture under
hypotheses that are (reasonably) straightforward to verify for algebras not given as
inductive limits.
Lin in [36] introduced the concept of tracial topological rank for C∗-algebras.
His definition, in spirit if not in letter, is this: a unital simple tracial C∗-algebra A
has tracial topological rank at most n ∈ N if for any finite set F ⊆ A, tolerance
ǫ > 0, and positive element a ∈ A there exist unital subalgebras B and C of A such
(i) 1A = 1B ⊕ 1C ,
(ii) F is almost (to within ǫ) contained in B ⊕ C,
(iii) C is isomorphic to F⊗C(X), where dim(X) ≤ n and F is finite-dimensional,
(iv) 1B is dominated, in the sense of Cuntz subequivalence, by a.
One denotes by TR(A) the least integer n for which A satisfies the definition above;
this is the tracial topological rank, or simply the tracial rank, of A.
The most important value of the tracial rank is zero. Lin proved that simple
unital separable amenable C∗-algebras of tracial rank zero satisfy the Elliott con-
jecture, modulo the ever present UCT assumption ([37]). The great advantage of
this result is that its hypotheses can be verified for a wide variety of C∗-dynamical
systems and all simple non-commutative tori, without ever having to prove that the
latter have tractable inductive limit decompositions (see [44], for instance). Indeed,
the existence of such decompositions is a consequence of Lin’s theorem! One can
also verify the hypotheses of Lin’s classification theorem for many real rank zero
C∗-algebras with unique trace ([3]), always with the assumption, indirectly, of strict
comparison.
REGULARITY PROPERTIES FOR AMENABLE C
-ALGEBRAS 11
Simple unital C∗-algebras of tracial rank zero can be shown to have stable rank
one and weakly unperforated K0-groups, whence they have strict comparison of
positive elements by a theorem of Perera ([41]). (There is a classification theorem
for algebras of tracial rank one ([38]), but this has been somewhat less useful—it
is difficult to verify tracial rank one in natural examples. Also, Niu has recently
proved a classification theorem for some C∗-algebras which are approximated in
trace by certain subalgebras of Mn ⊗ C[0, 1] ([40]).)
And what of our regularity properties? Lin proved in [36] that every unital simple
C∗-algebra of tracial rank zero has stable rank one and weakly unperforated K0-
group. These facts, by the results reviewed at the end of the preceding subsection,
entail strict comparison, and are not nearly so difficult to prove as the tracial rank
zero classification theorem. In a further analogy with the case of AH algebras,
finite decomposition rank and Z-stability can only be verified by applying Lin’s
classification theorem—a consequence of this theorem is that the algebras it covers
are in fact AH algebras of bounded dimension!
4.4. Villadsen’s algebras. Until the mid 1990s we had no examples of simple
separable amenable C∗-algebras where one of our regularity properties failed. To
be fair, two of our regularity properties had not yet even been defined, and strict
comparison was seen as a technical version of the more attractive Second Fun-
damental Comparability Question for projections (this last condition, abbreviated
FCQ2, asks for strict comparison for projections only). This all changed when Vil-
ladsen produced a simple separable amenable and stably finite C∗-algebra which
did not have FCQ2, answering a long-standing question of Blackadar ([59]). The
techniques introduced by Villadsen were subsequently used by him and others to
answer many open questions in the theory of nuclear C∗-algebras including the
following:
(i) Does there exist a simple separable amenable C∗-algebra containing a finite
and an infinite projection? (Solved affirmatively by Rørdam in [46].)
(ii) Does there exist a simple and stably finite C∗-algebra with non-minimal
stable rank? (Solved affirmatively by Villadsen in [60].)
(iii) Is stability a stable property for simple C∗-algebras? (Solved negatively by
Rørdam in [48].)
(iv) Does a simple and stably finite C∗-algebra with cancellation of projections
necessarily have stable rank one? (Solved negatively by the second named
author in [54].)
Of the results above, (i) was (and is) the most significant. In addition to showing
that simple separable amenable C∗-algebras do not have a factor-like type classifi-
cation, Rørdam’s example demonstrated that the Elliott invariant as it stood could
not be complete in the simple case. This and other examples due to the second
named author have necessitated a revision of the classification program. It is to
the nature of this revision that we now turn.
5. The way(s) forward
5.1. New assumptions. (EC) does not hold in general, and this justifies new as-
sumptions in efforts to confirm it. In particular, one may assume any combination
of our three regularity properties. We will comment on the aptness of these new
assumptions in the next subsection. For now we observe that, from a certain point
12 GEORGE A. ELLIOTT AND ANDREW S. TOMS
of view, we have been making these assumptions all along. Existing classification
theorems for C∗-algebras of real rank zero are accompanied by the crucial assump-
tions of stable rank one and weakly unperforated K-theory; as has already been
pointed out, unperforated K-theory can be replaced with strict comparison in this
setting.
How much further can one get by assuming the (formally) stronger condition
of Z-stability? What role does finite decomposition rank play? As it turns out,
these two properties both alone and together produce interesting results. Let RR0
denote the class of simple unital separable amenable C∗-algebras of real rank zero.
The following subclasses of RR0 satisfy (EC):
(i) algebras which satisfy the UCT, have finite decomposition rank, and have
tracial simplex with compact and zero-dimensional extreme boundary;
(ii) Z-stable algebras which satisfy the UCT and are approximated locally by
subalgebras of finite decomposition rank.
These results, due to Winter ([61], [62]), showcase the power of our regularity
properties: included in the algebras covered by (ii) are all simple separable unital
Z-stable ASH (approximately subhomogeneous) algebras of real rank zero.
Another advantage to the assumptions of Z-stability and strict comparison is
that they allow one to recover extremely fine isomorphism invariants for C∗-algebras
from the Elliott invariant alone. (This recovery is not possible in general.) We will
be able to give precise meaning to this comment below, but first require a further
dicussion of Cuntz semigroup.
5.2. New invariants. A natural reaction to an incomplete invariant is to enlarge
it: include whatever information was used to prove incompleteness. This is not
always a good idea. It is possible that one’s distinguishing information is ad hoc,
and unlikely to yield a complete invariant. Worse, one may throw so much new
information into the invariant that the impact of its potential completeness is se-
verely diminished. The revision of an invariant is a delicate business. In this light,
not all counterexamples are equal.
Rørdam’s finite-and-infinite-projection example is distinguished from a simple
and purely infinite algebra with the same K-theory by the obvious fact that the
latter contains no finite projections. The natural invariant which captures this dif-
ference is the semigroup of Murray-von Neumann equivalence classes of projections
in matrices over an algebra A, denoted by V(A). After the appearance of Rørdam’s
example, the second named author produced a pair of simple, separable, amenable,
and stably finite C∗-algebras which agreed on the Elliott invariant, but were not
isomorphic. In this case the distinguishing invariant was Rieffel’s stable rank. It
was later discovered that these algebras could not be distinguished by their Murray-
von Neumann semigroups, but it was not yet clear which data were missing from
the Elliott invariant. More dramatic examples were needed, ones which agreed on
most candidates for enlarging the invariant, and pointed the way to the “missing
information”.
In [52], the second named author constructed a pair of simple unital AH algebras
which, while non-isomorphic, agreed on a wide swath of invariants including the
Elliott invariant, all continuous (with respect to inductive sequences) and homotopy
invariant functors from the category of C∗-algebras (a class which includes the
Murray-von Neumann semigroup), the real and stable ranks, and, as was shown
REGULARITY PROPERTIES FOR AMENABLE C
-ALGEBRAS 13
later in [], stable isomorphism invariants (those invariants which are insensitive to
tensoring with a matrix algebra or passing to a hereditary subalgebra). It was thus
reasonable to expect that the distinguishing invariant in this example—the Cuntz
semigroup—might be a good candidate for enlarging the invariant. At least, it
was an object which after years of being used sparingly as a means to other ends,
merited study for its own sake.
Let us collect some evidence supporting the addition of the Cuntz semigroup to
the usual Elliott invarariant. First, in the biggest class of algebras where (EC) can
be expected to hold—Z-stable algebras, as shown by Theorem 3.2—it is not an
addition at all! Recent work of Brown, Perera, and the second named author shows
that for a simple unital separable amenable C∗-algebra which absorbs Z tensorially,
there is a functor which recovers the Cuntz semigroup from the Elliott invariant
([4], [42]). This functorial recovery also holds for simple unital AH algebras of slow
dimension growth, a class for which Z-stability is not known and yet confirmation of
(EC) is expected. (It should be noted that the computation of the Cuntz semigroup
for a simple approximately interval (AI) algebra was essentially carried out by
Ivanescu and the first named author in [23], although one does require [11, Corollary
4] to see that the computation is complete.) Second, the Cuntz semigroup unifies
the counterexamples of Rørdam and the second named author. One can show that
the examples of [45], [51], and [52] all consist of pairs of algebras with different Cuntz
semigroups; there are no counterexamples to the conjecture that simple separable
amenable C∗-algebras will be classified up to ∗-isomorphism by the Elliott invariant
and the Cuntz semigroup. Third, the Cuntz semigroup provides a bridge to the
classification of non-simple algebras. Ciuperca and the first named author have
recently proved that AI algebras—limits of inductive sequences of algebras of the
Mmi(C[0, 1])
— are classified up to isomorphism by their Cuntz semigroups. This is accomplished
by proving that the approximate unitary equivalence classes of positive operators
in the unitization of a stable C∗-algebra of stable rank one are determined by the
Cuntz semigroup of the algebra, and then appealing to a theorem of Thomsen
([50]). (These approximate unitary equivalence classes of positive operators can
be endowed with the structure of a topological partially ordered semigroup with
functional calculus. This invariant, known as Thomsen’s semigroup, is recovered
functorially from the Cuntz semigroup for separable algebras of stable rank one,
and so from the Elliott invariant in algebras which are moreover simple, unital,
exact, finite, and Z-stable by the results of [4]. This new semigroup is the fine
invariant alluded to at the end of subsection 5.1.)
There is one last reason to suspect a deep connection between the classification
program and the Cuntz semigroup. Let us first recall a theorem of Kirchberg, which
is germane to the classification of purely infinite C∗-algebras (cf. Theorem 4.1).
Theorem 5.1 (Kirchberg, c. 1994; see [33]). Let A be a separable amenable C∗-
algebra. The following two properties are equivalent:
(i) A is purely infinite;
(ii) A⊗O∞ ∼= A.
14 GEORGE A. ELLIOTT AND ANDREW S. TOMS
A consequence of Kirchberg’s theorem is that among simple separable amenable
C∗-algebras which merely contain an infinite projection, there is a two-fold char-
acterisation of the (proper) subclass which satisfies the original form of the Elliott
conjecture (modulo UCT). If one assumes a priori that A is simple and unital
with no tracial state, then a theorem of Rørdam (see [47]) shows that property (ii)
above — known as O∞-stability—is equivalent to Z-stability. Under these same
hypotheses, property (i) is equivalent to the statement that A has strict compar-
ison. Kirchberg’s theorem can thus be rephrased as follows in the simple unital
case:
Theorem 5.2. Let A be a simple separable unital amenable C∗-algebra without a
tracial state. The following two properties are equivalent:
(i) A has strict comparison;
(ii) A⊗Z ∼= A.
The properties (i) and (ii) in the theorem above make perfect sense in the presence
of a trace. We moreover have that (ii) implies (i) even in the presence of traces (this
is due to Rørdam—see [47]). It therefore makes sense to ask whether the theorem
might be true without the tracelessness hypothesis. Remarkably, this appears to be
the case. Winter and the second named author have proved that for a substantial
class of stably finite C∗-algebras, strict comparison and Z-stability are equivalent,
and that these properties moreover characterise the (proper) subclass which satisfies
(EC) ([58]). In other words, Kirchberg’s theorem is quite possibly a special case
of a more general result, one which will give a unified two-fold characterisation of
those simple separable amenable C∗-algebras which satisfy the original form of the
Elliott conjecture.
It is too soon to know whether the Cuntz semigroup together with Elliott in-
variant will suffice for the classification of simple separable amenable C∗-algebras,
or indeed, whether such a broad classification can be hoped for at all. But there
is already cause for optimism. Zhuang Niu has recently obtained some results on
lifting maps at the level of the Cuntz semigroup to ∗-homomorphisms. This type of
lifting result is a key ingredient in proving classification theorems of all stripes. His
results suggest the algebras of [52] as the appropriate starting point for any effort
to establish the Cuntz semigroup as a complete isomorphism invariant, at least in
the absence of K1.
We close our survey with a few questions for the future, both near and far.
(i) When do natural examples of simple separable amenable C∗-algebras satisfy
one or more of the regularity properties of Section 3? In particular, do
simple unital inductive limits of recursive subhomogeneous algebras have
strict comparison whenever they have strict slow dimension growth?
(ii) Can the classification of positive operators up to approximate unitary equiv-
alence via the Cuntz semigroup in algebras of stable rank one be extended
to normal elements, provided that one accounts for K1?
(iii) Let A be a simple, unital, separable, and amenable C∗-algebra with strict
comparison of positive elements. Is A Z-stable? Less ambitiously, does A
have stable rank one whenever it is stably finite?
(iv) Can one use Thomsen’s semigroup to prove new classification theorems?
(The attraction here is that Thomsen’s semigroup is already implicit in the
Elliott invariant for many classes of C∗-algebras.)
REGULARITY PROPERTIES FOR AMENABLE C
-ALGEBRAS 15
References
[1] Blackadar, B., Dădărlat, M., and Rørdam, M.: The real rank of inductive limit C∗-algebras,
Math. Scand. 69 (1991), 211-216
[2] Blackadar, B., and Handelman, D.:Dimension Functions and Traces on C∗-algebras, J.
Funct. Anal. 45 (1982), 297-340
[3] Brown, N.: Invariant means and finite representation theory of C∗-algebras, Mem. Amer.
Math. Soc. 184, no. 865, viii + 105 pp.
[4] Brown, N. P., Perera, F., and Toms, A. S.: The Cuntz semigroup, the Elliott conjecture,
and dimension functions on C∗-algebras, arXiv preprint math.OA/0609182 (2006)
[5] Brown, N. P., and Toms, A. S.: Three applications of the Cuntz semigroup, preprint
[6] Ciuperca, A., and Elliott, G. A.: A remark on invariants for C∗-algebras of stable rank one,
preprint
[7] Choi, M.-D., and Effros, E. G.: Nuclear C∗-algebras and the approximation property, Amer.
J. Math. 100 (1978), 61-79
[8] Choi, M.-D., and Effros, E. G.: Nuclear C∗-algebras and injectivity: the general case,
Indiana Univ. Math. J. 26 (1977), 443-446
[9] Connes, A.: On the cohomology of operator algebras, J. Funct. Anal. 28 (1978), 248-253
[10] Connes, A.: Classification of injective factors. Cases II1, II∞, IIIλ, λ 6= 1., Ann. of Math.
(2) 104 (1976), 73-115
[11] Coward, K. T., Elliott, G. A., and Ivanescu, C.: The Cuntz semigroup as an invariant for
C∗-algebras, preprint
[12] Cuntz, J.: Dimension Functions on Simple C∗-algebras, Math. Ann. 233 (1978), 145-153
[13] Dădărlat, M.: Reduction to dimension three of of local spectra of real rank zero C∗-algebras,
J. Reine Angew. Math. 460 (1995), 189-212
[14] Dădărlat, M.: Nonnuclear subalgebras of AF algebras, Amer. J. Math. 122 (2000), 581-597
[15] Elliott, G. A.: On the classification of inductive limits of sequences of semi-simple finite-
dimensional algebras, J. Algebra 38 (1976), 29-44
[16] Elliott, G. A.: The classification problem for amenable C∗-algebras, Proceedings of ICM
’94, Zurich, Switzerland, Birkhauser Verlag, Basel, Switzerland, 922-932
[17] Elliott, G. A.: On the classification of C∗-algebras of real rank zero, J. Reine Angew. Math.
443 (1993), 179-219
[18] Elliott, G. A.: Towards a theory of classification, preprint
[19] Elliott, G. A., and Evans, D. E.: The structure of irrational rotation C∗-algebras, Ann. of
Math. (2) 138 (1993), 477-501
[20] Elliott, G. A., and Gong, G.: On the classification of C∗-algebras of real rank zero. II. Ann.
of Math. (2) 144 (1996), 497-610
[21] Elliott, G. A., Gong, G., and Li, L.: Approximate divisibility of simple inductive limit
C∗-algebras, Contemp. Math. 228 (1998), 87-97
[22] Elliott, G. A., Gong, G., and Li, L.: On the classification of simple inductive limit C∗-
algebras, II: The isomorphism theorem, to appear in Invent. Math.
[23] Elliott. G. A.: The classification of separable simple C∗-algebras which are inductive limits
of continuous-trace C∗-algebras with spectrum homeomorphic to [0, 1], preprint
[24] Glimm, J.: On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960),
318-340
[25] Gong, G.: On the classification of simple inductive limit C∗-algebras I. The reduction
theorem., Doc. Math.7 (2002), 255-461
[26] Gong, G., Jiang, X., and Su, H.: Obstructions to Z-stability for unital simple C∗-algebras,
Canad. Math. Bull. 43 (2000), 418-426
[27] Haagerup, U.: All nuclear C∗-algebras are amenable, Invent. Math. 74 (1983), 305-319
[28] Haagerup, U.: Connes’ bicentralizer problem and uniqueness of the injective factor of type
III1, Acta. Math. 158 (1987), 95-148
[29] Haagerup, U.: Quasi-traces on exact C∗-algebras are traces, preprint (1991)
[30] Jiang, X. and Su, H.: On a simple unital projectionless C∗-algebra, Amer. J. Math. 121
(1999), 359-413
[31] Kirchberg, E.: The classification of Purely Infinite C∗-algebras using Kasparov’s Theory,
to appear in Fields Inst. Comm.
[32] Kirchberg, E.: C∗-nuclearity implies CPAP, Math. Nachr. 76 (1977), 203-212
http://arxiv.org/abs/math/0609182
16 GEORGE A. ELLIOTT AND ANDREW S. TOMS
[33] Kirchberg, E., and Phillips, N. C.: Embedding of exact C∗-algebras in the Cuntz algebra
O2, J. Reine Angew. Math. 525 (2000), 17-53
[34] Kirchberg, E., and Winter, W.: Covering dimension and quasidiagonality, Intern. J. Math.
15 (2004), 63-85
[35] Lance, E. C.: Hilbert C∗-modules, London Mathematical Society Lecture Note Series 210,
Cambridge University Press, 1995
[36] Lin, H.: The tracial topological rank of C∗-algebras, Proc. London Math. Soc. (3) 83 (2001),
199-234
[37] Lin, H.: Classification of simple C∗-algebras of tracial topological rank zero, Duke Math. J.
125 (2004), 91-119
[38] Lin, H.: Simple nuclear C∗-algebras of tracial topological rank one, preprint
[39] Manuilov, V. M., and Troitsky, E. V.: Hilbert C∗-modules, Translations of Mathematical
Monographs 226, American Mathematical Society, 2001
[40] Niu, Z.: A classification of certain tracially approximately subhomogeneous C∗-algebras,
Ph.D. thesis, University of Toronto, 2005
[41] Perera, F.: The structure of positive elements for C∗-algebras of real rank zero, Int. J. Math.
8 (1997), 383-405
[42] Perera, F., and Toms, A. S.: Recasting the Elliott conjecture, to appear in Math. Ann.,
arXiv preprint math.OA/0601478 (2006)
[43] Phillips, N. C.: A classification theorem for nuclear purely infinte simple C∗-algebras, Doc.
Math. 5 (2000), 49-114
[44] Phillips, N. C.: Every simple higher dimensional noncommutative torus is an AT-algebra,
arXiv preprint math.OA/0609783 (2006)
[45] Rørdam, M.: Classification of Nuclear C∗-Algebras, Encyclopaedia of Mathematical Sci-
ences 126, Springer-Verlag, Berlin, Heidelberg 2002
[46] Rørdam, M.: A simple C∗-algebra with a finite and an infinite projection, Acta Math. 191
(2003), 109-142
[47] Rørdam, M.: The stable and the real rank of Z-absorbing C∗-algebras, Int. J. Math. 15
(2004), 1065-1084
[48] Rørdam, M.: Stability of C∗-algebras is not a stable property, Doc. Math. 2 (1997), 375-386
[49] Rørdam, M.: On the structure of simple C∗-algebras tensored with a UHF-algebra, II, J.
Funct. Anal. 107 (1992), 255-269
[50] Thomsen, K.: Inductive limits of interval algebras: unitary orbits of positive elements,
Math. Ann. 293 (1992), 47-63
[51] Toms, A. S.: On the independence of K-theory and stable rank for simple C∗-algebras, J.
Reine Angew. Math. 578 (2005), 185-199
[52] Toms, A. S.: On the classification problem for nuclear C∗-algebras, to appear in Ann. of
Math. (2), arXiv preprint math.OA/0509103 (2005)
[53] Toms, A. S.: An infinite family of non-isomorphic C∗-algebras with identical K-theory,
arXiv preprint math.OA/0609214 (2006)
[54] Toms, A. S.: Cancellation does not imply stable rank one, Bull. London Math. Soc. 38
(2006), 1005-1008
[55] Toms, A. S.: Stability in the Cuntz semigroup of a commutative C∗-algebra, to appear in
Proc. London Math. Soc., arXiv preprint math.OA/0607099 (2006)
[56] Toms, A. S. and Winter, W.: Strongly self-absorbing C∗-algebras, to appear in Trans. Amer.
Math. Soc., arXiv preprint math.OA/0502211 (2005)
[57] Toms, A. S. and Winter, W.: Z-stable ASH algebras, to appear in Canad. J. Math., arXiv
preprint math.OA/0508218 (2005)
[58] Toms. A. S. and Winter, W.: The Elliott conjecture for Villadsen algebras of the first type,
arXiv preprint math.OA/0611059 (2006)
[59] Villadsen, J.: Simple C∗-algebras with perforation, J. Funct. Anal. 154 (1998), 110-116
[60] Villadsen, J.: On the stable rank of simple C∗-algebras, J. Amer. Math. Soc. 12 (1999),
1091-1102
[61] Winter, W.: On topologically finite-dimensional simple C∗-algebras, Math. Ann. 332 (2005),
843-878
[62] Winter, W.: Simple C∗-algebras with locally finite decomposition rank, arXiv preprint
math.OA/0602617 (2006)
http://arxiv.org/abs/math/0601478
http://arxiv.org/abs/math/0609783
http://arxiv.org/abs/math/0509103
http://arxiv.org/abs/math/0609214
http://arxiv.org/abs/math/0607099
http://arxiv.org/abs/math/0502211
http://arxiv.org/abs/math/0508218
http://arxiv.org/abs/math/0611059
http://arxiv.org/abs/math/0602617
REGULARITY PROPERTIES FOR AMENABLE C
-ALGEBRAS 17
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada,
M5S 2E4
E-mail address: elliott@math.toronto.edu
Department of Mathematics and Statistics, York University, 4700 Keele St., Toronto,
Ontario, Canada, M3J 1P3
E-mail address: atoms@mathstat.yorku.ca
	1. Introduction
	2. Preliminaries
	2.1. The Elliott invariant and the original conjecture
	2.2. Amenability
	2.3. The Cuntz semigroup
	3. Three regularity properties
	3.1. Strict comparison
	3.2. Finite decomposition rank
	3.3. Z-stability
	3.4. Relationships
	4. A brief history
	4.1. Purely infinite simple algebras
	4.2. The stably finite case, I: inductive limits
	4.3. The stably finite case, II: tracial approximation
	4.4. Villadsen's algebras
	5. The way(s) forward
	5.1. New assumptions
	5.2. New invariants
	References
