The classification ofseparable simple C*-algebras which are inductive
  limits of continuous-trace C*-algebraswith spectrum homeomorphic to the
  closed interval [0,1]
THE CLASSIFICATION OF SEPARABLE SIMPLE
C*-ALGEBRAS WHICH ARE INDUCTIVE LIMITS OF
CONTINUOUS-TRACE C*-ALGEBRAS WITH SPECTRUM
HOMEOMORPHIC TO THE CLOSED INTERVAL [0,1]
GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
Abstract. A classification is given of certain separable nuclear C*-
algebras not necessarily of real rank zero, namely, the class of sep-
arable simple C*-algebras which are inductive limits of continuous-
trace C*-algebras whose building blocks have spectrum homeo-
morphic to the closed interval [0, 1], or to a disjoint union of copies
of this space. Also, the range of the invariant is calculated.
1991 Mathematics Subject Classification. 46L35, 46L06.
Key words and phrases. K-theory, classification, C*-algebras, inductive limits, real rank one.
Date: September 2005.
http://arxiv.org/abs/0704.1449v1
2 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
1. Introduction
It is shown in [23] that an important class of separable simple crossed
product C*-algebras are approximately subhomogeneous. Recall that a C*-
algebra is said to be subhomogeneous if it is isomorphic to a sub-C*-algebra
ofMn(C0(X)) for some natural number n and for some locally compact Haus-
dorff space X. An approximately subhomogeneous C*-algebra, abbreviated
ASH algebra, is an inductive limit of subhomogeneous algebras.
This article contains a partial result in the direction of classifying all simple
ASH algebras by their Elliott invariant.
The first result on the classification of C*-algebras not of real rank zero
was the classification by G. Elliott of unital simple approximate interval al-
gebras, abbreviated AI algebras (see [12]). This result was extended to the
non-unital case independently by I. Stevens ([30]) and K. Thomsen ([34]).
Also, an interesting partial extension of this result to the non-simple case
was given by K. Stevens ([32]). It is worth mentioning that all these algebras
are what are referred to as approximately homogeneous algebras, abbreviated
AH algebras, and that the most general classification result for simple AH
algebras was obtained by Elliott, Gong and Li in [16].
One of the first isomorphism results for ASH algebras was the proof by H.
Su of the classification of C*-algebras of real rank zero which are inductive
limits of matrix algebras over non-Hausdorff graphs; see [33]. The classifica-
tion of ASH algebras was also considered in [19], [27] and [29]. (This list of
contributions is intended to be representative rather than complete for the
classification of ASH algebras.)
An important work on the classification of ASH algebras not of real rank
zero, and in fact one of the first ones, is due to I. Stevens ([31]). The main
result of the present paper is a substantial extension of Stevens’s work, to
the class consisting of all simple C*-algebras which are inductive limits of
continuous-trace C*-algebras with spectrum homeomorphic to the closed in-
terval [0, 1] (or to a finite disjoint union of closed intervals). In particular,
the spectra of the building blocks considered here are the same as for those
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 3
considered by Stevens. The building blocks themselves are more general.
The isomorphism theorem is proved by applying the Elliott intertwining
argument.
Inspired by I. Stevens’s work, the proof proceeds by showing an Existence
Theorem and a Uniqueness Theorem for certain special continuous trace C*-
algebras. (As can be seen from the proofs, it is convenient to have a special
kind of continuous trace C*-algebra as the domain algebra in both these
theorems. By special we mean having finite dimensional irreducible repre-
sentations and such that the dimension of the representation, as a function
on the interval, is a finite (lower semicontinuous) step function.)
The present Existence Theorem, Theorem 5.1, differs in an important way
from that of [31], Theorem 29.4.1. In fact Theorem 29.4.1 of [31] is false, as
is shown in Section 5.1 below.
The proof of the present Existence Theorem is an eigenvalue pattern per-
turbation, as shown in Section 5, which is similar to the approach used in
[31]. (Indeed, once the statement of Theorem 29.4.1 of [31] is corrected, the
argument given in [31] does not need to be essentially changed.)
The proof of the present Uniqueness Theorem is different from the one in
[31]. It uses the finite presentation of special continuous trace C*-algebras
that was given in [17] and [18]. Also the present Uniqueness Theorem has
the advantage that both the statement and the proof are intrinsic, i.e., there
is no need to say that the building blocks are hereditary sub-C*-algebras of
interval algebras as in [31].
In order to apply the Existence and Uniqueness Theorems, it is necessary to
approximate the general continuous trace C*-algebras appearing in a given
inductive limit decomposition by special continuous trace C*-algebras, as
described in [18], Theorem 4.15. This is admissible since in [18] (and also
more generally in [17]), it is shown that these special C*-algebras are weakly
semiprojective, i.e., have stable relations. (A result of T. Loring, Lemma
15.2.2, [24], allows one to conclude that the original inductive limit decom-
position can be replaced by an inductive limit of special continuous trace
4 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
C*-algebras.)
An important step of the proof of the isomorphism theorem is the pulling
back of the invariant from the inductive limit to the finite stages. The invari-
ant has roughly two major components: a stable part and a non-stable part.
The pulling back of the stable part is contained in [12] or [31] and is performed
in the present situation with respect to the unital hereditary sub-C*-algebras.
The intertwining which is obtained at the level of the stable invariant will
approximately respect the non-stable part of the invariant on finitely many
elements, as pointed out in [31]. To be able to apply the Existence Theo-
rem it is crucial to ensure that the non-stable part of the invariant is exactly
preserved on finitely many elements (actually, just a single element). It is
possible to obtain an exact preservation of the non-stable invariant on finitely
many elements because one can change the given finite stage algebras in the
inductive limit decomposition in such a way that a non-zero gap arises at the
level of the affine function spaces; see Section 8 below. It is this non-zero gap
that will ultimately guarantee (after passing to subsequences in a convenient
way) the exact intertwining on finte sets of the non-stable invariant, as shown
in Section 9. It is worth mentioning that in the pulling back of the stable
invariant, we must ensure, at the same time that the maps at the affine func-
tion space level are given by eigenvalue patterns. This is necessary in order
to apply the Existence Theorem and is possible by the Thomsen-Li theorem.
Now all the hypotheses of the Elliott intertwining argument are fulfilled
and in this way the proof of the isomorphism Theorem 3.1 is completed.
I. Stevens’s description of the range of the invariant is also extended to
include the case of unbounded traces (Theorem 3.2).
To conclude, the class of simple inductive limits of continuous-trace C*-
algebras under consideration is compared with the class of simple AI alge-
bras.
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 5
2. The invariant
The invariant is similar to the invariant I. Stevens has used in [31], usually
summed up as the Elliott invariant, namely, (K0(A),AffT
+A, Aff ′A), where
K0(A) is a partially ordered abelian group, AffT
+A is a partially ordered vec-
tor space consisting of linear and continuous functions defined on the cone of
traces T+A, Aff′A is a certain special subset of AffT+A. The special subset
Aff ′A is the most important part of the invariant for our purposes, and in
an informal way it might be said to be the non-stable part of the AffT+A.
Formally, the special subset Aff′A is the convex set obtained as the closure
of {â ∈ AffT+A| a ≥ 0, a ∈ Ped(A) and ||a|| ≤ 1} inside AffT+A, with
respect to the topology naturally associated to a full projection. Here â is
the linear and continuous function defined by the positive element a from the
Pedersen ideal by â(τ) = τ(a) where τ ∈ T+A. As shown in [31], Remark
30.1.1 and Remark 30.1.2, the information given by Aff′A is equivalent with
that given by the trace-norm map, which is a lower semicontinuous function
µ : T+A→ R, µ(τ) = ||τ || and ∞ if τ is unbounded.
It is a crucial fact that the trace-norm map is equivalent to the dimension
function in the case of a building block algebra, cf. Section 4 below. The
dimension function of a building block (i.e. the function that assigns to each
point in the spectrum of the building block the dimension of the irreducible
representation) can be viewed as a lower semicontinuous function on the ex-
treme traces normalized on minimal projections in primitive quotiens and
hence we can compare it with functions from AffT+A. Then the subset Aff′A
is the closure of the set of all affine functions smaller than the dimension
function. Conversely by taking the supremum over all elements of Aff′A we
recover the dimension function in the case of the building blocks.
3. The results
Using the invariant described above it is possible to prove a complete iso-
morphism theorem, namely,
6 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
Theorem 3.1. Let A and B be two non-unital simple C*-algebras which are
inductive limits of continuous-trace C*-algebras with spectrum homeomorphic
to [0, 1]. Assume that
1. there is a order preserving isomorphism ψ0 : K0(A) → K0(B),
2. there is an isomorphism ψT : AffT
+ → AffT+B, such that
ψT (Aff
′A) ⊆ Aff ′,
3. the two isomorphisms are compatible:
ψ̂0([p]) = ψT ([̂p]), [p] ∈ K0(A).
Then there is an isomorphism of the algebras A and B that induces the given
isomorphism at the level of the invariant.
A description is given of the range of the invariant. More precisely, the
following theorem is proved:
Theorem 3.2. Suppose that G is a simple countable dimension group and
V is the cone associated to a metrizable Choquet simplex. Let λ : V →
Hom+(G,R) be a continuous affine map which takes extreme rays into ex-
treme rays. Let f : V → [0,+∞] be an affine lower semicontinuous map, zero
at zero and only at zero. Then (G, V, λ, f) is the invariant of some simple
non-unital inductive limit of continuous-trace C*-algebras whose spectrum is
the closed interval [0, 1].
4. Special continuous trace C*-algebras with spectrum the
interval [0,1]
In this section we will introduce some terminology. A very important piece
of data that we shall consider is a map that assigns, to each class of irreducible
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 7
representations, the dimension of a representation from that class. Roughly
speaking, the dimension function can be thought of as the non-stable part of
the invariant when restricted to the building blocks.
Definition 4.1. Let A be a C*-algebra and let Â denote the spectrum of A.
Then the dimension function is the map from Â to R ∪ +∞,
π 7→ dim(Hπ),
where by dim(Hπ) we mean the dimension of the irreducible representation
It was shown in [18], Theorem 4.13, that the dimension function is a com-
plete invariant for continuous trace C*-algebras with spectrum the closed
interval [0, 1]. Also concrete examples were constructed for each given di-
mension function, cf. Section 7 of [18].
Therefore given a lower semicontinuous integer valued (i.e., a “dimension
function”) which is finite-valued and bounded we can exhibit a continuous
trace C*-algebra

C0(An) C0(An) C0(An) . . . C0(An)
C0(An) C0(An−1) C0(An−1) . . . C0(An−1)
C0(An) C0(An−1) C0(An−2) . . . C0(An−2)
. . .
C0(An) C0(An−1) C0(An−2) . . . C[0, 1]

⊆Mn ⊗ C[0, 1].
whose dimension function is the given function. Here An ⊆ An−1 ⊆ · · · ⊆
[0, 1] and each Ai is an open subset of [0, 1]. Moreover any trace on such
an algebra is of the form tr ⊗ ν, where tr is the usual trace normalized on
minimal matrix projections and ν is a finite measure on [0, 1]. The extreme
traces are parameterized by t ∈ [0, 1], and are given as (tr ⊗ δt)t∈[0,1], where
8 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
δt is the normalized point mass at t. Then the trace norm map is equal to
the dimension function when restricted to the extreme traces. To see that
the trace norm map is equivalent to the special subset Aff ′() of the affine
function space AffT+() we repeat the proof of I. Stevens from [31], Remark
30.1.1 and Remark 30.1.2.
Inspired by a construction of I. Stevens in [31] we make
Definition 4.2. A continuous-trace C*-algebra whose spectrum is [0, 1] will
be called a special continuous-trace C*-algebra if its dimension function is a
finite-valued finite step function: there is a partition of [0, 1] into a finite
union of intervals such that the dimension function is finite and constant on
each such subinterval.
Remark 4.1. Let A be a continuous trace C*-algebra with spectrum [0, 1] and
with dimension function d : [0, 1] → N ∪ {+∞}. There exists a projection-
valued function that if composed with the rank function gives rise to the
dimension function d. To see this first we notice that because the Dixmier-
Douady invariant of A is trivial, the C*-algebra A is a continuous field
of elementary C*-algebras over [0, 1], where the fibers are hereditary sub-
C*-algebras of the algebra of compact operators. Then take the unit of
the hereditary sub-C*-algebra in each fiber. In this way we construct a
projection-valued function which is lower semicontinuous. By composing
this constructed projection-valued function with the rank function we get
the dimension function d.
Remark 4.2. A priori our definition for a special sub-C*-algebra is more general
than I. Stevens’s definition. As it is shown in [18], any special sub-C*-algebra
in our sense is isomorphic to a special sub-C*-algebra in I. Stevens’s sense.
Remark 4.3. It was shown in [18] that special continuous trace C*-algebras
are finite presented and weakly semiprojective. Also a stronger result was
proven in [8], namely that special continuous trace C*-algebras are strongly
semiprojective.
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 9
5. Balanced inequalities and the Existence Theorem
The proof of the isomorphism theorem 3.1 is based on the Elliott inter-
twining argument. Among the main ingredients of this procedure are the
Existence Theorem that will be described below as well as the Uniqueness
Theorem that is presented in Section 6.
It is worth noticing that for the Existence Theorem and the Uniqueness
Theorem we require that the inequalities are balanced, i.e., independent of
the choice we make for the normalization of the affine function space. We
normalize the affine function spaces with respect to a full projection. Even
though we fix a projection in the domain algebra for both the Existence The-
orem and the Uniqueness Theorem, this choice does not make any difference
when we apply the theorems to obtain an approximate commuting diagram.
As was pointed out to us by Andrew Toms, we only need to consider a com-
patible family of projections when we go through the whole proof, provided
that a corresponding projection is chosen in the codomain algebra. In fact,
we can state the theorems without mentioning the choices of the projections
as long as their K0 -classes are compatible with respect to the K0 -map under
consideration even though they exist and some choices of them will be used
during the proof.
To be able to focus on the new aspects of the present Existence Theorem
as opposed to the Existence Theorem for unital continuous trace C*-algebras
proved by Elliott in [12], we will both state the theorem and prove it in terms
of so-called eigenvalue pattern maps. In our situation an eigenvalue pattern
map is a positive unital map from C([0, 1]) to C([0, 1]) which is a finite sum of
*-homomorphisms from C([0, 1]) to C([0, 1]). Using the Gelfand theory each
such *-homomorphism is given by a continuous function from [0, 1] to [0, 1].
As follows from the intertwining of the invariant and will be explained below,
Section 9, one can always obtain a (non-necessarily compatible) eigenvalue
patterns maps.
The proof of the Existence Theorem is obtained by perturbing an eigen-
value pattern map between the affine function spaces in a such a way that it
10 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
defines an algebra map between the building blocks.
Theorem 5.1. Let A be a special building block and by dA denote the dimension
function of A. Let a finite subset F contained in AffT+A, and ǫ > 0 be given.
There is f ′ ∈ Aff ′A such that for any special building block B with dimension
function dB, and maps k : D(A) → D(B) and T : AffT
+A → AffT+B
verifying the conditions
1. k has multiplicity Mk,
2. T is given by an eigenvalue pattern and has the property
T (f ′) ≤ dB,
3. k and T are exactly compatible, i.e.,
k̂([r]) = T ( ˆ[r]),
there is a homomorphism ψ : A→ B such that k = ψ0 and
||(T − ψT )a||k̂(p) ≤ ǫ||a||p̂, a ∈ F.
Remark 5.2. Recall that AffT+A is a Banach space with a norm given by ||f ||p =
sup{|f(τ)| | τ(p) = 1, τ ∈ T+A}, where f ∈ AffT+A and p is a fixed
full projection of A. In addition, using the norm we just defined, AffT+A
is identified with C([0, 1]). This identification allows us to compare in the
supremum norm the dimension function and elements of AffT+A. Also the
norm of AffT+B is defined with respect to a projection from B which is
Murray-von Neumann equivalent to k(p). Since our inequalities at the level of
the affine function spaces are balanced, which is the only theorem that makes
sense, in particular they are independent of the choice of the projection p.
Proof. The idea of the proof is to choose in a clever way a function f ′ and then
change within the given tolerance the eigenvalue functions that appear in the
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 11
eigenvalue pattern T so that the image of the dimension function dA under
the new eigenvalue pattern is smaller than or equal the dimension function
of the algebra B, as desired.
Let ǫ > 0 and a finite set F ⊂ AffT+A be given.
As already mentioned it is a crucial step how f ′ is chosen. There is no
loss in generality if we assume that the dimension function dA has only one
discontinuity point, t0 ∈ [0, 1].
Figure 1. Dimension function dA.
Choose f ′ to be a continuous function such that f ′(t) = dA(t) for t ∈
[0, t0−δ]∪ [t0+δ, 1], f
′(t) ≤ dA(t) for t ∈ [0, 1], and f
′(t0) = dA(t0), where δ ≤
. Hence f ′ is a continuous function defined on the interval [0, 1] which
approximate dA, namely f
′ is equal to dA except on a small neighbourhood
around the discontinuity point.
12 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
Figure 2. Graph of f ′.
Next we proceed by showing how to change the eigenfunctions such that
a desired eigenvalue pattern is obtained. We will carry out this procedure
in a very special case, namely all the eigenfunctions are assumed to be the
identity function.
Figure 3. Eigenfunction λ.
In the above picture we have the original eigenvalue function λ which is
the identity map. We define a new eigenvalue function as the picture shows
below, Figure 4. More precisely the new eigenvalue function λ̂ : [0, 1] → [0, 1],
λ̂(t) = t for t ∈ [0, t0− δ)∪ (t0+ δ+ δt0, 1], λ̂(t) = t0− δ for t ∈ [t0− δ, t0+ δ],
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 13
the linear map λ̂(t) = t0 − δ + (t − t0 − δ)
2δ+δt0
for t ∈ [t0 + δ, t0 + δ + δρ],
where δt0 is a strictly positive number such that t0 + δ + δt0 ≤ 1.
Figure 4. Eigenfunction λ̂.
A short computation or a geometric argument shows that the difference
||λ− λ̂||∞ = 2δ.
Moreover the dimension function dA evaluated on the perturbed eigenvalue
λ̂ is smaller then f ′ evaluated on the given eigenvalue λ
dA(λ̂(t)) ≤ f
′(λ(t)).
Hence by hypothesis 2 we have
dA ◦ λ̂ ≤
f ′ ◦ λ ≤ dB.
Here we say that one dimension function is smaller than another one if the
relation holds pointwise.
The change of the eigenvalues is small because of the choice of δ
||(Tλ − T )(a)||k̂(p) =
||a ◦ (λ̂i − λi)||k̂(p)
14 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
sup{|a ◦ (λ̂i − λi)(τ)| | τ(k(p)) = 1, τ ∈ T
∣∣∣∣a ◦ (λ̂i − λi)
)∣∣∣∣ | τ(p) = Mk, τ ∈ T
Mk||a ◦ (λ̂i − λi)||p̂ ≤ 2δM
k ||a||p̂ ≤ ǫ||a||p̂, a ∈ F.
To obtain the inequality above we used the linearity of the function a◦(λ̂i−
λi) and that an extreme trace τ in T
+A has the property that τ(k(p)) = 1 if
and only if τ(p) =Mk.
We claim that the argument for the special case shown above can be ex-
tended to the case of piecewise linear eigenfunctions which is known to be
equivalent to the general case of continuous eigenfunctions that arise in the
inductive limits of interval algebras (see for instance [12]).
5.1. An exact inequality is necessary between the non-stable part
of the invariant. As mentioned in the introduction, the Theorem 29.4.1 of
[31] is false. To prove the Existence Theorem it is fundamental to have an
exact inequality between the non-stable part of the invariant at the level of
the affine function space, i.e., T (f) ≤ dB for some continuous affine function
f ≤ dA. A weaker inequality is required in the statement of the Existence
Theorem of [31], Theorem 29.4.1, i.e., T (f) ≤ dB(1+δ) for some small δ > 0.
Therefore it is possible to construct a counterexample to the I. Stevens Ex-
istence Theorem. This counterexample is already assuming that the positive
linear map T is given by an eigenvalue pattern. To reduce the proof of The-
orem 29.4.1 of [31] to an eigenvalue pattern problem, one needs an extra
assumption in hypothesis 2, for instance a positive gap η > 0 in the other
side of the inequality described above T (f) + η ≤ dB(1 + δ).
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 15
Next we describe the counterexample. Let dA be the lower semicontinu-
ous function defined on [0, 1] which is equal to 2 on the subintervals [0, 1/2)
and (1/2, 1], and equal to 1 at 1/2. Let ǫ0 be such that 0 < ǫ0 < 1/4 and
F = {a1(t) = t}. Let f be a continuous function which approximates dA.
Since they can not be equal everywhere around 1/2, we can assume that
f(t) < 2 = dA(t) for all t in (1/2− η, 1/2+ η), where η > 0 can be chosen as
small as needed.
Let δ > 0 be given. There exists a positive integerMk such that
2Mk−1
Then choose T to be defined by Mk eigenvalue functions (λi)i=1,...,Mk , all be-
ing the identity functions, λi(t) = t, for all i = 1, . . . ,Mk. Next choose B to
be a continuous trace C*-algebra with dimension function constant equal to
2Mk − 1.
Note that the hypothesis 2 of the Existence Theorem 29.4.1 from [31] holds
T (f)(t) =
f ◦ λi(t) ≤ 2Mk ≤ (1 + δ)dB(t).
Now we claim that among all perturbations of T which are within the given
ǫ0 with respect to the finite set F , the particular one P which is given by the
continuous eigenfunctions (µi)i=1,...,Mk that have the property µi(t) = 1/2 for
t ∈ (1/2− η, 1/2 + η), is the smallest in the sense that the value of P (dA) is
the smallest. Here it is important to notice that because ǫ0 < 1/4 it forces
that (µi)i(t) = λ(t) = t for t close to 0 and 1 including 0 and 1. In particular
we have (µi)(0) = λi(0) = 0. Therefore
P (dA)(0) =
dA(µi(0) = 2Mk > 2Mk − 1 = dB(0).
16 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
Therefore we cannot perturb the eigenfunctions to obtain a compatible
eigenvalue pattern and the Existence Theorem as stated in [31] cannot be
proved.
6. Uniqueness Theorem
It is important to notice that the conclusion of the Existence Theorem is
part of the hypothesis of the Uniqueness Theorem; this makes sense since all
inequalities are balanced (i.e. independent of the choice of projection with
respect to which the normalization is done).
Theorem 6.1. Let A be a special continuous-trace C*-algebra, F ⊂ A a finite
subset and ǫ > 0. Let B be a special continuous-trace C*-algebra and ψ, ϕ :
A→ B be maps with the following properties:
1. ϕ0 = ψ0 : K0(A) → K0(B),
2. ψ and ϕ have at least the fraction δ of their eigenvalues in each of the
d consecutive subintervals of length 1
of [0, 1], for some d > 0 such that for
r̂i the functions equal to 0 from 0 to
, equal to 1 on [ i+1
, 1] and linear in
between, for each 0 ≤ i ≤ d, ||(ϕT − ψT )(r̂i)||K(p) < δ||r̂i||p, with respect to
the norm of AffT+B,
Then there is an approximately inner automorphism of B, f , such that
||(ψ − fϕ)(a)|| < ǫ, a ∈ F
Proof. Because of the isomorphism theorem 4.13 from [18], there is no loss of
generality to assume that our building blocks are in a very special form
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 17

C0(A1) C0(A1) C0(A1) . . . C0(A1)
C0(A1) C0(A2) C0(A2) . . . C0(A2)
C0(A1) C0(A2) C0(A3) . . . C0(A3)
. . .
C0(A1) C0(A2) C0(A3) . . . C[0, 1]

Notice that the cancellation property holds for the unital sub-C*-algebra of
A and any projection of A is Murray-von Neumann equivalent to a projection
inside of the unital sub-C*-algebra. Therefore the cancellation property holds
for A. A similar argument shows that the cancellation property holds for any
continuous-trace C*-algebra with the spectrum the closed interval [0, 1].
Since ϕ0 = ψ0, we can assume that ϕ(p) = ψ(p), where p is the unit of the
sub-C*-algebra C([0, 1]) of A. In other words the restrictions of the maps to
the unital subalgebra share the same unit.
The stable part of the Elliott invariant (i.e., the K0 group and the affine
function space AffT+) of A and of C([0, 1]) is the same. Let us restrict the two
maps ϕ and ψ to the unital sub-C*-algebra C([0, 1]). The image of C([0, 1])
under ϕ and ψ is up to a unitary a full matrix algebra over the interval.
Then using assumptions 1 and 2 we notice that the hypotheses of the Elliott
Uniqueness Theorem ([12], Theorem 6), are fulfilled. Hence we get a partial
isometry V of B (a unitary inside of the full matrix sub-C*algebra of B) such
||ϕ(fAi ⊗ enn)− V ψ(fAi ⊗ enn)V
∗|| ≤ ǫ, i ∈ {1, . . . , n}.
We want this relation to hold for the case when the domain is A. We follow
a strategy already present in the case of full matrix over the interval. An
18 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
important data that we will use is that the domain algebra A has a finite
presentation. In fact we will use the concrete description of this presentation
that was given in [18], Section 8. The set of generators consists of elements
of the form fAi ⊗ ein which are certain positive functions tensor the matrix
units.
For each i let ui be a continuous function defined on [0, 1] which is equal
to 1 on Ai except near the end points of each open subinterval of Ai and
0 otherwise. One can think of ui as an approximate unit of the functions
fAi, i ∈ {1, . . . , n} and later estimates depend on the size of the subset of Ai
where ui is not equal to 1.
Define
ϕ(ui ⊗ eni)
∗V ψ(ui ⊗ eni).
Vψ(fAi ⊗ eni)V
ϕ(uk⊗enk)
∗V ψ(uk⊗enk))ψ(fAi⊗eni)(
ψ(ul⊗eln)
∗V ∗ϕ(ul⊗enl)) =
= ϕ(un ⊗ enn)V ψ(fAi ⊗ eni)(
ψ(ul ⊗ eln)
∗V ∗ϕ(ul ⊗ enl)) =
= ϕ(un ⊗ enn)V ψ(fAi ⊗ eni)(
ψ(ul ⊗ enl)V
∗ϕ(ul ⊗ enl)) =
= ϕ(un ⊗ enn)V ψ(fAi ⊗ eni)ψ(ui ⊗ ein)V
∗ϕ(ui ⊗ eni) =
= ϕ(un ⊗ enn)V ψ(fAi ⊗ enn)V
∗ϕ(ui ⊗ eni).
Now we have that
ϕ(fAi ⊗ eni) = ϕ(un ⊗ enn)ϕ(fAi ⊗ enn)ϕ(ui ⊗ eni).
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 19
Therefore
||ϕ(fAi ⊗ eni)− Vψ(fAi ⊗ eni)V
∗|| =
||ϕ(un ⊗ enn)(ϕ(fAi ⊗ enn)− V ψ(fAi ⊗ enn)V
∗)ϕ(ui ⊗ eni)|| ≤
≤ ||ϕ(un ⊗ enn)||ǫ||ϕ(ui ⊗ eni)||
i.e. it can be made as small as needed.
We want to argue that V gives rise to a partial isometry. Let us calculate
V∗V =
ψ(ul ⊗ enl)
∗V ∗ϕ(ul ⊗ enl)
ϕ(ui ⊗ eni)
∗V ψ(ui ⊗ eni) =
ψ(ul ⊗ eln)V
∗ϕ(ul ⊗ enl)
ϕ(ui ⊗ ein)V ψ(ui ⊗ eni) =
Assuming that each ui is equal to 1 on the open intervals Ai except small
neighbourhood around the end points of Ai we get
ψ(ui ⊗ ein)V
∗ϕ(ui ⊗ enn)V ψ(ui ⊗ eni)
which is very close to
ψ(ui ⊗ ein)ψ(ul ⊗ enn)ψ(ui ⊗ eni) =
ψ(ui ⊗ eii)
which is the value of the projection-valued map of the hereditary sub-C*-
algebra generated by ψ(A) inside B. In other words V∗V is as close as we
want to be a projection. It is important to notice that this is true if we
are not in a small neighbourhood of the singularity points of the dimension
function of the hereditary sub-C*-algebra generated by ψ(A) (i.e. whenever
20 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
ui = 1).
Similarly VV∗ is almost equal to the
i=1 ϕ(ui ⊗ eii) if we are not in a
small neighbourhood of the singularity points of the dimension function of
the hereditary sub-C*-algebra generated by ϕ(A). Notice that any singularity
point y0 of the dimension function of the hereditary sub-C*-algebra generated
by ϕ(A) or ψ(A) has the property that there is an eigenfunction λi such that
λi(y0) is a singularity point of the dimension function dA ofA. In addition λi is
uniform continuous function from [0, 1] to [0, 1]. Hence small neighbourhoods
of y0 correspond to small neighbourhoods of some singularity point of dA.
From the polar decomposition V = W|V| we get a partial isometry W . We
claim that W still intertwines approximately the two maps ϕ and ψ, i.e.,
||ϕ(fAi ⊗ eni)−Wψ(fAi ⊗ eni)W
∗|| < 3ǫ,
||W∗ϕ(fAi ⊗ eni)W − ψ(fAi ⊗ eni)|| < 3ǫ.
This is true because
||ϕ(fAi ⊗ eni)−Wψ(fAi ⊗ eni)W
∗|| =
= ||ϕ(fAi⊗eni)−Vψ(fAi⊗eni)V
∗+W|V|ψ(fAi⊗eni)|V|W
∗−Wψ(fAi⊗eni)W
∗|| ≤
≤ ||ϕ(fAi⊗eni)−Vψ(fAi⊗eni)V
∗||+||W|V|ψ(fAi⊗eni)|V|W
∗−Wψ(fAi⊗eni)W
∗|| ≤
≤ ǫ+ |||V|ψ(fAi ⊗ eni)|V| − ψ(fAi ⊗ eni)|| ≤
≤ ǫ+ |||V|ψ(fAi ⊗ eni)|V|− |V|ψ(fAi ⊗ eni)+ |V|ψ(fAi ⊗ eni)−ψ(fAi ⊗ eni)|| ≤
≤ ǫ+ ǫ+ ǫ = 3ǫ.
and similarly we get the other desired inequality.
Hence we have constructed a family of partial isometriesW from the hered-
itary sub-C*-algebra generated by ϕ(A) to the hereditary sub-C*-algebra
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 21
generated by ψ(A). In addition W induces an isomorphism between the two
above mentioned hereditary sub-C*-algebras. In particular it implies that
the two hereditary sub-C*-algebras have the same dimension function.
Next we will show how to approximateW with a unitary in the unitazation
of the codomain algebra.
Let us start by applying Theorem 4.12 of [18] to the projection-valued
function corresponding to the hereditary sub-C*-algebra generated by ϕ(A).
Hence we get a decomposition, possibly infinite, in terms of functions each
of which is projection-valued of rank 1 on a certain open subset of [0, 1] and
zero otherwise. Notice that the discontinuity points of the dimension func-
tion of the hereditary sub-C*-algebra generated by ϕ(A) correspond to the
discontinuity points of the functions appearing in the decomposition and the
open sets are increasing in a suitable sense.
Next we apply Lemma 6.2 for each point at singularity in the interval [0, 1],
or, in other words, to each function appearing on the decomposition. Thus,
we have a family of unitaries that preserves the continuity of the continuous
elements of the hereditary sub-C*-algebra ϕ(A) and at the same time has the
property that it still intertwines the two maps.
In the following lemma the hereditary sub-C*-algebras H1 and H2 are as-
sumed to be continuous bundles over [0, 1] (for more details about continuous
bundles of C*-algebras see [20]).
If A is a continuous bundle of C*-algebras over [0, 1] then At stands for the
fiber of A over t.
Lemma 6.2. Let H1 and H2 be hereditary sub-C*-algebra of M2(C[0, 1]) with
the same spectrum [0, 1] and identical dimension function equal to 1 on the
closed interval [0, t0] and equal to 2 on the half-open interval (t0, 1], t0 ∈ (0, 1).
Let W = (W (t))t∈[0,1] be a family of partial isometries indexed by the points
of [0, 1]. For each t ∈ [0, 1], Wt : M2(C) → M2(C) such that W (t)W (t)
the unit of H t1 and W (t)
∗W (t) = the unit of H t2. Then there exists a family
22 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
W⊥ of partial isometries indexed by [0, 1] such that W + W⊥ is a unitary
inside of M2(C[0, 1]) and (W +W
⊥)t(f)(t) = Wt(f)(t) for any continuous
function f ∈ H1 and t ∈ [0, 1].
Proof. Diagrammatically the dimension function of H1 and H2 can be pictured
as follows.
Figure 5. Dimension function of H1 and H2.
We construct the family W⊥ = (W⊥t )t∈[0,1] as follows. Fix a t in [0, 1],
t ≤ t0. W (t) is a partial isometry on some dimension-one subspace ofM2(C).
Hence Wt(M) = c(M)Mt where c(M) is a constant depending on M and Mt
is a projection matrix in M2(C). Let W
t = c(M)(I2 − Mt). Notice that
Wt +W
t is a unitary operator on M2(C). If t > t0 then W
t = 0.
The family of unitaries (Wt +W
t )t∈[0,1] is continuous except at the point
t0. Our work below shows that this family can be modified to be continuous
overall [0, 1].
Extend (Wt)t∈[0,t0] to be a continuous family (W
t )t∈[0,1] of partial isometries
on dimension-one subspaces of M2(C). W
and lim
t→t0,t>t0
(Wt −W
t ) are two
partial isometries on the same dimension one subspace of M2(C), hence they
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 23
differ by a constant of absolute value one, i.e.
W⊥t0 = c limt→t0,t>t0
(Wt −W
Define the continuous family of unitaries (Ut)t∈[0,1] to be Ut = Wt +W
t ≤ t0 and Ut = W
t + c(Wt −W
t ) if t > t0.
Hence the continuous family of unitaries Wt is given by Ut and (Ut(f)(t) =
Wt(f)(t) for any continuous function f ∈ H1 and t ∈ [0, 1].
7. Inductive limits of special continuous trace C*-algebras
Next let us show that the Existence Theorem and the Uniqueness Theorem
presented above can be applied, i.e., that the hypotheses of the theorems can
be fulfilled. As a first step in this direction let us show that an inductive
limit of continuous-trace C*-algebras with spectrum [0, 1] (or disjoint unions
of closed intervals) is isomorphic to an inductive limit of special continuous-
trace C*-algebras.
The basic tools in establishing this step are the fact that special continous
trace C*-algebras are semiprojective (cf. [18], Theorem 6.5) and a result by
T. Loring ([24], Lemma 15.2.2) which for the convenience of the reader we
state below:
Suppose that A is a C*-algebra containing a (not necessarily nested) se-
quence of sub-C*-algebras An with the property that for all ǫ > 0 and for
any finite number of elements x1, . . . , xk of A, there exist an integer n such
{x1, . . . , xk} ⊂ǫ An.
24 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
If each An is weakly semiprojective and finitely presented, then
A ∼= lim
(Ank, γk)
for some subsequence of (An) and some maps γk : Ank → Ank+1.
Proposition 7.1. Let A be a simple inductive limit of continuous-trace C*-
algebras whose building blocks have their spectrum homeomorphic to [0, 1].
Then A is an inductive limit of direct sums of special continuous-trace C*-
algebras with spectrum [0, 1].
Proof. In Proposition 5.4 and Theorem 6.5 of [18] it is proved that the class
of special continuous trace C*-algebras with spectrum [0, 1] are finitely pre-
sented and have weakly stable relations. Each building block from the in-
ductive limit decomposition of A can be approximated by special continuous
trace C*-algebras (cf. Theorem 6.14 of [18]). Then A satisfies Loring’s hy-
pothesis where the sequence of semiprojective algebras is given by the special
algebras from the approximation of the building blocks. Thus the Loring’s
lemma implies that A is an inductive limit of special continuous trace C*-
algebras. �
8. Getting a non-zero gap at the level of affine function
spaces
To be able to exactly intertwine the non-stable part of the invariant it is
useful to know that the dimension function of any building block Am or Bm
is taken by the homomorphism φm,m+1 respectively ψm,m+1 into a function
smaller than or equal to the dimension function of Am+1 or Bm+1 such that
a non-zero gap arises. In other words we want to exclude the possible cases
when the dimension function is taken into the next stage dimension function
such that equality holds at a point or at more points. We shall show this
in the following lemma. Recall that because of Proposition 7.1, the algebras
that we want to classify can be assumed to be inductive limits of special
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 25
continuous trace C*-algebras with spectrum [0, 1], i.e., A ∼= lim
(An, φnm) and
B ∼= lim
(Bn, ψnm), where An, Bn are special continuous trace C*-algebras.
Lemma 8.1. Let A = lim
(An, φnm) be a simple C*-algebra, where each An is
a special continuous trace C*-algebra with spectrum the closed interval [0, 1]
and the dimension function assumed to be a finite-valued bounded function.
Then there exists δ1 > 0, a subsequence (Ani)ni≥0 of (An)n and a sequence of
maps φi : Ani → Ani+1 such that
1. A ∼= lim
(Ani, φnimi),
2. (φn1n2)T (P̂An1) + δ1 < P̂An2 ,
where the inequality holds pointwise, (φnm)T is the induced map at the level
of the affine function spaces, PAn1 and PAn2 are the units of the biduals of An1
and An2, and P̂An1 and P̂An2 denote the corresponding lower semicontinuous
functions.
Proof. Let A be equal to lim
An with maps φn,m : An → Am.
The plan is to keep the same building blocks and to change slightly the
maps with respect to some given finite sets such that the desired property
holds. To do this we use the property that the building blocks that appear
in the inductive limit decomposition are weakly semiprojective.
Assume that the dimension function of φ12(A1) equals the dimension func-
tion of A2 at some point or even everywhere and let ǫ > 0, F1 ⊂ A1 be
given. Because the largest value of the dimension function of the hereditary
sub-C*-algebra generated by φ12(A1) inside A2 is attained on an open subset
U of [0, 1], let us construct another dimension function as follows: shrink one
of the open intervals of the open set U to get U ′ and in exchange enlarge the
interval adjacent to that discontinuity point. U ′ is constructed in a such a
way that is as close as necessary to the given U .
In this manner we find a sub-C*-algebra B which is as close as we want
to the hereditary sub-C*-algebra generated by φ12(A1) inside of A2. Next
we use that A1 is weakly semiprojective to find another *-homomorphism
ρ1 : A1 → B which is close within the given ǫ on the given finite set F1.
26 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
Then there exists some open interval between the dimension function of A2
and the dimension function of the B. This open interval corresponds to a
non-zero ideal I1 inside of A2. Now the image of I1 in the inductive limit is
also a non-zero ideal. Since the inductive limit is simple, it implies that the
ideal is the whole algebra. We know that there are full projections in the
inductive limit. Therefore there is a finite stage in the inductive limit of the
ideals coming from I1 that has a full projection. Assume that the finite stage
is inside of Ak. This means that at that stage the image of the ideal I1 is Ak.
Pick a strictly positive element a1 in I1. Then the image of a1 in Ak will be
strictly positive at each point from [0, 1], k > 1. This shows that the image
of the dimension function dB inside the dimension of Ak has a gap of at least
1 everywhere in [0, 1].
Because of the normalizations of the affine function, this gap of size 1 will
correspond to some strictly non-zero δ1. To complete the proof we relabel B
as An1, Ak as An2 etc.
Corollary 8.2. Let A = lim
(An, φn,m) be a simple C*-algebra. Then there ex-
ists a sequence (δi)i≥1, δi > 0, a subsequence of algebras (Ani)i≥1 of (Ai)i≥1
and a sequence of maps φ : Ani → Ani+1 such that:
1. A ∼= lim
(Ani, φni,mi),
2. φTni,ni+1(P̂Ani) + δi < P̂Ani+1 .
Proof. Follows by successively applying the previous lemma. �
9. Pulling back of the isomorphism between inductive limits at
the level of the invariant
Step 1 The intertwining between the stable part of the invariant
With no loss of generality we assume that the building blocks have the
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 27
following concrete representation

C0(A1) C0(A1) C0(A1) . . . C0(A1)
C0(A1) C0(A2) C0(A2) . . . C0(A2)
C0(A1) C0(A2) C0(A3) . . . C0(A3)
. . .
C0(A1) C0(A2) C0(A3) . . . C[0, 1]

One can distinguish a full unital hereditary sub-C*-algebra

0 0 0 . . . 0
. . . 0 . . . 0
0 . . . C[0, 1] . . . C[0, 1]
. . .
0 . . . C[0, 1] . . . C[0, 1]

The unital hereditary sub-C*-algebra has the same stable invariant (i.e.,
K0, AffT
+ and the pairing) as the given C*-algebra. Moreover the unital
hereditary sub-C*-algebra is a full matrix algebra over the closed interval
[0, 1]. Using this fact we derive an intertwining between the stable invariant,
as is shown in [31] or originally in [12].
It is important to mention the method of normalizing the affine function
spaces. Pick a full projection p1 ∈ A1. Normalize the affine space AffT
with respect to p1. Next consider a image of p1 in A2 under the map at the
dimension range level, call it p2. Normalize AffT
+A2 with respect to p2. Note
that the map which is induced at the affine level is a contraction. Continue
in this way so that we obtain an inductive limit sequence at the level of the
28 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
affine spaces, with all the maps being contractions:
AffT+A1 → AffT
+A2 → · · · → AffT
Let p∞ denote the image of p1 in the inductive limit A and denote by q∞ a
representative of φ0(p∞) in B. Then there exists q1 ∈ B1 such that the image
of q1 is q∞ in the inductive limit. Normalize the AffT
+B1 with respect to q1,
AffT+B2 with respect to a image of q1 in B2 and so on. Hence we obtain
another inductive limit of affine spaces with contractions maps
AffT+A1 → AffT
+A2 → · · · → AffT
AffT+B1 → AffT
+B2 · · · → AffT
As already mentioned above, we pull back the invariant for the unital hered-
itary sub-C*-algebras (i.e. full matrix algebras or the stable invariant). This
will give rise to an exact commuting diagram at the K0-level, an approximate
commuting diagram at the affine function spaces level and an exact pairing.
The compatibility can be made exact as shown in [11] by noticing that , be-
cause of simplicity, non-zero positive elements in both K0 and AffT
+ are sent
into strictly positive elements and then normalize the affine function spaces
in a suitable way.
To summarize, we now have a commutative diagram
C[0, 1]
−→ C[0, 1]
−→ . . . −→ (AffT+A,Aff ′A)
↓ τ1 ր τ
1 ↓ τ2 ր τ
2 ր l
C[0, 1]
−→ C[0, 1]
−→ . . . (AffT+B,Aff ′B)
where AffT+Ai and AffT
+Bi are identified with C([0, 1]) and each finite stage
algebra Ai and Bi is assumed to have only one direct summand.
For us it is very important to study the pulling back of the non-stable part
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 29
of the invariant.
Step 2. The intertwining of the non-stable part of the invariant
As I. Stevens mentioned in [31], at this moment we know that the non-
stable part of the invariant is only approximately mapped at a later stage
into the non-stable part of the invariant.
To be able to apply the Existence Theorem 5.1, one needs to check that
hypothesis 2 can be ensured. Otherwise, a counterexample can be given to the
Existence Theorem, as shown in Section 5.1 above. The special assumption
from the hypothesis of the isomorphism theorem, φT (Aff
′A) ⊆ Aff ′B, as well
as Corollary 8.2 will be used to prove the above mentioned claim.
By applying Corollary 8.2 to the given inductive limits A = lim
(An, φn,m),
B = lim
(Bn, φn,m) we get two sequences (δi)i≥1, δi > 0 and (δ
i)i≥1, δ
i > 0
respectively, and two subsequences of algebras such that after relabeling, we
can assume that φii+1(P̂Ai) + δi < P̂Ai+1 , ψii+1(P̂Ai) + δi < P̂Ai+1 , ψii+1(P̂Ai) +
δi < P̂Ai+1 and ψii+1(P̂Ai) + δi < P̂Ai+1 for all i ≥ 1.
Reworking the intertwining of the stable invariant for the new sequences
of algebras and the new maps that have gaps δi we obtain the following
intertwining
C([0, 1])
−→ C([0, 1])
−→ . . . −→ (AffT+A,Aff ′A)
↓ τ1 ր τ
1 ↓ τ2 ր τ
2 ր l
C[0, 1]
−→ C[0, 1]
−→ . . . (AffT+B,Aff ′B)
As a consequence of the Thomsen-Li theorem, which in the present case
states that the closed convex hull of the set of all unital *-homomorphisms of
C([0, 1]) in the strong operator topology is exactly the set of positive of unital
operators on C([0, 1]), we can assume that all the maps φii+1, ψii+1, τi, τ
i are
30 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
given by eigenvalue patterns. Because each such map takes the unit, say p̂,
into the unit, K̂(p), it follows that each map is an average of the eigenvalues,
i.e., φi,i+1(f) =
, etc.
Let P̂A1 be the image in the affine function space of the unit in the bidual
of A1. Take a continuous function f smaller than P̂A1. It is important to
say that there are no extra conditions on f , i.e., f can be any element of the
special set AffT′A1. Then there exists δ1 > 0 such that
φ12(P̂A1) + δ1 < P̂A2.
Since φ12(f) ≤ φ12(P̂A1) we have
φ12(f + δ1) ≤ φ12(P̂A1 + δ1) < P̂A2.
Since φT (Aff
′A) ⊆ Aff ′B, it follows that there exists a large N and ǫN ≤ δ1
such that
τN ◦ φN−2N−1 ◦ . . . φ12(f + δ1) < P̂BN + ǫN .
It is important to say that a different choice for f will give rise to pos-
sibly different N . This is not a difficulty because we can always pass to
subsequence. Equivalently we have
τN ◦ φN−2N−1 ◦ . . . φ12(f) + δ1 < P̂BN + ǫN .
Using δ1 ≥ ǫN we conclude
τN ◦ φN−2N−1 ◦ . . . φ12(f) < P̂BN ,
which is the desired strict inequality from the hypothesis 2 of the Existence
Theorem 5.1.
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 31
10. The Isomorphism Theorem
To complete the proof of the Isomorphism Theorem 3.1 for the algebras
Ai = A and lim
Bi = B, we have to construct an approximate commu-
tative diagram at the algebra level in the following sense, as was defined by
Elliott in [11],“for any fixed element in any Ai (or Bi), the difference of the
images of this element along two different paths in the diagram, starting at
Ai (or Bi) and ending at the same place, converges to zero as the number
of steps for which the two paths coincide, starting at the beginning, tends to
infinity.”
At this stage because of Step 2 of the previous section, Section 9, we can ap-
ply the Existence Theorem to generate a sequence of algebra homomorphisms
ν1, ν2, . . . and ν
2, . . . such that
||τi(f)−νi∗(f)||
||f ||
||τ ′i(f)−ν
i∗(f)||
||f ||
f ∈ Fi and g ∈ Gi, where νi∗, ν
i∗, are the induced afine maps by algebra maps
νi, ν
i, and Fi and Gi are finite sets.
After relabeling the indices of the inductive limit systems we now have a
(not necessarily approximately commutative) diagram of algebra homomor-
phisms
−→ A2
−→ . . . −→ A
↓ τ1 ր τ
1 ↓ τ2 ր τ
−→ B2
−→ . . . −→ B
that induces an approximately commutative diagram at the level of the in-
variant.
This will be done with respect to given arbitrary finite sets Fi ⊂ Ai and
Gi ⊂ Bi.
To make the diagram approximately commuting we modify the diagonal
maps by composing with approximately inner automorphisms and this will
32 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
be done with respect to a given arbitrary finite sets Fi ⊂ Ai and Gi ⊂ Bi
with dense union in A and B respectively.
Here we notice that we can apply the Uniqueness Theorem to the data ob-
tained from the the Existence Theorem because our inequalities are balanced.
For every ǫ > 0 we find an increasing sequence of integers 1 = M0 < L1 <
M2 < L2 < . . . and unitaries (UMi+1) ∈ A
, (V ni )n ∈ B
such that for
f ∈ FMi and g ∈ GLi we have
||UMi+1τ
(τMi(f))U
− φMiMi+1(f)||
||f ||
||VMi+1τLi(τ
(g))V ∗Li+1 − φLiLi+1(g)||
||g||
In other words passing to suitable subsequences of algebras, it is possible
to perturb each of the homomorphisms obtained in the Existence Theorem
by an approximately inner automorphism, in such a way that the diagram
becomes an approximate intertwining, in the sense of Theorem 2.1, [11].
Therefore, by the Elliott approximate intertwining theorem (see [11], The-
orem 2.1), the algebras A and B are isomorphic.
11. The range of the invariant
In this section we prove Theorem 3.2 which answers the question what are
the possible values of the invariant from the isomorphism theorem 3.1. It is
useful to notice that the invariant consists of two parts. One part is the stable
part, i.e., K0, AffT
+, λ : T+ 7→ S(K0) which was shown by K. Thomsen in
[34] to be necessary if one wants to construct an AI-algebra, and the other
part which one may call the non-stable part, namely Aff′ or equivalently, as
shown in [31], Remark 30.1.1 and Remark 30.1.2, the trace norm map. It is
the non-stable part of the invariant that one needs to investigate in its full
generality. Next the definition of the trace norm map is introduced.
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 33
Definition 11.1. Let A be a sub-C*-algebra of a C∗-algebra B. The trace
norm map associated to A is a function f : T+(A) → (0,∞] such that
f(τ) = ||τ |A||, ∞ if τ is unbounded.
Recall that:
Definition 11.2. T+(A) is the cone of positive trace functionals on A with the
inherited w*-topology.
Remark 11.1. The trace norm map is a lower semicontinuous affine map (being
a supremum of a sequence of continuous functions).
Remark 11.2. The dimension range can be determined using the values of the
trace norm map f , the simplex of tracial states S and dimension group G.
A formula for the dimension range D is:
D = {x ∈ G/v(x) < f(v), v ∈ S, v 6= 0}
I. Stevens has constructed a hereditary sub-C*-algebra of a simple (uni-
tal) AI-algebra which is obtained as an inductive limit of hereditary sub-C*-
algebras of interval algebras, and has as a trace norm map any given affine
continuous function; cf.[31], Proposition 30.1.7. Moreover she showed that
any lower semicontinuous map can be realized as a trace norm map in a spe-
cial case. Our result is a generalization to the case of unbounded trace norm
map when restricted to the base of the cone. It is worth mentioning that our
approach gives another proof in the case of any lower semicontinuous map as
a trace norm map. Still our approach is using the I. Stevens’s proof for the
case of continuous trace norm map.
Theorem 3.2 Suppose that G is a simple countable dimension group, V is
the cone associated to a metrizable Choquet simplex. Let λ : V → Hom+(G,R)
be a continuous affine map and taking extreme rays into extreme rays. Let
f : V → [0,∞] be an affine lower semicontinuous map, zero at zero and only
at zero. Then [G, V, λ, f ] is the Elliott invariant of some simple non-unital
34 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
inductive limit of continuous trace C*-algebras whose spectrum is the closed
interval [0, 1] or a finite disjoint union of closed intervals.
Proof. The proof is based on I. Stevens’s proof in a special case and consists
of several steps.
Step 0
We start by constructing a simple stable AI algebra A with its Elliott in-
variant: [(G,D), V, λ]. We know that this is possible (see [30]). By tensoring
with the algebra of compact operators we may assume A is a simple stable
AI algebra.
Step 1
We restrict the map f to some base S of the cone T+(A), where the cone
V is naturally identified with T+(A). Since any lower semicontinuous affine
map f : S → (0,+∞] is a pointwise limit of an increasing sequence of con-
tinuous affine positive maps, (see [2]), we can choose f = limfn, where fn are
continuous affine and strictly positive functions.
Moreover by considering the sequence of functions gn = fn+1 − fn if n > 1
and g1 = f1 we get that:
gn = f
Step 2
Next we use the results of Stevens ([31], Prop. 30.1.7), to realize each such
continuous affine map gn as the norm map of a hereditary sub-C*-algebra Bn
(which is an inductive limit of special algebra) of the AI algebra A obtained
at Step 0.
Consider the L∞ direct sum ⊕Bi as a sub-C*-algebra of A. The trace norm
map of the sub-C*-algebra ⊕Bi of A is equal to
i=1 gn = f .
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 35
To see that ⊕Bi is a sub-C*-algebra of A we use that A is a stable C*-
algebra:
⊕Bi =
. . .
 ⊆ A⊗K
∼= A.
Next denote with H the hereditary sub-C*-algebra generated by ⊕Bi inside
of A.
To prove that the trace norm map of H is f is enough to show that the
norm of a trace on ⊕Bi is the same as on H.
It suffices to prove that an approximate unit of the sub-C*-algebra ⊕Bi is
still an approximate unit for the hereditary sub-C*-algebra H.
We shall prove first that the hereditary sub-C*-algebra generated by ⊕Bi
coincides with the hereditary sub-C*-algebra generated by one of its approx-
imate units. Let (uλ)λ be an approximate unit of ⊕Bi. Denote by U the
hereditary sub-C*-algebra of H generated by {(uλ)λ}. We want to prove
that U is equal with H.
Since (uλ)λ is a subset of ⊕Bi we clearly have
U ⊆ H.
For the other inclusion, one can observe that
for all b ∈ ⊕Bi : b = lim
uλbuλ.
Now each uλbuλ is an element of the hereditary sub-C*-algebra generated
by (uλ)λ and hence b ∈ U . Therefore ⊕Bi ⊂ U which implies H ⊆ U .
We conclude thatH = U and hence the trace normmap ofH is f . Therefore
36 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
H is a simple hereditary sub-C*-algebra of an AI algebra with the prescribed
invariant. �
Remark 11.3. The approximate unit (uλ)λ of ⊕Bi is still an approximate unit
for the hereditary sub-C*-algebra U . To see why this is true let us consider
the sub-C*-algebra of A defined as follows: {h ∈ A | h = lim
uλh}.
This sub-C*-algebra of A is a hereditary sub-C*-algebra. Indeed let 0 ≤
k ≤ h with h = lim
uλh. We want to prove that k = lim
Consider the hereditary sub-C*-algebra hAh of A which clearly contains h
(because h2 = lim
huλh). Therefore k ∈ hAh.
Since h = lim
uλh we obtain that uλ is an approximate unit for hAh. In
particular
k = lim
and hence {h ∈ A |h = lim
uλh} is a hereditary sub-C*-algebra of A. Since
U is the smallest hereditary containing (uλ)λ we get that
U ⊆ {h ∈ A |h = lim
and uλ is an approximate unit for U .
12. Non-AI algebras which are inductive limits of
continuous-trace C*-algebras
In this section we present a necessary and sufficient condition on the invari-
ant for the algebra to be AI. We shall use this in the next section to construct
an inductive limit of continuous trace C*-algebras with spectrum [0, 1] which
is not an AI algebra.
With [G, V, λ, f ] as before we observe that for an AI algebra with Elliott
invariant canonically isomorphic to the given invariant the following equality
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 37
always holds:
f(v) = sup{v(g) : g ∈ D},
where D is the dimension range. This is seen by simply using the fact that
any AI algebra has an approximate unit consisting of projections.
Therefore a sufficient condition imposed on the invariant in order to get an
inductive limit of continuous trace C*-algebra with spectrum [0, 1] but not
an AI algebra is
f(v) 6= sup{v(g) : g ∈ D}.
This condition is also necessary. Namely assume that we have f(v) =
sup{v(g) : g ∈ D} and we have constructed a simple C*-algebra A which is
an inductive limit of continuous trace C*-algebras with spectrum [0, 1] and
with the invariant canonically isomorphic with the tuple [G, V, λ, f ]. Consider
D = {x ∈ G : v(x) < f(v), v ∈ S, v 6= 0}, where S is a base of the cone V .
For the tuple [G,D, V, S, λ] we can build (via the range of the invariant for
simple AI algebras, [30]) a simple AI-algebra B with the invariant naturally
isomorphic with the given tuple.
Note that the trace norm map which is defined starting from the tuple
[K0(B), D(B), T
+B, λB] is exactly f because of the equality
f(v) = sup{v(g) : g ∈ D}
and B is an AI algebra.
It is clear that B is an inductive limit of continuous trace C*-algebras with
spectrum [0, 1] and hence by the isomorphism theorem 2.1 we conclude that
A isomorphic to B. Hence A is a simple AI algebra as desired and we have
proved the following theorem:
38 GEORGE A. ELLIOTT AND CRISTIAN IVANESCU
Theorem 12.1. Let A be a simple C*-algebra which is an inductive limit of
continuous-trace C*-algebras whose spectrum is homeomorphic to [0, 1]. A
necessary and sufficient condition for A to be a simple AI algebra is
f(v) = sup{v(g) : g ∈ D}.
13. The class of simple inductive limits of continuous trace
C*-algebras with spectrum [0, 1] is much larger than the
class of simple AI algebras
To see this consider the simple AI algebra necessarily not of real rank zero
with scaled dimension group (Q,Q+) and cone of positive trace functionals a
2-dimensional cone; see [30]. Then the set of possible stably AI algebras, or
equivalently the set of possible trace norm maps, may be represented as the
extended affine space shown in the following schematic diagram:
Figure 6.
Each off-diagonal point in the diagram is the trace norm map of one of I.
Stevens’s algebras. The boundary points of the first quadrant are removed
(dotted lines) and the points with infinite coordinates are allowed. The di-
mension range is embedded in a canonical way in the extended affine space
as the main diagonal consisting of the points with rational coordinates.
The two bold lines represent the cases of inductive limits of continuous
CERTAIN SIMPLE APPROXIMATELY SUBHOMOGENEOUS ALGEBRAS 39
trace C*-algebras with unbounded trace norm map (points on these two lines
have at least one coordinate infinity).
If the point is off the diagonal and in the first quadrant, by Theorem 12.1
we get that the corresponding C*-algebra is an inductive limit of continuous
trace C*-algebras which is not AI-algebra. It is clear that the size of the set
of points off the diagonal is much larger then the size of the set of points on
the diagonal. (For instance in terms of the Lebesgue measure.)
This picture shows that the class of simple AI algebras sits inside the class
of inductive limits of continuous trace C*-algebras in the same way that the
main diagonal sits inside the first quadrant.
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Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4
E-mail address : elliott@math.toronto.edu
Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S 2E4
E-mail address : cristian@math.toronto.edu
	1. Introduction
	2. The invariant
	3. The results
	4. Special continuous trace C*-algebras with spectrum the interval [0,1]
	5. Balanced inequalities and the Existence Theorem
	5.1. An exact inequality is necessary between the non-stable part of the invariant
	6. Uniqueness Theorem
	7. Inductive limits of special continuous trace C*-algebras
	8. Getting a non-zero gap at the level of affine function spaces
	9. Pulling back of the isomorphism between inductive limits at the level of the invariant
	10. The Isomorphism Theorem
	11.  The range of the invariant
	12. Non-AI algebras which are inductive limits of continuous-trace C*-algebras
	13.  The class of simple inductive limits of continuous trace C*-algebras with spectrum [0,1] is much larger than the class of simple AI algebras
	References
