The Classification Problem for Amenable C*-Algebras
G E O R G E A. ELLIOTT
Matematisk Institut
K0benhavns Universitet
DK-2100 K0benhavn 0
Denmark
and
Department of Mathematics
University of Toronto
Toronto, Ontario
Canada M5S 1A1
1. For thirty-five years, one of the most interesting and rewarding classes of operator algebras to study has been the approximately finite-dimensional C*-algebras
of Glimm and Bratteli ([39], [7]).
Recall that a separable C*-algebra A is said to be approximately finitedimensional (or AF) if it is generated by an increasing sequence
M Ç A2 Ç . • • Ç A
of finite-dimensional sub-C*-algebras: A = (UAn)~.
It should be recalled that a finite-dimensional C*-algebra is isomorphic to a
finite direct sum of matrix algebras over the complex numbers.
One of the striking properties of AF algebras is that they can be classified.
A classification in terms of the multiplicity data involved in a given increasing
sequence of finite-dimensional subalgebras was obtained in [39] and [7]. A classification in terms of an invariant — the ordered group K 0 — was obtained in [21]
and [22]. (Of course, the ordered Ko-group can be computed from the multiplicity
data. It can in fact be seen directly to contain the same asymptotic information
concerning this data that appears in the classification of [39] and [7].)
The classification works equally well for the AF algebra — the norm closure
of the union of the increasing sequence — and for the union itself.
2. For thirty years, the only known AF algebras were those actually given in terms
of an increasing sequence of finite-dimensional C*-algebras.
An example for which this was not the case was given by Blackadar in
[3]. Blackadar's construction still involved giving an increasing sequence of C*algebras, of a rather special form, but these were no longer finite dimensional.
The algebras Blackadar used were direct sums of matrix algebras over C(T), the
algebra of continuous functions on the circle.
The AF algebra constructed by Blackadar was not itself new. It was in fact
one of the most familiar such algebras — one of the infinite tensor products of
matrix algebras over the complex numbers considered by Glimm (the case of the
algebra of 2 x 2 matrices repeated infinitely often).
Exploiting the symmetry of his construction, though, Blackadar was able to
construct a finite-order automorphism of this C*-algebra that was essentially new
Proceedings of the International Congress
of Mathematicians, Zürich, Switzerland 1994
© Birkhäuser Verlag, Basel, Switzerland 1995
The Classification Problem for Amenable C*-Algebras
923
— it could not arise through any construction involving only finite-dimensional
C*-algebras. More precisely, it could not leave invariant any increasing sequence
of finite-dimensional subalgebras with dense union — or even leave invariant the
union of such a sequence. This was because the fixed point subalgebra could be
shown — by K-theoretical considerations — not to be an AF algebra. (The fixed
point subalgebra could be easily calculated, as the closure of the increasing union
of the fixed point subalgebras of the algebras in Blackadar's construction, because
the automorphism was of finite order.)
One way of putting this, perhaps, is as follows. With Blackadar's construction, the theory of AF algebras changed from being basically algebraic (with a
little spectral theory thrown in — which was almost optional) to being essentially
topological in nature. In view of the circles appearing in the construction, one
could even say that an AF algebra was revealed as a topological object itself, in
a real sense. It would no longer be sufficient to view an AF algebra as just a
topological completion of a locally finite-dimensional algebra.
3. Topological constructions of AF algebras similar to Blackadar's were soon given
by other authors ([48], [8], [37], [11], [13], and [65]). In addition to revealing more
and more structure in various AF algebras, this work had other consequences.
Perhaps most surprisingly, it led in a natural way to the question of classifying
C*-algebras constructed in terms of increasing sequences, but which did not happen
to be AF algebras. To begin with, if some algebras were not AF, why weren't
they? In certain constructions, involving a choice of embedding at each stage, the
non-vanishing of the Ki-group of the resulting algebra appeared to be the only
obstruction to its being AF. (Other obstructions were observed later, but these
were also K-theoretical in nature.)
It was just a small step from this to the idea that the complete K-theoretical
data, i.e., the (pre-) ordered group K 0 , the group Ki, and the space of traces on
the algebra (at least in the simple, stable case), should determine the algebra,
within a certain class (indeed, that this data might be all there was to see).
This idea has now been borne out in a number of investigations, beginning
with [24]. Some of these will be summarized below. So successful has it been, in
fact, that one now expects these invariants to determine isomorphism within the
class of all stable, non type I, separable, amenable, simple C*-algebras.
4. Recall that a C*-algebra is amenable (equivalently, nuclear — see [17], [42])
if, and only if, its bidual is an amenable von Neumann algebra (see [16], [15]).
Amenability itself, in either setting, is defined in terms of a fixed point property
(innerness of certain derivations). It is equivalent to amenability of the unitary
group — with the weak topology in the C*-algebra setting and the weak* topology
in the von Neumann algebra setting ([44], [55]).
Accordingly, on very general grounds, a classification of separable amenable
C*-algebras might be hoped for in analogy with the classification of amenable von
Neumann algebras with separable pre-dual due to Connes, Haagerup, Krieger, and
Takesaki.
5. Until recently, perhaps the main reason for considering the class of amenable C*algebras as the target for classification was that on account of the Choi-Effros lift-
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George A. Elliott
ing theorem (together with Voiculescu's theorem — which does not use amenability), two homomorphisms from a separable amenable C*-algebra into the Calkin
algebra are unitarily equivalent if, and only if, they give rise to the same Kasparov
KK-element. This uniqueness theorem resembles very much results that have already proved useful for classifying special classes of amenable C*-algebras.
6. Recently, Kirchberg has shown that any separable, amenable, unital, simple C*algebra that contains a sequence of copies of the Cuntz algebra ö2 approximately
commuting with each element of the algebra must be isomorphic to ö2 ([47]).
This very strong isomorphism theorem — a characterization of ö2 — although
not formulated in terms of K-theoretical invariants, comes very close to being so.
The desired K-theoretical form of the characterization would be that a
nonzero, separable, amenable, unital, simple C*-algebra with the same K-groups
as 02, namely, zero, and with no traces, is isomorphic to ö2. One would hope
to deduce the existence of a sequence of embeddings of 02 as above from the
K-theoretical hypotheses.
7. A more explicit description of the invariant under consideration for stable
simple C*-algebras is as follows.
(i) The Ko-group, with its natural pre-order structure. (The positive cone,
Kg", consists of the elements arising from projections in the algebra or, in the
nonstable case, in matrix algebras over the algebra.)
(ii) The Ki-group.
(iii) The space T+ of densely defined, lower semicontinuous, positive traces,
with its natural structure of topological convex cone. (This structure is most easily
introduced by identifying the traces with the positive tracial linear functionals on
the Pedersen ideal — the smallest dense two-sided ideal — and considering the
pointwise convex operations and the topology of pointwise convergence.)
(iv) The natural pairing of the cone of traces with the Ko-group. (Any positive
tracial functional on the Pedersen ideal can be restricted to a hereditary sub-C*algebra contained in the Pedersen ideal — on which it must be bounded — and
then extended to the algebra with unit adjoined and to matrix algebras over this
algebra. In this way, by Brown's stabilization theorem, one gets a functional on
Ko of the original algebra, which can be shown to be independent of the choice of
hereditary sub-C*-algebra.)
The nonstable case would involve additional information — but in the unital
case presumably just the Ko-class of the unit. The nonsimple case would involve
even more information — one should keep track of the ideals and also the associated
K-theory and KK-theory data. We shall not consider these cases here.
8. As well as the question of the completeness of the invariant (now established
for a fairly large class of algebras), there is also the question of its range. What
properties characterize the objects arising as above from separable, amenable,
stable, simple C*-algebras?
There are certain properties that the invariant is known to have, in the separable, amenable, stable, simple case.
(i) The Ko-group is countable (and abelian), and the pre-order structure is
simple (if g > 0 and h is any element, then — rig < h <ng for some n = 1,2,... ).
The Classification Problem for Amenable C*-Algebras
925
(ii) The Ki-group is countable (and abelian).
(iii) The cone of traces is nonzero whenever Ko ^ KQ~ and, when nonzero,
has a compact base that is a Choquet simplex.
(iv) The pairing of the cone of traces with K 0 gives rise to all positive functional on Ko, unless Kg" = 0.
It is an interesting question whether these properties characterize what arises
as the invariant in the stable, simple case.
In particular, it is not known whether KQ must be weakly unperforated. (If
g E Ko and ng E KQ~ \ 0 for some n = 2 , 3 , . . . , then must g E KQ"?)
9. If G is any simple pre-ordered abelian group (i.e., G + + G + Ç G + and for any
g E G + \ 0 and h E G there exists n = 1,2,... such that —ng < h < ng), then
each of the two subgroups G + D — G + and G + — G + is equal either to 0 or to G.
It follows that there are three cases:
(1)
(2)
G+=0;
G+ n - G + = 0 and G+ - G+ = G (i.e., G is an ordered group);
(3) G+ = G.
The intersection of any two of these cases is the case G = 0.
10. If A is any amenable simple C*-algebra, then the three cases enumerated
above for the simple pre-ordered group KoA may be combined in a natural way
with the two main cases for T + A as follows.
Case (1)
K+=0;
Case (2)
K + n - K + = 0, K + - K + = K o / 0 ;
Case (3)
T+^0.
T+ ^ 0.
K+=K0; T+=0.
These cases are now clearly disjoint. In fact, they are also exhaustive. To see
this, assume that A does not belong either to Case (1) or to Case (2), and let us
verify that it belongs to Case (3). First, let us show that T + = 0. By the "zero-one
law" of Section 9, it is enough to consider the case that KQ~ n — K j ^ 0. If r E T +
then r is zero on KQ~ H — KQ~, and as this is not zero, in particular r is zero on a
nonzero projection. Because A is simple, r = 0.
Second, let us show that KQ = K 0 . By what we have just proved, it is enough
to show that this follows from the property T + = 0. If K^ ^ Ko, then Ko "/= 0,
and by Section 9 either KQ" = 0 or KQ is an ordered group. In the case that
K 0 is a (nonzero) ordered group, by [6] A has a nonzero quasi-trace, which by
[43] is a trace. (In dealing with this case, we may assume for convenience that
A is unital.) In the case that KQ" = 0, but Ko ^ 0, A is stably projectionless.
(Otherwise, on passing to the stabilization of A, i.e., to A 0 / C , where JC is the C*algebra of compact operators on a separable infinite-dimensional Hilbert space, as
A is simple, by Brown's stabilization theorem, A is isomorphic to B 0 /C for some
unital C*-algebra B — any nonzero unital hereditary sub-C*-algebra of A. Hence,
KQA = (K 0 A)+ - (K 0 A)+, in contradiction with K0A ^ 0, (K 0 ^4) + = 0.) On the
other hand, from T + = 0 it follows by [5] (on using [43] again) that A&IC has an
infinite projection — in particular, a nonzero projection. This contradiction shows
that the hypotheses T + = 0 and KQ" ^ KQ are incompatible, as desired.
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George A. Elliott
This argument also establishes statements 8 (iii) and 8(iv). Statements 8(i)
and 8 (ii) follow from the fact that close projections or unitaries are homotopic,
together with the use of Brown's stabilization theorem as above.
11. In Case (1), the invariant consists of the countable abelian groups K 0 and Ki,
together with the cone T + , in duality with K 0 .
Examples of Case (1), which consists of the stably projectionless simple
amenable C*-algebras, were first constructed by Blackadar in [1].
Additional examples are constructed in [36], [63], and [27].
12. In Case (2), the invariant consists of the countable ordered abelian group K 0 ,
the countable abelian group Ki, and the simplicial cone T+, in duality with K 0 .
The pairing of T+ with K 0 is positive on KQ", and all positive functionals on K 0
arise from this pairing ([6], [43]).
This case consists of the stably finite unital simple amenable C*-algebras,
and the C*-algebras stably isomorphic to these.
Constructions in [2], [4], [36], [63], and [27] realize all possibilities for the
invariant in which Ko is weakly unperforated. (The algebra /C need not be used.)
Whether the ordered group K 0 must be weakly unperforated (i.e., whether
ng > 0 for some n = 2 , 3 , . . . always implies g > 0) is an interesting question.
13. In Case (3), the invariant consists of the countable abelian groups KQ and Ki.
Examples exhausting all pairs of such groups are now known. (The algebra
0 need not be used.) (See [57], [59], and [35].)
The first such examples are of course the Cuntz algebras, for which the invariant was computed in [18].
14. In Case (1), no isomorphism results have yet been obtained.
15. In Case (2), the isomorphism theorem of [22] for AF algebras was succeeded
(twenty years later) by the following result ([24], [25], [26]).
Let A and B be stable separable amenable simple C*-algebras in Case (2).
Assume that each of A and B is the closure of the union of an increasing sequence
of sub-C*-algebras isomorphic to finite direct sums of matrix algebras over C(T)
(cf. above). Suppose that the invariant for A is isomorphic to the invariant for
B. More explicitly, suppose that there are isomorphisms, of ordered groups, tpo :
Ko A —> KQB, of groups, tpi : K\A —• K\B, and of topological convex cones,
ip?'- T+B —> T + A such that tpo and C^T respect the pairing of T+ with Ko, i.e.,
< r, ip0 g > = < tpr T, g >,
g E K0A, r E T+B.
It follows that A and B are isomorphic. Furthermore, there exists an isomorphism
ip: A^B
giving rise to a given triple of isomorphisms (tpo, <^i, <*?T) as above.
More recently, this isomorphism theorem has been generalized to the case
that the circle T is replaced by an arbitrary compact metrizable space of finite
dimension; the space may vary, but the dimension must (so far) be assumed to be
bounded. This result was proved by Gong, Li, and the author in [31]. As well as
using the result of [26] (the case of circles), this theorem uses results in [60], [50],
[29], [33], [32], [51], [33], [20], [40], [49], [19], and [41].
The Classification Problem for Amenable C*-Algebras
927
All examples obtained in this way can in fact be obtained using spaces of
dimension three or less.
The algebras obtained by such a construction are still rather special; in particular, the ordered K 0 -group has the Riesz decomposition property and is weakly
unperforated. If the algebra /C is excluded from consideration, then the group KQ
is noncyclic. Such ordered groups were considered in [23], and using the decomposition result for such ordered groups proved in [23] (due to Effros, Handelman,
and Shen in the torsion-free case), it was shown in [30] that every simple countable
ordered abelian group with these properties arises as ordered Ko in this class of
algebras; at the same time, every countable abelian group arises as Ki.
The question of what can arise as the tracial cone in this construction, and
as the pairing of this with Ko, has been answered by Villadsen in [64] (building
on results of Thomsen in [61] and [62]). Villadsen showed that, in the case that
only circles are used, the only special properties of the invariant, in addition to
those described for Case (2) in Section 8, and in addition to the properties of KQ
described above, are that, first, Ko and Ki are torsion free, and, second, every
extreme ray of the tracial cone gives rise to an extreme ray of the cone of positive
functionals on KQ.
Combining Villadsen's methods with the result mentioned before, one sees
that the range of the invariant for the class of algebras described above (with
spaces of dimension three in place of circles) is the same as in the case considered
by Villadsen (circles) except with torsion allowed in Ko and Ki- In other words,
beyond the specifications for Case (2) in Section 8, KQ is weakly unperforated,
noncyclic, and has the Riesz property, and extreme rays of T + yield extreme rays
in the cone of positive functionals on KQ .
When more general examples are considered (for instance, based on subhomogeneous building blocks), it will be necessary to exclude the algebra /C. As
pointed out in Section 12, all possibilities for the invariant in Case (2) allowed in
Section 8, with Ko weakly unperforated, are realizable — without using /C.
16. In Case (3), no isomorphism results were known until very recently.
Three years ago, in a tour de force of mathematical physics, Bratteli, Kishimoto, R0rdam, and St0rmer showed in [14], using the anticommutation relations
of quantum field theory, that the shift on the infinite tensor product of 2 x 2 matrix algebras, M2oo (cf. Section 2), has the noncommutative Rokhlin property of
Voiculescu. Using this, they were able to conclude that the tensor product of the
Cuntz algebra ö2 with M200 is isomorphic to ö2.
Using the known connection between the Rokhlin property and stability (first
appearing in the work of Connes, and studied in the C*-algebra context by Herman
and Ocneanu in [45]), a number of authors — most notably, R0rdam — pushed
rapidly forward to classify a very large natural class of algebras in Case (3), so large
that it could very well consist of everything — i.e., all stable, separable, amenable,
nonzero, simple C*-algebras in Case (3). (See [57], [9], [53], [59], and [35].)
More precisely, the class of algebras in Case (3) for which the invariant considered — the Ko- and Ki-groups — has been shown to be complete exhausts the
invariant and has certain natural properties. First of all, the class can be described
in a rather abstract way (introduced in [59]): within the class of stable, separa-
928
George A. Elliott
ble, amenable, nonzero, simple C*-algebras in Case (3), consider those that are
purely infinite, i.e., in which every nonzero hereditary sub-C*-algebra contains a
projection that is infinite in the sense of Murray and von Neumann — equivalent
to a proper subprojection. (This property may be automatic.) Consider those C*algebras in this class such that for every other algebra in the class, any KK-element
into it is realized by a nonzero homomorphism, and a unique one up to approximate unitary equivalence (to within an arbitrarily small tolerance on each finite
subset). Within this subclass, the Ko-group and Ki-group determine an algebra up
to isomorphism. This subclass contains algebras with arbitrary Ko- and Ki-groups.
Second, the preceding class is closed under the operation of passing to the
closure of an increasing sequence A\ Ç A2 Ç • • • with Ai belonging to the class. In
fact, each Ai may be allowed to be a direct sum of members of the class, provided
that the closure of the union is assumed to be simple, which is no longer automatic.
Although no stable, separable, amenable, nonzero, simple C*-algebra in Case
(3) is known not to belong to the class, and many well-known ones are known to
belong — for instance, the Cuntz algebra On with n finite and even [57], or n = oo
[54], and the tensor products On (8) Om with both n and m finite and even [9] —
already the algebra Ö3 is not known to belong.
17. The isomorphism theorems described in Sections 15 and 16 have been applied
in a number of different ways.
The AF algebra case of the isomorphism theorem was used by Blackadar in
his constructions of stably projectionless C*-algebras in [1] and of what he called
unital projectionless C*-algebras in [2].
This case of the result was also used in the thermodynamical phase diagram
construction of [10] (the construction of a C*-algebraic dynamical system with
a prescribed bundle of simplices as the bundle of KMS states at various inverse
temperatures — and with prescribed ground and ceiling state spaces at inverse
temperatures ±00 — as well as a generalization of this to include more thermodynamical variables).
What might perhaps be called the AT case of the theorem (inductive limits of
sequences of direct sums of matrix algebras over C(T) — instead of C(pt)) has been
used in computing the automorphism group of the irrational rotation C*-algebra
AQ for each irrational number 0 between 0 and 1. Using the fact that this simple
C*-algebra is AT (established in [28]), it was shown in [34] that the automorphism
group of AQ is an extension of a topologically simple group by GL(2,Z) — the
latter group arising as the image under the action of the automorphism group on
Ki AQ = Z 2 . Although the question of the triviality of this extension was left open,
the realization of the full automorphism group of K \AQ by automorphisms of the
algebra answered a well-known question. (Earlier, only the automorphisms of Z 2
with determinant + 1 had been realized in this way.)
Before it was known that the irrational rotation C*-algebras are AT, it had
been proved by Putnam in [56] that the simple C*-algebras arising (as crossed
products) from actions of Z on the Cantor set are AT. Therefore, these algebras
come under the purview of the isomorphism theorem. Because these algebras all
have the same Ki-group (namely, Z), and their traces are separated by Ko, the
invariant reduces just to the ordered group KQ (together with the class of the unit
The Classification Problem for Amenable C*-Algebras
929
of the algebra). In [46], using a construction based on a Bratteli diagram for the
ordered group (a representation of it as an inductive limit of finite direct sums of
copies of Z — which always exists in the AT case), it was shown that every simple
unital AT algebra with the property that the traces are separated by Ko arises as
above from an action of Z on the Cantor set. Thus, the simple C*-algebras arising
from actions of Z on the Cantor set can be both characterized and classified. The
classification, viewed at the level of the Z-action, was shown in [38] to amount
to a refinement of orbit equivalence — called strong orbit equivalence. Ordinary
orbit equivalence was shown also to be determined by the K-theoretical invariant:
quite remarkably, as shown in [38], the Ko-group of the crossed product modulo
the elements zero on traces (together with the class of the unit) is a complete
invariant for orbit equivalence.
It is easy to see that the C*-algebras arising (as crossed products) from tensor
product actions of Z on M2oc (cf. above) are AT. In [12], it was shown that for a
simple C*-algebra arising in this way (the most likely outcome), there are only two
possibilities for the K-theoretical invariant. KQ and Ki are always the same, and
either there is a unique tracial state, or the extreme tracial states form a circle.
Hence by the isomorphism theorem (or, rather, the unital variant of it), precisely
two simple C*-algebras arise in this way.
Recently, using methods also used in the isomorphism theorem of Section 16,
Lin showed in [52] that two almost commuting self-adjoint matrices are close to
commuting ones, thus solving a well-known problem. This result turns out to have
important implications for the classification question.
There have been fewer applications so fax of the isomorphism theorem in
Case (3), described in Section 16, but as R0rdam pointed out in [57], it was not
even known before that 0% 0 M3 is isomorphic to 0%.
An interesting subclass of Case (3) consists of the so-called Cuntz-Krieger
algebras. Because these include all ön with n finite (and not just n even), as
mentioned above this class is not known to be contained in the class for which
the isomorphism theorem has been established. In spite of this, R0rdam showed
in [58] that the isomorphism theorem could be applied to two particular CuntzKrieger algebras, which were known (by earlier work of Cuntz) to be critical for
deciding the classification question. As a consequence, two simple C*-algebras in
the Cuntz-Krieger class are isomorphic if and only if they have the same K-groups
(together with the class of the unit in K 0 ). In particular, Ö5 0 M3 = Ö5.
18. In addition to the isomorphism theorem of Sections 15 and 16, one has the
following homomorphism theorem (proved by similar methods).
Let A and B be two stable, separable, amenable, simple, non type I C*algebras belonging to either of the classes considered in Sections 15 and 16. Then
any homomorphism between the invariants (in the appropriate sense) arises from
a homomorphism between the algebras. (Presumably, this result also holds more
generally.)
Hence by [28], one recovers the Pimsner-Voiculescu embedding of the irrational rotation C*-algebra in an AF algebra with the same ordered K 0 -group.
(This approach shows that the embedding is unique — up to approximate unitary
equivalence.)
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George A. Elliott
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x
It should be noted that the result of [43] is stated only in the unital case. E. Kirchberg
has informed the author that for simple C*-algebras he has been able to extend the result to the
nonunital case.
932
[49'
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[51
[52
[53
[54
George A. Elliott
L. Li, Ph.D. thesis, University of Toronto, 1994.
H. Lin, Approximation by normal elements with finite spectra in C* -algebras of real
rank zero, preprint.
H. Lin, Homomorphisms from C(X) into (T-algebras, preprint.
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H. Lin and N. C. Phillips, Classification of direct limits of even Cuntz-circle algebras,
preprint.
H. Lin and N. C. Phillips, Approximate unitary equivalence of homomorphisms from
C?oc, preprint.
[55;
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[581
[59^
[60
[61
[62
[63
[64'
[65
A. L. T. Paterson, Nuclear (T-algebras have amenable unitary groups, Proc. Amer.
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Math. 440 (1993), 175-200.
M. R0rdam, Classification of Cuntz-Krieger algebras, K-Theory, to appear.
M. R0rdam, Classification of certain infinite simple (T-algebras, J. Funct. Anal.,
to appear.
H. Su, Ph.D. thesis, University of Toronto, 1992.
K. Thomsen, Inductive limits of interval algebras: The tracial state space, Amer. J.
Math. 116 (1994), 605-620.
K. Thomsen, On the range of the Elliott invariant, J. Funct. Anal., to appear.
K. Thomsen, On the ordered Ko-group of a simple (T-algebra, preprint.
J. Villadsen, The range of the Elliott invariant, J. Reine Angew. Math., to appear.
S. G. Walters, Inductive limit automorphisms of the irrational rotation (T-algebra,
Comm. Math. Phys., to appear.
Note added in proof on December 1, 1994:
Building on his results announced in [47] — basically, t h a t for any separable,
amenable, purely infinite, simple C*-algebra A, the tensor product with öoo is
isomorphic to A, and t h e tensor product with 02 0 JC is isomorphic to ö2 ® /C
— Kirchberg has now almost solved the classification problem in Case (3). More
precisely, the problem is solved for the class of separable amenable simple algebras
in Case (3) t h a t are purely infinite (cf. Section 16) and t h a t satisfy the so-called
universal coefficient theorem in KK. (In other words, these algebras are classified
by their Ko- and Ki-groups — together with the Ko-class of the unit in the unital
case.) T h e remaining problem, therefore, is to show t h a t these two properties
always hold.
T h e same result has also been obtained by Phillips — using the results of
Kirchberg announced in [47] (stated above).
Note added in proof on July 14, 1995:
Recently, Villadsen has constructed a simple amenable C*-algebra in Case (2) with
perforated positive cone K j . (Cf. Sections 8, 12 above.)