The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension
Introduction
The classification of unital simple separable C*-algebras with finite nuclear dimension which satisfy the UCT has been completed (see, for example, [16], [24], [29], [40], and [50]). As is well known, the case that there exists a non-zero projection in the stabilization of the algebra follows. In the remaining case, that the algebra is stably projectionless (i.e., if the algebra is finite, the case ), a number of classification results are known (see [41], [43], [53]).
In this paper we consider the general (axiomatically determined) case assuming trivial K-theory. Recall that a C*-algebra is said to be KK-contractible if it is KK-equivalent to . In the presence of the UCT, it is equivalent to say that , . From the order structure of the -group, one sees that the case of stably projectionless simple C*-algebras is very different from the unital case. In particular, the proofs in this paper do not depend on the unital results—and require rather different techniques.
We obtain the following classification theorem:
Theorem Theorem 7.5 The class of KK-contractible stably projectionless simple separable C*-algebras with finite nuclear dimension is classified by the invariant . Any C*-algebra in this class is a simple inductive limit of Razak algebras.
Here, is the cone of lower semicontinuous traces finite on the Pedersen ideal of , with the topology of pointwise convergence (on ), and is the norm function (the lower semicontinuous extended positive real-valued function on defined by ).
Consider the C*-algebra , the (unique) simple inductive limit of Razak algebras with a unique trace (up to a multiple), which is furthermore bounded (see [41] and [28]; is also sometime called the Razak–Jacelon algebra). We will show that is the unique separable simple C*-algebra with a unique tracial state and with finite nuclear dimension which is KK-contractible. Hence is KK-contractible for any amenable C*-algebra (see Lemma 3.17). Thus, if has finite nuclear dimension, so that the C*-algebra has finite nuclear dimension as well (see Proposition 2.3(ii) of [57]), then is classifiable (whether it is finite – Theorem 7.5 – or infinite—in which case by [40] it must be ).
Corollary Corollary 6.7 Let be a simple separable C*-algebra with finite nuclear dimension. Then the C*-algebra is classifiable. In particular, .
Section snippets
The reduction class , the tracially approximate point-line class , and model algebras
Let be a C*-algebra. Denote by the Pedersen ideal. Denote by the topological cone of lower semicontinuous positive traces defined (i.e., finite) on , with the topology of pointwise convergence (on the elements of ). Denote by the set of all tracial states of . Denote by the weak* closure of in the space of all positive linear functionals on . Let be a topological convex set, or a topological cone. Denote by the cone of all continuous
A stable uniqueness theorem
The following lemma, concerning extensions with non-unital quotient, is a consequence of, and in fact equivalent to, the second part of Corollary 16 of [18] and Theorem 2.1 of [20], in the case of a trivial extension (which is all that we need – this restriction can easily be removed, in the nuclear setting, by working with Choi–Effros liftings). The analogous, purely unital setting – both quotient and extension unital—is dealt with in Theorem 6 of [18]. As pointed out in [20], the mixed case,
An isomorphism theorem
Recall that a non-unital C*-algebra is said to have almost stable rank one if the closure of the set of invertible elements in contains , and if this holds also for each hereditary sub-C*-algebra of in place of (see [44]).
Recall also that if is a separable simple C*-algebra, then has (Blackadar) strict comparison for positive elements, has stable rank one, and the map from to is an isomorphism of ordered semigroups (for any non-zero element )
Tracial approximation and non-unital versions of some results of Winter
Lemma 5.1 Let be a simple C*-algebra (with or without unit) belonging to the reduction class , and assume that has strict comparison. Let be a finite dimensional C*-algebra, and let be c.p.c. order-zero maps such that for each and , It follows that there are contractions such that Prop. 2.1 of [55]
(See
The C*-algebra and UHF-stability
Definition 6.1 Let be a non-unital separable C*-algebra. Suppose that . Recall that was said to be a -trace in [17] if there exists a sequence of completely positive contractive maps from into such that where is the unique tracial state on .12.1 of [17]
The following two statements (6.2, and 6.3) are taken from [17] (and the proofs are straightforward).
Proposition 6.2 Let be a separable simple C*-algebra with a -tracial state . Let 12.4 of [17]
The case of finite nuclear dimension
Let be a non-unital separable C*-algebra. Since is unital, we may view as a sub-C*-algebra of with the unit . In the following corollary we use for the embedding from to as well as from to . Since , from the six-term exact sequence in K-theory, one concludes that the homomorphism is injective.
We will use this fact and identify with for all in the following corollary.
Lemma 7.1 Let be a non-unital separable
Acknowledgments
The greater part of this research was done while the second and third named authors were at the Research Center for Operator Algebras in East China Normal University in the summer of 2016. Much of the revision from the initial work was done while the last three authors were there in the summer of 2017. All authors acknowledge the support of the Research Center which is partially supported by Shanghai Key Laboratory of PMMP, The Science and Technology Commission of Shanghai Municipality (STCSM),
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