The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension

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Abstract

The class of simple separable KK-contractible (KK-equivalent to {0}) C*-algebra s which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebra s AW is classifiable, where A is a simple separable C*-algebra  with finite nuclear dimension and W is the simple inductive limit of Razak algebras with unique trace, which is bounded (see Razak (2002) and Jacelon (2013)).

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 A is stably projectionless (i.e., if the algebra is finite, the case K0(A)+={0}), 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 A is said to be KK-contractible if it is KK-equivalent to {0}. In the presence of the UCT, it is equivalent to say that Ki(A)={0}, i=0,1. From the order structure of the K0-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 (T˜(A),ΣA). Any C*-algebra A in this class is a simple inductive limit of Razak algebras.

Here, T˜(A) is the cone of lower semicontinuous traces finite on the Pedersen ideal Ped(A) of A, with the topology of pointwise convergence (on Ped(A)), and ΣA is the norm function (the lower semicontinuous extended positive real-valued function on T˜(A) defined by ΣA(τ)=sup{τ(a):aPed(A)+,a1}).

Consider the C*-algebra W, the (unique) simple inductive limit of Razak algebras with a unique trace (up to a multiple), which is furthermore bounded (see [41] and [28]; W is also sometime called the Razak–Jacelon algebra). We will show that W is the unique separable simple C*-algebra with a unique tracial state and with finite nuclear dimension which is KK-contractible. Hence AW is KK-contractible for any amenable C*-algebra A (see Lemma 3.17). Thus, if A has finite nuclear dimension, so that the C*-algebra AW has finite nuclear dimension as well (see Proposition 2.3(ii) of [57]), then AW is classifiable (whether it is finite – Theorem 7.5 – or infinite—in which case by [40] it must be O2K).

Corollary Corollary 6.7

Let A be a simple separable C*-algebra with finite nuclear dimension. Then the C*-algebra AW is classifiable. In particular, WWW.

Section snippets

The reduction class R, the tracially approximate point-line class D, and model algebras

Let A be a C*-algebra. Denote by Ped(A) the Pedersen ideal. Denote by T˜(A) the topological cone of lower semicontinuous positive traces defined (i.e., finite) on Ped(A), with the topology of pointwise convergence (on the elements of Ped(A)). Denote by T(A) the set of all tracial states of A. Denote by T(A)¯w the weak* closure of T(A) in the space of all positive linear functionals on A. Let X be a topological convex set, or a topological cone. Denote by Aff+(X) 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 A is said to have almost stable rank one if the closure of the set of invertible elements in A˜ contains A, and if this holds also for each hereditary sub-C*-algebra of A in place of A (see [44]).

Recall also that if AD is a separable simple C*-algebra, then A has (Blackadar) strict comparison for positive elements, A has stable rank one, and the map from Cu(A) to LAff0+(T(aAa¯)¯w) is an isomorphism of ordered semigroups (for any non-zero element aPed(A))

Tracial approximation and non-unital versions of some results of Winter

Lemma 5.1

Prop. 2.1 of [55]

Let A be a simple C*-algebra (with or without unit) belonging to the reduction class R, and assume that A has strict comparison.

Let F be a finite dimensional C*-algebra, and let ϕ:FAandϕi:FAforalliNbe c.p.c. order-zero maps such that for each cF+ and fC0+((0,1]), limisupτT(A)|τ(f(ϕ)(c)f(ϕi)(c))|=0andlim supif(ϕi)(c)f(ϕ)(c). It follows that there are contractions siM4AforalliNsuch that limisi(14ϕ(c))(e1,1ϕi(c))si=0forallcF+andlimi(e1,1ϕi(c))sisie1,1ϕi(c)=0.

(See

The C*-algebra W and UHF-stability

Definition 6.1

12.1 of [17]

Let A be a non-unital separable C*-algebra. Suppose that τT(A). Recall that τ was said to be a W-trace in [17] if there exists a sequence of completely positive contractive maps (ϕn) from A into W such that limnϕn(ab)ϕn(a)ϕn(b)=0foralla,bA,andτ(a)=limnτW(ϕn(a))forallaA, where τW is the unique tracial state on W.

The following two statements (6.2, and 6.3) are taken from [17] (and the proofs are straightforward).

Proposition 6.2

12.4 of [17]

Let A be a separable simple C*-algebra with a W-tracial state τT(A). Let 0

The case of finite nuclear dimension

Let A be a non-unital separable C*-algebra. Since A˜Q is unital, we may view AQ˜ as a sub-C*-algebra of A˜Q with the unit 1A˜Q. In the following corollary we use ı for the embedding from AQ to A˜Q as well as from AQ˜ to A˜Q. Since K1(Q)={0}, from the six-term exact sequence in K-theory, one concludes that the homomorphism ı0:K0(AQ)K0(A˜Q) is injective.

We will use this fact and identify x with ı0(x) for all xK0(AQ) in the following corollary.

Lemma 7.1

Let A 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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