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

Abstract

The class of simple separable KK-contractible (KK-equivalent to {0}) C*-algebras which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras A ⊗ W 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 [40] and [26]).

A characterization of the Razak-Jacelon algebra

“…Combing Elliott, Gong, Lin and Niu's result [13] and Castillejos and Evington's result [4] (see also [5]), we see that if A is a simple separable nuclear monotracial C * -algebra, then A⊗W is isomorphic to W. This can be considered as a Kirchberg-Phillips type theorem [22] for W. In this paper, we give another proof of this. In our proof, we do not consider tracial approximations of C * -algebras with finite nuclear dimension.…”

“…In particular, W is expected to play a central role in the classification theory of simple separable nuclear stably projectionless C * -algebras as O 2 played in the classification theory of Krichberg algebras (see, for example, [43] and [17]). We refer the reader to [12], [13] and [18] for recent progress in the classification of simple separable nuclear stably projectionless C * -algebras. Note that there exist many interesting examples of simple stably projectionless C * -algebras.…”

“…We studied properties of F (W) in [33] and [34] by using Elliott-Gong-Lin-Niu's stable uniqueness theorem in [13]. In particular, we showed that F (W) has many projections and satisfies a certain comparison theory for projections.…”

Nuclear dimension of simple stably projectionless C*-algebras

“…This second condition has a geometric flavour and generalises the notation of finite covering dimension for topological spaces. Recent results ( [29,23,24,30]) are now converging on a similar classification result in the stably projectionless case; the state of the art will be discussed below.…”

Moves on $k$-graphs preserving Morita equivalence

“…For separable simple unital C * -algebras A which have finite nuclear dimension and satisfy the Universal Coefficient Theorem of [RS87], the Elliott invariant (consisting of the ordered K-theory of A, its trace simplex, and the pairing between traces and K 0 (A)) is a classifying invariant [TWW17, EGLN15,GLN15]: two such C * -algebras A, B are isomorphic if and only if their Elliott invariants are isomorphic. Work has already begun [EGLN17,GL18] on expanding these results to the non-unital setting.…”