Nuclear dimension of simple stably projectionless C∗-algebras

Abstract

We prove that Z-stable, simple, separable, nuclear, non-unital C * -algebras have nuclear dimension at most 1. This completes the equivalence between finite nuclear dimension and Zstability for simple, separable, nuclear, non-elementary C * -algebras.1 i.e. (a) A is a simple, separable, nuclear and Z-stable C * -algebra if and only if A ⊗ K is likewise, and (b) dim nuc A ≤ 1 if and only if dim nuc (A ⊗ K) ≤ 1; see Proposition 2.3 for details and references.

The Classification of Rokhlin Flows on $$\mathrm {C}^{*}$$-algebras

“…It is an observation of Robert [74] that every projectionless Jiang-Su stable C * -algebras has this property, and hence it will automatically hold for the C * -algebras of interest in this section. In this context recall from [87,83,10,9] that any separable non-elementary nuclear simple C * -algebra is Jiang-Su stable if and only if it has finite nuclear dimension.…”

“…Recall that the class of C * -algebras covered by this question has recently been classified by Gong-Lin [30]. 9 More specifically, let us ask: Question 5.17. Does every trace-scaling flow on W ⊗ K have the Rokhlin property?…”

“…By Takai duality, this would imply that 2) α (2) R) α (2) R, and would hence recover Haagerup's theorem. 9 Note that any flow acts trivially on the K-theory of A, so the existence of a trace-scaling flow on A implies that the pairing between K0(A) and T (A) must be trivial.…”

Equivariant property (SI) revisited

“…In stark comparison to the factor case, simple nuclear C * -algebras are not automatically well-behaved, which may stem from too high-dimensional topological information being encoded in their structure [86,64,83,80]. The Toms-Winter conjecture [22,87,91,88], which may by now almost be called a theorem [65,55,57,81,71,5,10,9], is postulating that various (a priori) different concepts of being well-behaved all coincide for (non-elementary) separable simple nuclear C * -algebras. A particularly prominent condition on a C * -algebra A to be well-behaved is to ask that it shall be Z-stable, i.e., isomorphic to the tensor product A ⊗ Z.…”

“…Secondly (and more importantly), the current state of the literature only allows one to apply property (SI) to unital C * -algebras, or perhaps algebraically simple ones with some amount of direct modification; see for example [59,Section 5]. In the context of non-equivariant property (SI) and its applications to the Toms-Winter conjecture, this is not so problematic because the verification of ordinary C * -algebraic Z-stability or finite nuclear dimension may be performed at the level of some carefully chosen hereditary subalgebra, as is thorougly explained in [9]. In stark contrast, if one is investigating the structure of trace-scaling Γ-actions on stable C * -algebras, then one can quickly observe that there may be no Γ-invariant hereditary subalgebras with (non-trivial) bounded traces, which means that the current iterations of (equivariant) property (SI) are no longer applicable.…”