Nuclear dimension and classification of 𝐶*-algebras associated to Smale spaces

Abstract

We show that the homoclinic C * -algebras of mixing Smale spaces are classifiable by the Elliott invariant. To obtain this result, we prove that the stable, unstable, and homoclinic C * -algebras associated to such Smale spaces have finite nuclear dimension. Our proof of finite nuclear dimension relies on Guentner, Willett, and Yu's notion of dynamic asymptotic dimension.2000 Mathematics Subject Classification. 46L35, 37D20.

Smale space $C^*$-algebras have nonzero projections

“…In addition, assuming existence of nonzero projections, they are proven to be approximately subhomogeneous. From these results in [8], Theorem 4.4 and Lemma 2.1 we deduce the next corollary. We are additionally using the fact that the C * -algebras associated with a mixing Smale space are simple (see [25,Theorem 1.3]) and stable (see [9]).…”

“…That (X, ϕ) is mixing implies that His simple is shown in [25]. Finally H is Z-stable by the main result of [8]. Proposition 5.3 shows that the range of K 0 (τ H ) is dense in R. Thus we can apply Lemma 5.4, which implies that H has real rank zero.…”

“…Given any nonwandering Smale space one can associate three C * -algebras, the stable, unstable and homoclinic, see [23,Section 2]. These algebras are each separable, nuclear, stably finite and the main result of [8] states that they have finite nuclear dimension (see [35] for more on nuclear dimension). Moreover, Smale's decomposition theorem reduces many questions about the nonwandering case to the mixing case, see [23,Section 2] for details.…”

Group actions on Smale space -algebras

“…In [12], the authors showed that for a mixing Smale space, C * (H), C * (S) and C * (U ) have finite nuclear dimension and hence are Z-stable. (There it was not noted that C * (S) and C * (U ) are Z-stable; however this follows from [66,Corollary 8.7].…”

“…Proof. Since (X, ϕ) is mixing, C * (H) is simple and so the classification results of [12] imply that C * (H) is Z-stable. Since C * (H) has unique trace, it must be fixed by the action.…”