Crossed Products of the Cantor Set by Free Minimal Actions of ?d

Abstract

Let d be a positive integer, let X be the Cantor set, and let Z d act freely and minimally on X. We prove that the crossed product C * (Z d , X) has stable rank one, real rank zero, and cancellation of projections, and that the order on K 0 (C * (Z d , X)) is determined by traces. We obtain the same conclusion for the C*-algebras of various kinds of aperiodic tilings.

The homoclinic and heteroclinic C*-algebras of a generalized one-dimensional solenoid

“…Since α is a free action the crossed product C(Σ) ⋊ α D ∞ is * -isomorphic to the reduced groupoid C * -algebra C * r (R) where R is theétale equivalence relation given by orbit equivalence under α, with the topology obtained from the product topology of Σ × D ∞ via the bijection (x, g) → (x, α g (x)) ∈ R. This follows from [Re] and/or [Ph1]. Since Λ : Σ → Z is a local homeomorphism and Σ is totally disconnected there is an open and closed subset U ⊆ Σ such that Λ(U) ⊆ Z is open and Λ : U → Λ(U) is a homeomorphism.…”

Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property

“…The next three sections treat, in order, the order on projections, real rank zero, and stable rank one. These are the analogs of Sections 3, 4, and 5 of [27]. It is in Section 4 that the weaker conditions satisfied by the subalgebras cause the greatest additional difficulty.…”

“…Second, we assume that the order on projections over A is determined by all traces, but only the invariant traces extend to the crossed product. Third, in [27] we relied on previous work to get from the Rokhlin property to the existence of appropriate subalgebras, but in the present paper we must do the analogous construction from scratch.…”

“…Then C * (Z, A, α) also has these three properties. In fact, we will see that not all the hypotheses on A are needed for all the conclusions.The proofs are adapted from [27]. The arguments here are more difficult for several reasons.…”

“…Second, we assume that the order on projections over A is determined by all traces, but only the invariant traces extend to the crossed product. Third, in [27] we relied …”