Boundary Quotient -algebras of Products of Odometers

Abstract

Abstract. In this paper, we study the boundary quotient C*-algebras associated to products of odometers. One of our main results shows that the boundary quotient C*-algebra of the standard product of k odometers over ni-letter alphabets (1 ≤ i ≤ k) is always nuclear, and that it is a UCT Kirchberg algebra if and only if {ln ni : 1 ≤ i ≤ k} is rationally independent, if and only if the associated single-vertex kgraph C*-algebra is simple, To achieve this, one of our main steps is to construct a topological k-graph such that its associated Cuntz-Pimsner C*-algebra is isomorphic to the boundary quotient C*-algebra. Some relations between the boundary quotient C*-algebra and the C*-algebra Q N introduced by Cuntz are also investigated. As an easy consequence of our main results, it settles a boundary quotient C*-algebra constructed by Brownlowe-Ramagge-Robertson-Whittaker.

KMS states of self-similar k-graph C*-algebras

“…Theorem 3. 19. Let B be a C*-algebra, (G, Λ) be an aperiodic self-similar P -graph with Property (FV), and π : O G,Λ → B be a homomorphism such that π(s v ) = 0 for all v ∈ Λ 0 .…”

“…Those examples include (1) self-similar k-graphs with underlying k-graphs strongly aperiodic (cf. [15]), and (2) the motivating example of our series of research [19,20,21] -the product of odometers, which has connections to many other areas like number theory.…”

The ideal structures of self-similar -graph C*-algebras

“…(Product of odometers) Let k < ∞, G = Z, and be a single-vertex kgraph. Consider the product of odometers in [19] Proof. Fix a cycline triple (µ, g, ν) of (G, ( 0 \H )).…”

“…| < ∞, as silently done in [21] (also cf. [19]), we can always assume that v∈ 0 S v = 1 A . Otherwise, let P := v∈ 0 S v .…”

“…However, some important self-similar k-graphs do satisfy this property. These include the product of odometers studied in [19] and a class of selfsimilar 1-graphs constructed by Exel-Pardo (see [9,Example 3.4]), whose C*-algebras embrace the Nekrashevych algebra and all (unital) Katsura algebras and hence the class of all (unital) simple separable amenable purely infinite C*-algebras with the UCT.…”