Simple ${\bi C}^*$-algebras arising from ${\beta}$-expansion of real numbers

Abstract

Given a real number $\beta > 1$, we construct a simple purely infinite $C^*$-algebra ${\cal O}_{\beta}$ as a $C^*$-algebra arising from the $\beta$-subshift in the symbolic dynamics. The $C^*$-algebras $\{{\cal O}_{\beta} \}_{1<\beta \in {\Bbb R}}$ interpolate between the Cuntz algebras $\{{\cal O}_n\}_{1 < n \in {\Bbb N}}$. The K-groups for the $C^*$-algebras ${\cal O}_{\beta}$, $1 < \beta \in {\Bbb R}$, are computed so that they are completely classified up to isomorphism. We prove that the KMS-state for the gauge action on ${\cal O}_{\beta}$ is unique at the inverse temperature $\log \beta$, which is the topological entropy for the $\beta$-shift. Moreover, ${\cal O}_{\beta}$ is realized to be a universal $C^*$-algebra generated by $n-1=[\beta]$ isometries and one partial isometry with mutually orthogonal ranges and a certain relation coming from the sequence of $\beta$-expansion of $1$.

On graphC*-algebras

“…Let us also mention that results similar to Theorem 2.6 appear in [10,11], albeit in a different setting, and that (7 r can be realized as a Pimsner algebra & x , for a suitable choice of bimodule X (see [16]). …”

Dimension groups associated to $\beta$-expansions

“…We need to point out a minor inconsistency in [12] of relevance to our work. In the proof of Case 1 of [12,Theorem 1] it is stated, correctly as proved in [12,Corollary 4.6], that the dimension group in the non-sofic case is always isomorphic to a sum of infinitely many copies of Z.…”

“…Here, a dimension group ([9], [10]) is an ordered abelian group which is unperforated and has the Riesz properties, and β-expansions ( [18], [17], [1], [19] and we shall work mainly with the closure of the set of all such β-expansions which is denoted as the β-shift X β , thinking of this as a symbolic representation of orbits under T β as indicated in Figure 1. The first such construction, considered in [12], involves the fixed point algebra F ∞ β for the so-called gauge action of the C * -algebras associated by Matsumoto to any shift space ( [14]). As noted in [5,Corollary 3.3], in this case the two different ways to build such C * -algebras coincide, but the reader 0 1 2…”

“…This is seen by comparing explicit computations of the dimension groups in [21,Proposition 10.3] and [12,Theorem 6.1]. But since the dimension group of the fixed point algebras of the C * -algebras associated by Matsumoto to the β-shift X β is only determined as a group in [12], the problem of determining whether the dimension groups coincide in general has been left open. It is the purpose of this note to compute the ordered dimension group in the non-sofic case, thus proving that the two dimension groups coincide in this case as well.…”

Dimension groups for interval maps II: the transitive case

“…In the latter case, Katayama, and Watatani [17] have associated C*-algebras F ∞ β and O β with the β-transformation, and K 0 (F ∞ β ) is a dimension group. We show that K 0 (F ∞ β ) is isomorphic to DG(τ ) as a group (and is order isomorphic if the orbit of 1 is eventually periodic.…”