Some remarks on the $C^*$-algebras associated with subshifts

Abstract

We point out incorrect lemmas in some papers regarding the C * -algebras associated with subshifts written by the second named author. To recover the incorrect lemmas and the affected main results, we will describe an alternative construction of C * -algebras associated with subshifts. The resulting C * -algebras are generally different from the originally constructed C * -algebras associated with subshifts and they fit the mentioned papers including the incorrect results. The simplicity conditions and the K-theory formulae for the originally constructed C * -algebras are described. We also introduce a condition called ( * ) for subshifts such that under this condition the new C * -algebras and the original C * -algebras are canonically isomorphic to each other. We finally present a subshift for which the two kinds of algebras have different K-theory groups.

Dimension groups associated to $\beta$-expansions

“…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…”

C*-algebras associated with presentations of subshifts ii. ideal structure and lambda-graph subsystems

“…For a finite set E, a subshift (A, a) is a topological dynamical system defined by a closed shiftinvariant subset A of the compact set S z of all bi-infinite sequences of E with shift transformation a. In [21] (compare [25,5]), the author generalized the class of the Cuntz-Krieger algebras to a class of C*-algebras associated with subshifts. He also introduced several topological conjugacy invariants and presentations for subshifts by using K-theory and algebraic structure of the associated C*-algebras with the subshifts in [23].…”

“…The C*-algebras O £ are generalization of the C*-algebras associated with subshifts. That is, if the A.-graph system is the canonical X-graph system for a subshift A, the constructed C*-algebra coincides with the C*-algebra O K associated with the subshift A in [26] (compare [5]). Let £ = (V, E,k, i) be a left-resolving X-graph system over E. We denote by A the presented subshift A £ by £ .…”

“…. , m(l), the second one, J^°, is [5] C*-algebras associated with presentations of subshifts 373 + . generated by T' k , k < I, I € Z + , and the third one, Tc, is generated by T™, k e Z There exist two embeddings £,,;+i : k°lv k e N yield the AF-algebra TcFor a vertex v\ e V h set there exists an edge e n/!+1 e £","+!…”

Augmenting dimension group invariants for substitution dynamics

“…The original approach in [24] based on a Fock space construction may in some cases lead to a different algebra, see [12]. Each of these approaches have independent virtues and add to the accumulated value of this concept.…”