The $K$-Theory of Some Reduced Inverse Semigroup $C^*$-Algebras

Abstract

Abstract. We use a recent result by Cuntz, Echterhoff and Li about the K-theory of certain reduced C * -crossed products to describe the K-theory of C * r (S) when S is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that C * r (S) is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.

Independent resolutions for totally disconnected dynamical systems. II. $C^*$-algebraic case

“…This is for instance possible for C*algebras of certain 0-F inverse semigroups and certain quotients of these. This has already been observed in [Nor2], but we present a slightly different approach which is more explicit and better suited for our purposes. More concretely, we discuss graph C*-algebras and one dimensional tiling C*-algebras, and derive crossed product descriptions for these.…”

“…Reduced C*-algebras of 0-F-inverse semigroups which admit gradings injective on maximal elements (in the sense of [Nor2]) can be described up to Morita equivalence as crossed products of totally disconnected dynamical systems which admit an invariant regular basis. This was observed in [Nor2]. Now we consider quotients of such inverse semigroup C*-algebras, for instance tight C*-algebras of these inverse semigroups.…”

“…Let S be 0-F-inverse, and let G be a group and σ : (S 1 ) × → G a morphism injective on the set of maximal elements M (S 1 ) as in [Nor2,§ 1]. Let s g be the maximal element of σ −1 (g) if the latter set is non-empty, and let s g := 0 otherwise.…”

“…Remark 4.12. The dual system ( Env (E), G,τ ) in our setting can be canonically identified with the dynamical system (Ω, G, τ ) from [Nor2,§ 2].…”

“…We show that using independent resolutions, it is easy to compute K-theory for graph C*-algebras. We use the same notation as in [Nor2, § 5]: Let E = (E 0 , E 1 , σ, ρ) be a graph, and let S E be its graph inverse semigroup. S 1 E is strongly 0-F-inverse with universal grading (S 1 E ) × → F , where F is the free group on E 1 .…”