On the K-Theory of Crossed Products by Automorphic Semigroup Actions

Abstract

Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies the Toeplitz condition of [24] and that the Baum-Connes conjecture holds for G. We prove a formula describing the Ktheory of the reduced crossed product A ⋊α,r P by any automorphic action of P . This formula is obtained as a consequence of a result on the K-theory of crossed products for special actions of G on totally disconnected spaces. We apply our result to various examples including left Ore semigroups and quasilattice ordered semigroups. We also use the results to show that for certain semigroups P , including the ax + b-semigroup R ⋊ R × for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras C * λ (P ) and C * ρ (P ) coincide, although the structure of these algebras can be very different. Preliminaries on totally disconnected spacesRecall that a locally compact Hausdorff space Ω is totally disconnected if and only if its topology has a basis of compact open subsets. The corresponding algebras C 0 (Ω) of continuous functions which vanish at infinity are precisely the commutative AF-Algebras. In what follows, if V ⊆ Ω, then 1 V : Ω → C denotes the characteristic function of V . Definition 2.1. Let Ω be a totally disconnected locally compact Hausdorff space and let V be a family of compact open subsets in Ω. Moreover, let U c (Ω) denote the set of all compact open subsets of Ω. Then we say that V is a generating family of the compact open sets of Ω if U c (Ω) coincides with the smallest family U of compact open sets in Ω which contains V and which is closed under finite intersections, finite unions, and under taking differences U W with U, W ∈ U . Lemma 2.2. Suppose that V is a family of compact open sets in the totally disconnected space Ω. Then the following are equivalent (1) The set {1 V : V ∈ V} generates C 0 (Ω) as a C*-algebra. (2) The set V generates U c (Ω) in the sense of Definition 2.1. Moreover, if V is closed under taking finite intersections, then (1) and (2) are equivalent to(3) span{1 V : V ∈ V} is a dense subalgebra of C 0 (Ω) containing span{1 U : U ∈ U c (Ω)}.

K ‐homology and K ‐theory for the lamplighter groups of finite groups

“…On the analytical side, for the case of classical lamplighter groups computations for the K ‐theory of have implicitly appeared in different contexts (see, for instance [, Example 6.10; , Theorem 15; , Theorem 4.12; , Section 6.5], or [, Example 3]). Our approach is different from the previous ones in two ways: first, we treat lamplighter groups of finite groups in full generality, and second, we specify generators for the K ‐groups and show their relevance for the Baum–Connes assembly map.…”

𝐾-theory for generalized Lamplighter groups

“…Now both D F and F eAe are finite dimensional commutative C*-algebras with an H-action, so that we are exactly in the setting of [4,Appendix]. It is straightforward to check that…”

C∗-Algebras Associated with Algebraic Actions

“…In particular, it can then be used to compute the K-theory of the left regular C*-algebra for a large class of semigroups as well as for crossed products by automorphic actions by such semigroups. Moreover this more general method also allows to compute the K-theory for crossed products for an action of a group on a totally disconnected space that admits a regular basis as in Definition 4.2, [8], [7]. For instance, one obtains Theorem 4.6.…”

“…This way of defining the relations also guided Xin Li in his description of the left regular C*-algebras for more general semigroups [19]. The (non-trivial) problem of computing the K-theory of C * λ (R ⋊ R × ) turned out to be particularly fruitful [8], [7]. It led to a powerful new method for computing the K-groups, for regular C*-algebras of more general semigroups and of crossed products by automorphic actions of such more general semigroups, as well as for crossed products of certain actions of groups on totally disconnected spaces.…”