On the K-theory of crossed products by automorphic semigroup actions

TL;DR

This paper derives a K-theory formula for crossed products by automorphic semigroup actions under certain conditions, extending understanding of semigroup C*-algebras and their K-theoretic properties.

Contribution

It provides a new K-theory formula for crossed products by automorphic actions of semigroups satisfying Toeplitz and Baum-Connes conditions, with applications to various semigroups.

Findings

K-theory formula for crossed products derived

K-theory of left and right regular semigroup C*-algebras coincide in certain cases

Applications to semigroups like ax + b over Dedekind domains

Abstract

Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies a certain Toeplitz condition and that the Baum-Connes conjecture holds for G. We prove a formula describing the K- theory of the reduced crossed product A \rtimes{\alpha},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 quasi-lattice ordered semigroups. We also use the results to show that for certain semigroups P, including the ax + b-semigroup for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras of P coincide, although the structure of these algebras can be very different.