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 recently introduced by the third named author and that the Baum–Connes conjecture holds for G. We prove a formula describing the K-theory of the reduced crossed product A⋊α, rP 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 R⋊ R× for a Dedekind domain R, the K-theory of the left and right regular semigroup C*-algebras Cλ*(P) and Cρ*(P) coincide, although the structure of these algebras can be very different.