$K$-theory for the crossed products of infinite tensor product of $C^*$-algebras by the shift

TL;DR

This paper computes the K-theory of crossed products formed by infinite tensor products of certain nuclear $C^*$-algebras under shift actions, including the lamplighter group $C^*$-algebra, extending K"unneth theorem applications.

Contribution

It extends K-theory calculations to crossed products of infinite tensor products of nuclear $C^*$-algebras by shift actions, including the lamplighter group case.

Findings

K-theory for the crossed product of infinite tensor products by shift is computed.

K-theory of lamplighter group $C^*$-algebra is explicitly determined.

Application of Schochet's K"unneth theorem to this class of $C^*$-algebras.

Abstract

C. Schochet shows K\"unneth theorem for the $C^*$-algebras in the smallest class of nuclear $C^*$-algebras which contains the separable Type I algebras and is closed under some operations. We calculate the $K$-theory for the crossed product of the infinite tensor product of a unital $C^*$-algebra in this class by the shift. In particular, we calculate the $K$-theory of the lamplighter group $C^*$-algebra.