K-theory for the $C^*$-algebras of the solvable Baumslag-Solitar groups

TL;DR

This paper computes the K-theory of the group C*-algebra for solvable Baumslag-Solitar groups using the Pimsner-Voiculescu sequence, providing a new proof of the Baum-Connes conjecture for these groups.

Contribution

It introduces a novel computation method for the K-theory of these algebras and offers a new proof of the Baum-Connes conjecture with trivial coefficients.

Findings

K-theory of the group C*-algebra for BS(1,n) computed.

New proof of the Baum-Connes conjecture for BS(1,n).

Method based on Pimsner-Voiculescu exact sequence.

Abstract

We provide a new computation of the K-theory of the group $C^*$-algebra of the solvable Baumslag-Solitar group $BS(1,n)\;(n\neq 1)$; our computation is based on the Pimsner-Voiculescu 6-terms exact sequence, by viewing $BS(1,n)$ as a semi-direct product $\mathbb{Z}[1/n]\rtimes\mathbb{Z}$. We deduce from it a new proof of the Baum-Connes conjecture with trivial coefficients for $BS(1,n)$.