Twisting the Baum-Connes morphism by a non-unitary representation

TL;DR

This paper explores twisted versions of the Baum-Connes morphism by non-unitary representations, defining new Banach algebras and computing their K-theory for groups satisfying the Baum-Connes conjecture.

Contribution

It introduces a novel approach to twisting the Baum-Connes morphism using non-unitary representations and calculates the K-theory of associated Banach algebras.

Findings

K-theory of twisted Banach algebras computed for many groups

Extension of Baum-Connes conjecture to non-unitary representation context

New algebraic structures analogous to group C*-algebras

Abstract

Let G be a locally compact group and rho a non-unitary finite dimensional representation of G. We consider tensor products of rho by some unitary representations of G in order to define two Banach algebras analogous to the group C*-algebras, C*(G) and C*_r(G). We calculate the K-theory of such algebras for a large class of groups satisfying the Baum-Connes conjecture.