Non-unitary representations, Baum-Connes morphism and unconditional completions

TL;DR

This paper proves that the Baum-Connes morphism twisted by certain non-unitary representations is an isomorphism for many groups, including real semi-simple Lie groups and hyperbolic groups, expanding the scope of the conjecture.

Contribution

It introduces a framework for twisting the Baum-Connes morphism with non-unitary representations and demonstrates its isomorphism for a broad class of groups.

Findings

Baum-Connes morphism twisted by non-unitary representations is an isomorphism for many groups.

Defines tensorisation with non-unitary finite dimensional representations in the context of Baum-Connes.

Shows the analogue of the twisted morphism in K-theory on twisted group algebras.

Abstract

We show that the Baum-Connes morphism twisted by a non-unitary representation, defined in [GA08], is an isomorphism for a large class of groups satisfying the Baum-Connes conjecture. Such class contains all the real semi-simple Lie groups, all hyperbolic groups and many infinite discret groups having Kazhdan's property (T). We define a tensorisation by a non-unitary finite dimensional representation on the left handside of the Baum-Connes morphism and we show that its analogue in K-theory must be defined on the K-theory of the twisted group algebras introduced in [GA07b].