Notes on C*-algebras, representations, and Morita equivalence (with a view toward C*-algebras of reductive groups)

TL;DR

This paper explains a proof of Wassermann's theorem relating the reduced C*-algebra of a real reductive group to simpler C*-algebras, using representation theory and Morita equivalence, aimed at advanced students and mathematicians.

Contribution

It provides an accessible exposition of the proof of Wassermann's theorem, connecting representation theory, Morita equivalence, and C*-algebras of reductive groups.

Findings

Identification of the reduced C*-algebra with a direct sum of simpler algebras

Introduction of Morita equivalence concepts relevant to the theorem

Clarification of the representation theory involved in the proof

Abstract

These notes give an expanded account of my lectures at the CIRM-IHP research school on 'Methods in representation theory and operator algebras', January 6-10, 2025. Their main goal is to explain a proof of a theorem of A. Wassermann, that identifies the reduced C*-algebra of a real reductive group, up to Morita equivalence, with a direct sum of much simpler C*-algebras. Along the way we introduce the basic theory of representations and Morita equivalence of C*-algebras that is needed to understand the theorem and its proof. The target audience is Masters- and PhD-level students, and other mathematicians who are not specialists in operator algebras.