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

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.