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

Tyrone Crisp

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.

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.
Paper Structure (24 sections, 25 theorems, 63 equations)

This paper contains 24 sections, 25 theorems, 63 equations.

Table of Contents

  1. Representations of $C^*$-algebras
  2. $C^*$-algebras: basic definitions and examples
  3. Representations of $C^*$-algebras
  4. Aside: the bicommutant theorem and irreducibility
  5. Representations and ideals
  6. Irreducible representations of $C^*_r(G)$
  7. Irreducible representations of $C_0(X)$
  8. Irreducible representations of $\operatorname{K}(H)$
  9. Irreducible representations of $C_0(X,\operatorname{K}(H))$
  10. Irreducible representations of direct sums
  11. Irreducible representations of $\operatorname{K}(H)^W$
  12. Irreducible representations of $C_0(X,\operatorname{K}(H))^W$
  13. CCR $C^*$-algebras
  14. Kaplansky's Stone--Weierstrass theorem
  15. The reduced $C^*$-algebra of a real reductive group I
  16. ...and 9 more sections

Key Result

Theorem 1

For each real reductive group $G$ there is a Morita equivalence of $C^*$-algebras

Theorems & Definitions (96)

  • Theorem 1
  • Definition 1.1
  • Definition 1.6
  • Theorem 1.7
  • Example 1.8
  • Example 1.9
  • Definition 1.10
  • Theorem 1.11: Cohen--Hewitt factorisation
  • Definition 1.12
  • Theorem 1.13
  • ...and 86 more