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Notes on $C^{*}$-algebras

S. Sundar

TL;DR

These notes provide a concise, structured survey of $C^{*}$-algebras, starting with foundational definitions and the Gelfand–Naimark framework and progressing through spectrum, positivity, and the GNS construction. They develop universal constructions (including universal and Toeplitz algebras, group $C^{*}$-algebras, and crossed products) and fundamental techniques (GNS, unitisation, spectral theory, functional calculus, and the double commutant) that underpin modern operator algebra theory. The text then surveys key examples—finite-dimensional algebras, compact operators, and crossed products—before turning to crossed products and Hilbert $C^{*}$-modules, Morita equivalence, and a compact treatment of $K$-theory (including Bott periodicity). The emphasis on universal constructions and standard tools equips readers to engage with current topics in noncommutative geometry, quantum groups, and semigroup $C^{*}$-algebras, with a practical, didactic approach suitable for readers building the language of the subject.

Abstract

These lecture notes on $C^{*}$-algebras were prepared for a couple of courses given by the author at IMSc and also at IIT Gandhinagar. The topics covered are: Gelfand-Naimark theorems, universal C*-algebras, Hilbert C*-modules, crossed products, Morita equivalence, K-theory.

Notes on $C^{*}$-algebras

TL;DR

These notes provide a concise, structured survey of -algebras, starting with foundational definitions and the Gelfand–Naimark framework and progressing through spectrum, positivity, and the GNS construction. They develop universal constructions (including universal and Toeplitz algebras, group -algebras, and crossed products) and fundamental techniques (GNS, unitisation, spectral theory, functional calculus, and the double commutant) that underpin modern operator algebra theory. The text then surveys key examples—finite-dimensional algebras, compact operators, and crossed products—before turning to crossed products and Hilbert -modules, Morita equivalence, and a compact treatment of -theory (including Bott periodicity). The emphasis on universal constructions and standard tools equips readers to engage with current topics in noncommutative geometry, quantum groups, and semigroup -algebras, with a practical, didactic approach suitable for readers building the language of the subject.

Abstract

These lecture notes on -algebras were prepared for a couple of courses given by the author at IMSc and also at IIT Gandhinagar. The topics covered are: Gelfand-Naimark theorems, universal C*-algebras, Hilbert C*-modules, crossed products, Morita equivalence, K-theory.

Paper Structure

This paper contains 30 sections, 152 theorems, 458 equations.

Table of Contents

  1. Preliminaries on $C^{*}$-algebras
  2. Gelfand-Naimark theorems
  3. Spectrum and Resolvent
  4. Commutative $C^{*}$-algebras
  5. Unitisation
  6. Positivity
  7. GNS construction
  8. Double commutant theorem
  9. A few more facts
  10. Examples of $C^{*}$-algebras
  11. Finite dimensional $C^{*}$-algebras
  12. The $C^{*}$-algebra of compact operators
  13. Universal $C^{*}$-algebras
  14. The Toeplitz algebra and the unilateral shift
  15. Measure theoretic preliminaries
  16. ...and 15 more sections

Key Result

Proposition 1.1.5

For $T \in B(\mathcal{H})$, $||T^{*}T||=||T||^{2}$.

Theorems & Definitions (238)

  • Definition 1.1.1
  • Example 1.1.2
  • Example 1.1.3
  • Example 1.1.4
  • Proposition 1.1.5
  • Example 1.1.6
  • Definition 1.1.7
  • Remark 1.1.8
  • Theorem 1.1.9: Gelfand-Naimark theorem: commutative version
  • Theorem 1.1.10: Gelfand-Naimark theorem: non-commutative version
  • ...and 228 more