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Trinity College Dublin


Last updated: 27 November 2009

Lecture notes

  • Chapter 1: Introduction to Banach algebras
    Definition and examples of Banach algebras; invertibility; the spectrum; the Gelfand-Mazur theorem; the spectral mapping theorem for polynomials; the spectral radius formula; ideals, quotients and homomorphisms.
  • Chapter 2: A topological interlude
    Recap of topological spaces, compactness, subspaces, continuity, homeomorphisms, Hausdorff spaces. Subbases; the weak topology induced by a family of maps. The product topology; Tychonoff's theorem. The weak* topology and the Banach-Alaoglu theorem.
  • Chapter 3: Unital abelian Banach algebras
    Characters and the Gelfand topology on the character space. Maximal ideals as kernels of characters. Examples. Characters and the spectrum. The Gelfand representation.
  • Chapter 4: C*-algebras
    Definitions and examples; elementary properties. *-homomorphisms. The Stone-Weierstrass theorem. Abelian C*-algebras and the continuous functional calculus. Positivity, states and the Gelfand-Naimark-Segal theorem.
  • All the notes in a single file with clickable theorem numbers
  • Errata
    Errors in the printed notes and exercises that have been handed out (hopefully corrected in the PDF files above).