Notes
Last updated: 27 November 2009
Lecture notes

Chapter 1: Introduction to Banach
algebras
Definition and examples of Banach algebras;
invertibility; the spectrum; the GelfandMazur 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
BanachAlaoglu 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 StoneWeierstrass
theorem. Abelian C*algebras and the continuous functional
calculus. Positivity, states and the GelfandNaimarkSegal
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).
