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Notes
Last updated: 27 November 2009
Lecture notes
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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.
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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.
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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.
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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.
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All the notes in a single file with
clickable theorem numbers
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Errata
Errors in the printed
notes and exercises that have been handed out (hopefully corrected
in the PDF files above).
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