Banach Algebra Notes
نویسنده
چکیده
These notes are for a series of lectures given in functional analysis during Winter term of 2015. The two influences of the presentation here are Banach Algebra Techniques in Operator Theory (2e) by Ronald G. Douglas and Lecture Notes on the Spectral Theorem by Dana P. Williams. The notes start with a presentation of the two facts about nets that are required for the subject matter at hand. We then move to (unital) Banach algebras (over C), the Gelfand Transform and Spectral theory. The only facts that are proven for Banach algebras and the Gelfand transform are those that seem necessary to eventually arrive at the Gelfand-Naimark theorem. This is then used as the primary tool to define the continuous functional calculus, which is used to prove a number of spectral theory results.
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