Spectral theory and complex analysis, Volume 4 (North-Holland Mathematics Studies) - ISBN:0444104291

Spectral theory and complex analysis, Volume 4 (North-Holland Mathematics Studies)

Date: 08 June 2011, 03:32

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The theory of several complex variables is very different from that of one variable, and many of the techniques used to study complex functions of several variables involve some fairly sophisticated tools from functional analysis and even algebraic topology. One of the main results in this area is the Oka-Weil theorem, which allows one to approximate a function holomorphic on a campact, polynomially convex set in complex n-space uniformly by polynomials. This result is an example of the 'holomorphic functional calculus' in the theory of Banach algebras.
The condition of polynomial convexity in the Oka-Weil theorem is the starting point for the author, for the holomorphic functional calculus works when only convexity is assumed for the compact set. He wants to get away from the dependence on Banach algebras by considering approximators that satisfy certain growth conditions. The strategy he uses consists of the spectral theory of (Waelbroeck) b-algebras, which is the topic of this book.

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