Applications of Sheaf Theory to Function Algebras

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Department of Mathematics, Stanford University., 1961 - Functional analysis - 198 pages
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A representation theory is developed for an arbitrary commutative algebra with identity in terms of al algebra of continuous functions on a suitable topological space. A brief survey is presented of the theory of coherent analytic sheaves. No attempt at completeness is made; the object being to collect in one place the definitions and results. The sheaf-theoretic results are then used to investigate algebras of holomorphic functions on the special class of complex manifolds, the Stein manifolds. A simple example is presented to point out the error of a theorem stating that the functions on the maximal ideal space of a Banach algebra, which came from elements of the algebra via the Gelfand respresentation, enjoy a certain local characterization, much as do analytic functions on a complex manifold, or continuous functions on a topological space. The appendix contains a proof of the generalization of the Stone-Weierstrass theorem. (Author).

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Contents

INTRODUCTION
1
ALGEBRAS OF HOLOMORPHIC FUNCTIONS
53
LOCAL ANALYTICITY OF BANACH ALGEBRAS
74

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