## Applications of Sheaf Theory to Function AlgebrasA 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 |

1 other sections not shown

### Common terms and phrases

algebra H(M algebra of continuous algebra of functions algebra with identity algebraic homomorphism algebras of holomorphic analytic functions analytic with respect Banach algebra Cartan theory Chapter co-dimension coherent analytic sheaves common zeros commutative algebras compact subset complex manifolds complex variables continuous functions continuous inverse converges absolutely defined Definition denote entire functions equivalence class evaluation homomorphism finite number free ideals function algebra fundamental system Gelfand topology Hausdorff holomorphic and non-vanishing holomorphic functions homrad homspace ideals in H(M ideals of co-dimension isomorphic it(M Lemma let f locally analytic mapping 9 Math maximal ideal maximal ideal space module morphic non-zero homomorphisms one-to-one open Riemann surfaces open set power series Proof quotient topology regular algebras relative topology restriction Riemann surfaces ring Royden semigroup ideals separate points sequence sheaf Shilov Stein manifold strongly regular subalgebra sup norm Suppose Theorem III.l theory of coherent topological algebra topological space uniform closure