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Some Aspects of the Theory of Commutative Banach Algebras and Holomorphic Functions of Several Complex Variables

Some Aspects of the Theory of Commutative Banach Algebras and Holomorphic Functions of Several... SOME ASPECTS OF THE THEORY OF COMMUTATIVE BANACH ALGEBRAS AND HOLOMORPHIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES G. R. ALLAN In this article we shall give some account of the interplay between the theory of holomorphic functions of several complex variables and the general theory of commutative Banach algebras over the complex field. This interplay began in the early days of Banach algebra theory with the papers of Shilov [1] and of Waelbroeck [1, 3]; it continues today especially in the study so-called uniform algebras. The field of uniform algebras comprises the theory of uniformly closed algebras of continuous complex-valued functions defined on compact spaces (and subject to certain further restrictions to avoid triviality). This theory makes a good deal of use of the theory which we shall describe and has also been considerably influenced by holomorphic function theory in the formulation of certain of its concepts and in providing motivation for some of its questions. However, we specifically exclude topics peculiar to the field of uniform algebras from this survey. The most compre- hensive account of uniform algebras at present existing is in Gamelin [1]. (See also the survey by Hoffmann [2]). We shall attempt firstly to give a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Some Aspects of the Theory of Commutative Banach Algebras and Holomorphic Functions of Several Complex Variables

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/3.1.1
Publisher site
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Abstract

SOME ASPECTS OF THE THEORY OF COMMUTATIVE BANACH ALGEBRAS AND HOLOMORPHIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES G. R. ALLAN In this article we shall give some account of the interplay between the theory of holomorphic functions of several complex variables and the general theory of commutative Banach algebras over the complex field. This interplay began in the early days of Banach algebra theory with the papers of Shilov [1] and of Waelbroeck [1, 3]; it continues today especially in the study so-called uniform algebras. The field of uniform algebras comprises the theory of uniformly closed algebras of continuous complex-valued functions defined on compact spaces (and subject to certain further restrictions to avoid triviality). This theory makes a good deal of use of the theory which we shall describe and has also been considerably influenced by holomorphic function theory in the formulation of certain of its concepts and in providing motivation for some of its questions. However, we specifically exclude topics peculiar to the field of uniform algebras from this survey. The most compre- hensive account of uniform algebras at present existing is in Gamelin [1]. (See also the survey by Hoffmann [2]). We shall attempt firstly to give a

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Mar 1, 1971

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