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Several Complex Variables and Banach Algebras

  • Textbook
  • © 1998

Overview

Authors:
  1. Herbert Alexander

    1. Department of Mathematics, University of Illinois at Chicago, Chicago

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    You can also search for this author in PubMed   Google Scholar

  2. John Wermer

    1. Department of Mathematics, Brown University, Providence

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    You can also search for this author in PubMed   Google Scholar

  • Includes a wealth of new material developed since the previous edition
  • Raises new questions and new methods of proofs in function theory
  • The authors are well known and have written other Springer titles

Part of the book series: Graduate Texts in Mathematics (GTM, volume 35)

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Table of contents (27 chapters)

  1. Front Matter

    Pages i-xii
    Download chapter PDF

  2. Preliminaries and Notation

    Pages 1-4

  3. Classical Approximation Theorems

    Pages 5-16

  4. Operational Calculus in One Variable

    Pages 17-22

  5. Differential Forms

    Pages 23-26

  6. The \( \bar \partial \) -Operator

    Pages 27-30

  7. The Equation \( \bar \partial u = f \)

    Pages 31-35

  8. The Oka-Weil Theorem

    Pages 36-42

  9. Operational Calculus in Several Variables

    Pages 43-49

  10. The Šilov Boundary

    Pages 50-56

  11. Maximality and Radó’s Theorem

    Pages 57-63

  12. Maximum Modulus Algebras

    Pages 64-83

  13. Hulls of Curves and Arcs

    Pages 84-91

  14. Integral Kernels

    Pages 92-101

  15. Perturbations of the StoneȔWeierstrass Theorem

    Pages 102-111

  16. The First Cohomology Group of a Maximal Ideal Space

    Pages 112-119

  17. The \( \bar \partial \) -Operator in Smoothly Bounded Domains

    Pages 120-133

  18. Manifolds Without Complex Tangents

    Pages 134-145

  19. Submanifolds of High Dimension

    Pages 146-154

  20. Boundaries of Analytic Varieties

    Pages 155-169
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Keywords

About this book

Many connections have been found between the theory of analytic functions of one or more complex variables and the study of commutative Banach algebras. While function theory has often been employed to answer algebraic questions such as the existence of idempotents in a Banach algebra, concepts arising from the study of Banach algebras including the maximal ideal space, the Silov boundary, Geason parts, etc. have led to new questions and to new methods of proofs in function theory. This book is concerned with developing some of the principal applications of function theory in several complex variables to Banach algebras. The authors do not presuppose any knowledge of several complex variables on the part of the reader and all relevant material is developed within the text. Furthermore, the book deals with problems of uniform approximation on compact subsets of the space of n complex variables. The third edition of this book contains new material on; maximum modulus algebras and subharmonicity, the hull of a smooth curve, integral kernels, perturbations of the Stone-Weierstrass Theorem, boundaries of analytic varieties, polynomial hulls of sets over the circle, areas, and the topology of hulls. The authors have also included a new chapter containing commentaries on history and recent developments and an updated and expanded reading list.

Authors and Affiliations

  • Department of Mathematics, University of Illinois at Chicago, Chicago

    Herbert Alexander

  • Department of Mathematics, Brown University, Providence

    John Wermer

Bibliographic Information

  • Book Title: Several Complex Variables and Banach Algebras

  • Authors: Herbert Alexander, John Wermer

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/b97591

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1998

  • Hardcover ISBN: 978-0-387-98253-3Published: 20 November 1997

  • Softcover ISBN: 978-1-4757-7163-3Published: 20 April 2013

  • eBook ISBN: 978-0-387-22586-9Published: 17 January 2008

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 3

  • Number of Pages: XII, 256

  • Topics: Analysis

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