Modular representation theory and commutative Banach algebras / David J. Benson.

Format:

Book

Author/Creator:

Benson, D. J. (David J.), 1955- author.

Series:

Memoirs of the American Mathematical Society ; no. 1488.

Memoirs of the American Mathematical Society, 0065-9266 ; no. 1488

Language:

English

Subjects (All):

Functional analysis.

Physical Description:

v, 118 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, 2024.

Summary:

In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition is then a commutative Banach algebra, and the techniques of Geldand from the 1940s are applied in order to study the space of algebra homomorphisms to C. One surprising consequence of this investigation is that the Jacobson radical and the nil radical of a (complexified) representation ring always coincide. These notes are intended for representation theorists. So background material on commutative Banach algebras is given in detail, whereas representation theoretic background is more condensed.

Contents:

Introduction

Chapter 1. Abstract representation rings

Chapter 2. Commutative Banach algebras

Chapter 3. Completions of representation rings

Chapter 4. Representation rings of finite groups

Chapter 5. Examples and problems

Bibliography

Index.

Notes:

Includes bibliographical references (pages 109-113) and index.

ISBN:

1470470292

9781470470296

OCLC:

1449623721