Preface

Summary

This volume provides a gentle introduction to most of the main areas of research on general Banach algebras. It also serves the more specific purpose of providing the background for Volume II which will deal more intensively with *-algebras (i.e. algebras with fixed involutions, normally denoted by *). The focus is on the algebraic, and sometimes the geometric, underpinnings of the analytic theory. The subject is rich with aesthetic appeal, and many topics are pursued just as far as I found them attractive. References are given to more thorough expositions when they are available or to original sources. I have tried to make the book readable for beginning graduate students. Towards this end, I sometimes include a bit of undergraduate level material when it may not have been absorbed by such readers. There are also generous comments and historical remarks. They are all intended to serve a pedagogic purpose. I have tried to document the original source of most ideas, but sometimes I have failed. I apologize to those thus slighted. The knowledgeable reader will also find numerous previously unpublished results and technical improvements.

Readers should note the Symbol Index at the end of the volume. I have chosen notation carefully and used it consistently throughout the work. For instance, A always represents an algebra and A with a subscript always represents a subset of that algebra. Each entry in the bibliography displays the numbers of the sections in this volume to which it is related.