Preface

Summary

Thirty years ago, when I first thought of writing a book, the book I wanted to write was the first three chapters of this Volume II. At that time I was rapidly discovering the results that now constitute Chapter 10. For me, at least, these essentially algebraic ideas explain why the subject of Banach *-algebras works so smoothly.

Back then, Jacques Dixmier's book was the only substantial exposition of the subject of C*-algebras. Thus my first manuscript, written between 1970 and 1978, contained many of the results in the present Chapters 9, 10 and 11 plus a complete exposition of C*-algebras. I have always regarded the category of C*-algebras as a purely algebraic category: any *-homomorphism is necessarily contractive. Hence the information coming from the complete norm is geometric, but this information is already completely encoded in the *-algebraic structure. Any of the several purely algebraic formulae for the complete norm provide a Rosetta stone. Not only is the norm determined by the algebraic structure, it is also very rigid. So much so, that one can tell whether a unital Banach algebra (without any involution) is a C*-algebra merely by examining the infinitesimal shape of its unit ball near 1 (cf. Theorem 9.5.9).