Banach algebras / Wieslaw Zelazko

Bibliogr. p. 169-175. Index

From the preface to the Polish edition: "Banach algebras are Banach spaces equipped with a continuous binary operation of multiplication. Numerous spaces considered in functional analysis are also algebras, e.g., the space C(0,1) with pointwise multiplication of functions, or the space l1 with convolution multiplication of sequences. Theorems of the general theory of Banach algebras, applied to those spaces, yield several classical results of analysis, e.g., the Wiener theorem and Wiener-Lévy theorem on trigonometric series (see Exercises 9.4.b and 16.6.a), or theorems on the spectral theory of operators. The foundations of the theory of Banach algebras are due to Gelʹfand. It was his astonishingly simple proof of the Wiener theorem that first turned the attention of mathematicians to the new theory. Certain specific algebras had been studied before, e.g., algebras of endomorphisms of Banach spaces, or weakly closed subalgebras of the algebra of endomorphisms of Hilbert spaces (the so-called von Neumann algebras or W∗-algebras); also certain particular results had been obtained earlier. But the first theorem of the general theory of Banach algebras was the theorem on the three possible forms of normal fields, announced by Mazur in 1938. This result, now known as the Gelʹfand-Mazur theorem, is the starting point of Gelʹfand's entire theory of Banach algebras. We give Mazur's original proof in this book; it is its first publication.
"The reader of this book is supposed to have some knowledge of functional analysis, algebra, topology, analytic functions and measure theory. The book consists of five chapters. The first two give an outline of the general theory of Banach algebras. Chapter III deals with analytic functions in Banach algebras, Chapter IV is devoted to algebras with an involution and Chapter V to function algebras. Some parts of the book may be omitted at the first reading without impairing the comprehension of the general theory. Most of the theorems and lemmas and also some of the definitions are followed by exercises. They not only provide an illustration for the preceding material, but also contain results of which use is made in subsequent considerations. Therefore they should be done or, at least, read through.
"Chapters I-IV developed from lectures given by the author at the University of Warsaw in 1965/66. The reader will certainly notice the almost complete lack of information on group algebras and also the rather small amount of material concerning algebras of operators. These omissions are due to shortage of space and may be justified by the existence of extensive literature concerning the above topics. However, it should be stressed that both abstract harmonic analysis and the theory of operators are the main source of inspiration for the general theory of Banach algebras, and the principal domain of its applications.'' ... (MathSciNet)