Rickart Charles. General Theory of Banach Algebras

Robert E. Krieger Publishing Co., 1960. — 407 p.A Banach algebra is a linear associative algebra which, as a vector space, is a Banach space with norm satisfying the multiplicative inequality ||xy|| < || x || || y ||. Many of the Banach spaces which occur in analysis are at the same time Banach algebras under a multiplication operation which is itself important for the analysis. A good example is the space of absolutely integrable functions on the infinite real line under convolution as multiplication. In spite of the fact that examples from analysis have always provided the main impetus to the study of Banach spaces, a comparable interest in Banach algebras was rather late in coming. Some of the reason lies no doubt in the absence of appropriate algebraic tools, since a large part of the earlier work in algebra was based on finiteness conditions which rule out the most interesting examples from analysis. There were, of course, a number of early papers in which some of the additional structure given in a Banach space by an operation of multiplication was exploited, a few of the authors being Nagumo [1] and Yosida [1] (metric rings), von Neumann [1] and Murray and von Neumann [1] (rings of operators), and Stone [1]. Also, Wiener [1] and Beurling [1], although not explicitly drawing attention to the algebraic methods they were using, made systematic use of the algebraic properties of convolution in establishing certain deep theorems in analysis. However, it remained for Gelfand to lay the foundation for a general theory of Banach algebras in his now classical paper [4] on normed rings which was announced in 1939 [1] and appeared in 1941. Gelfand's innovation was a systematic use of elementary ideal theory coupled with the Mazur-Gelfand theorem which states that a normed division algebra (over the complex field) must be isomorphic with the complex field. His fundamental result was that a semi-simple commutative Banach algebra with an identity element is isomorphic to an algebra of continuous functions on a compact Hausdorff space. At the same time, Gelfand [7] used his theory to give an elegant proof of the well-known Wiener lemma that the reciprocal of a non-vanishing absolutely convergent Fourier series is also an absolutely convergent Fourier series. This proof attracted a great deal of attention to Banach algebras.