General Theory of Banach Algebras

Description

Since the appearance of Gelfand's 1941 papers, there has been a rapid growth of interest in Banach algebras. The resulting development of the subject has brought the theory to a point where it is no longer just a promising tool in analysis but is an important field of research in its own right. Standing, as it were, between analysis and algebra (or perhaps more accurately, with feet in analysis and head in algebra), the theory of Banach algebras has developed roughly along two main lines representing respectively the analytic and algebraic influences. The analytic emphasis has been on the study of certain special Banach algebras, along with some generalizations of these algebras, and on extending certain portions of function theory and harmonic analysis to the more general situations offered by Banach algebras. On the other hand, the algebraic emphasis has naturally been on various aspects of structure theory. Of great importance here has been the growing interest of algebraists in algebras without finiteness restrictions. This development, which has been much stimulated by the study of Banach algebras, has supplied important new algebraic methods which are profitably applied to Banach algebras. It becomes increasingly evident that, in spite of the deep and continuing influence of analysis on the theory of Banach algebras, the essence of the subject as an independent discipline is to be found in its algebraic development.