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The axioms of a complex Banach algebra were very happily chosen.
They are simple enough to allow wide ranging fields of application,
notably in harmonic analysis, operator theory and function
algebras. At the same time they are tight enough to allow the
development of a rich collection of results, mainly through the
interplay of the elementary parts of the theories of analytic
functions, rings, and Banach spaces. Many of the theorems are
things of great beauty, simple in statement, surprising in content,
and elegant in proof. We believe that some of them deserve to be
known by every mathematician. The aim of this book is to give an
account of the principal methods and results in the theory of
Banach algebras, both commutative and non commutative. It has been
necessary to apply certain exclusion principles in order to keep
our task within bounds. Certain classes of concrete Banach algebras
have a very rich literature, namely C*-algebras, function algebras,
and group algebras. We have regarded these highly developed
theories as falling outside our scope. We have not entirely avoided
them, but have been concerned with their place in the general
theory, and have stopped short of developing their special
properties. For reasons of space and time we have omitted certain
other topics which would quite naturally have been included, in
particular the theories of multipliers and of extensions of Banach
algebras, and the implications for Banach algebras of some of the
standard algebraic conditions on rings."
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