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 the Wiener-Levy theorem on trigonometric series, or theorems on the spectral theory of operators.
The foundations of the theory of Banach algebras are due to Gelfand. 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 weak-closed subalgebras of the algebra of endomorphisms of Hilbert spaces (the so-called von Neumann algebras or Wx 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 Gelfand-Mazur theorem, is the starting point of Gelfand's entire theory of Banach algebras. Mazur's original proof is included in this volume; 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 is intelligible to first year university students, though certain sections are based on material which grew beyond the usual program of university mathematics, e.g. Chapter III makes use of the theory of analytic functions of several variables. However, the necessary notions and facts are always given explicitly and provided with references.
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