## Theory of Operator Algebras I, Volume 1Mathematics for infinite dimensional objects is becoming more and more important today both in theory and application. Rings of operators, renamed von Neumann algebras by J. Dixmier, were first introduced by J. von Neumann fifty years ago, 1929, in [254] with his grand aim of giving a sound founda tion to mathematical sciences of infinite nature. J. von Neumann and his collaborator F. J. Murray laid down the foundation for this new field of mathematics, operator algebras, in a series of papers, [240], [241], [242], [257] and [259], during the period of the 1930s and early in the 1940s. In the introduction to this series of investigations, they stated Their solution 1 {to the problems of understanding rings of operators) seems to be essential for the further advance of abstract operator theory in Hilbert space under several aspects. First, the formal calculus with operator-rings leads to them. Second, our attempts to generalize the theory of unitary group-representations essentially beyond their classical frame have always been blocked by the unsolved questions connected with these problems. Third, various aspects of the quantum mechanical formalism suggest strongly the elucidation of this subject. Fourth, the knowledge obtained in these investigations gives an approach to a class of abstract algebras without a finite basis, which seems to differ essentially from all types hitherto investigated. Since then there has appeared a large volume of literature, and a great deal of progress has been achieved by many mathematicians. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Fundamentals of Banach Algebras and CAlgebras 0 Introduction 1 Banach Algebras | 1 |

Spectrum and Functional Calculus | 2 |

Gelfand Representation of Abelian Banach Algebras | 3 |

Copyright | |

34 other sections not shown

### Other editions - View all

### Common terms and phrases

abelian admits approximate assertion assume Banach algebra Banach space belongs bounded C*-algebra called central projection choose closed closure commutes compact completely condition consider contains converges strongly convex Corollary corresponding countable cyclic decomposition define Definition denote dense direct element equivalent exists extended fact factor faithful field finite fixed follows given Hence Hilbert space ideal identity implies increasing inequality infinite invariant isometry isomorphism Lemma locally lower Math maximal means measure Neumann algebra nonzero norm normal normal trace o-finite o-weakly obtain operator orthogonal positive positive linear functional projection PROOF Proposition prove representation resp respectively self-adjoint semicontinuous semifinite separable sequence Show space H spectrum structure subset subspace Suppose tensor product Theorem theory topology unique unit ball unitary universal valued vector von Neumann algebra