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. |
Contents
Fundamentals of Banach Algebras and CAlgebras 0 Introduction 1 Banach Algebras | 1 |
Chapter | 2 |
Spectrum and Functional Calculus | 6 |
Copyright | |
46 other sections not shown
Other editions - View all
Common terms and phrases
A₁ A₂ abelian abelian von Neumann B₁ Banach algebra Banach space Borel set Borel space bounded C*-algebra C*-subalgebra central projection closure commutes completely positive conjugate space continuous function converges strongly convex countable cross-norm cyclic representation define denote dense dµ(v dµ(w e₁ E₂ element enveloping von Neumann equivalent exists f₁ faithful semifinite normal finite follows H₁ Hence Hilbert space homomorphism infinite invariant isometry isomorphism LC(H lower semicontinuous M₁ M₂ Math maximal abelian measure minimal projection N₁ Neumann algebra norm o-finite o-strong o-strongly orthogonal polar decomposition Polish space positive linear functional PROOF Q.E.D. Corollary Q.E.D. Definition Q.E.D. Proposition Q.E.D. Theorem resp self-adjoint semifinite normal trace separable sequence Show space H subalgebra Suppose tensor product topology unique unit ball unitary vector von Neumann algebra w₁ x₁