Theory of Operator Algebras I

ISBN-13: 9781461261902 / Angielski / Miękka / 2011 / 418 str.

Masamichi Takesaki

Opis

Mathematics 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."

Spis treści

I Fundamentals of Banach Algebras and C*-Algebras.- 0. Introduction.- 1. Banach Algebras.- 2. Spectrum and Functional Calculus.- 3. Gelfand Representation of Abelian Banach Algebras.- 4. Spectrum and Functional Calculus in C*-Algebras.- 5. Continuity of Homomorphisms.- 6. Positive Cones of C*-Algebras.- 7. Approximate Identities in C*-Algebras.- 8. Quotient Algebras of C*-Algebras.- 9. Representations and Positive Linear Functional.- 10. Extreme Points of the Unit Ball of a C*-Algebra.- 11. Finite Dimensional C*-Algebras.- Notes.- Exercises.- II Topologies and Density Theorems in Operator Algebras.- 0. Introduction.- 1. Banach Spaces of Operators on a Hilbert Space.- 2. Locally Convex Topologies in ?(?).- 3. The Double Commutation Theorem of J. von Neumann.- 4. Density Theorems.- Notes.- III Conjugate Spaces.- 0. Introduction.- 1. Abelian Operator Algebras.- 2. The Universal Enveloping von Neumann Algebra of a C*-Algebra.- 3. W*-Algebras.- 4. The Polar Decomposition and the Absolute Value of Functionals.- 5. Topological Properties of the Conjugate Space.- 6. Semicontinuity in the Universal Enveloping von Neumann Algebra*.- Notes.- IV Tensor Products of Operator Algebras and Direct Integrals.- 0. Introduction.- 1. Tensor Product of Hilbert Spaces and Operators.- 2. Tensor Products of Banach Spaces.- 3. Completely Positive Maps.- 4. Tensor Products of C*-Algebras.- 5. Tensor Products of W*-Algebras.- Notes.- 6. Integral Representations of States.- 7. Representation of L2(?,?) ? ?, L1(?,?) ?y? *, and L(?,?) ?? ?.- 8. Direct Integral of Hubert Spaces, Representations, and von Neumann Algebras.- Notes.- V Types of von Neumann Algebras and Traces.- 0. Introduction.- 1. Projections and Types of von Neumann Algebras.- 2. Traces on von Neumann Algebras.- Notes.- 3. Multiplicity of a von Neumann Algebra on a Hilbert Space.- 4. Ergodic Type Theorem for von Neumann Algebras*.- 5. Normality of Separable Representations*.- 6. The Borel Spaces of von Neumann Algebras.- 7. Construction of Factors of Type II and Type III.- Notes.- Appendix Polish Spaces and Standard Borel Spaces.- Monographs.- Papers.- Notation Index.