Integrals and Operators

TO THE SECOND EDITION Since publication of the First Edition several excellent treatments of advanced topics in analysis have appeared. However, the concentration and penetration of these treatises naturally require much in the way of technical preliminaries and new terminology and notation. There consequently remains a need for an introduction to some of these topics which would mesh with the material of the First Edition. Such an introduction could serve to exemplify the material further, while using it to shorten and simplify its presentation. It seemed particularly important as well as practical to treat briefly but cogently some of the central parts of operator algebra and higher operator theory, as these are presently represented in book form only with a degree of specialization rather beyond the immediate needs or interests of many readers. Semigroup and perturbation theory provide connections with the theory of partial differential equations. C*-algebras are important in har monic analysis and the mathematical foundations of quantum mechanics. W*-algebras (or von Neumann rings) provide an approach to the theory of multiplicity of the spectrum and some simple but key elements of the gram mar of analysis, of use in group representation theory and elsewhere. The v vi Preface to the Second Edition theory of the trace for operators on Hilbert space is both important in itself and a natural extension of earlier integration-theoretic ideas.

Inhalt
I. Introduction.- 1.1 General preliminaries.- 1.2 The idea of measure.- 1.3 Integration as a technique in analysis.- 1.4 Limitations on the concept of measure space.- 1.5 Generalized spectral theory and measure spaces.- Exercises.- II. Basic Integrals.- 2.1 Basic measure spaces.- 2.2 The basic Lebesgue-Stieltjes spaces.- Exercises.- 2.3 Integrals of step functions.- Exercises.- 2.4 Products of basic spaces.- 2.5* Coin-tossing space.- Exercises.- 2.6 Infinity in integration theory.- Exercises.- III. Measurable Functions and Their Integrals.- 3.1 The extension problem.- 3.2 Measurability relative to a basic ring.- Exercises.- 3.3 The integral.- Exercises.- 3.4 Development of the integral.- Exercises.- 3.5 Extensions and completions of measure spaces.- Exercises.- 3.6 Multiple integration.- Exercises.- 3.7 Large spaces.- Exercises.- IV. Convergence and Differentiation.- 4.1 Linear spaces of measurable functions.- Exercises.- 4.2 Set functions.- Exercises.- 4.3 Differentiation of set functions.- Exercises.- V. Locally Compact and Euclidean Spaces.- 5.1 Functions on locally compact spaces.- Exercises.- 5.2 Measures in locally compact spaces.- Exercises.- 5.3 Transformation of Lebesgue measure.- Exercises.- 5.4 Set functions and differentiation in euclidean space.- Exercises.- VI. Function Spaces.- 6.1 Linear duality 152 Exercises.- Exercises.- 6.2 Vector-valued functions.- Exercises.- VII. Invariant Integrals.- 7.1 Introduction.- 7.2 Transformation groups.- Exercises.- 7.3 Uniform spaces.- Exercises.- 7.4 The Haar integral.- 7.5 Developments from uniqueness.- Exercises.- 7.6 Function spaces under group action.- Exercises.- VIII. Algebraic Integration Theory.- 8.1 Introduction.- 8.2 Banach algebras and the characterization of function algebras.- Exercises.- 8.3 Introductory features of Hilbert spaces.- Exercises.- 8.4 Integration algebras.- Exercises.- IX. Spectral Analysis in Hilbert Space.- 9.1 Introduction.- 9.2 The structure of maximal Abelian self-adjoint algebras.- Exercises.- X. Group Representations and Unbounded Operators.- 10.1 Representations of locally compact groups.- 10.2 Representations of Abelian groups.- Exercises.- 10.3 Unbounded diagonalizable operators.- Exercises.- 10.4 Abelian harmonic analysis.- Exercises.- XI. Semigroups and Perturbation Theory.- 11.1 Introduction.- 11.2 The Hille-Yosida theorem.- 11.3 Convergence of semigroups.- 11.4 Strong convergence of self-adjoint operators.- 11.5 Rellich-Kato perturbations.- Exercises.- 11.6 Perturbations in a calibrated space.- Exercises.- XII. Operator Rings and Spectral Multiplicity.- 12.1 Introduction.- 12.2 The double-commutor theorem.- Exercises.- 12.3 The structure of abelian rings.- Exercises.- XIII. C*-Algebras and Applications.- 13.1 Introduction.- 13.2 Representations and states.- Exercises.- XIV. The Trace as a Non-Commutative Integral.- 14.1 Introduction.- 14.2 Elementary operators and the trace.- Exercises.- 14.3 Hilbert algebras.- Exercises.- Selected references.