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Representations of *-algebras, locally compact groups, and Banach *- algebraic bundles. Vol. 1: Basic representation theory of groups and algebras. (English) Zbl 0652.46050
Pure and Applied Mathematics, 125. Nosten, MA etc.: Academic Press, Inc. xviii, 746 p. \$ 99.00 (1988).
“This two-volume work serves a double purpose... The first purpose (which occupies volume 1) is to provide a leisurely introduction to the basic functional analysis underlying the theory of infinite-dimensional unitary representations of locally compact groups and its generalization to representations of Banach algebras... The second purpose (which occupies chapters VIII, IX and XII of volume 2) is to bring the reader to the frontiers of present-day knowledge in one limited but important area, namely the Mackey normal subgroup analysis and its generalization to the context of Banach *-algebraic bundles... In its classical form developed by Frobenius for finite groups and by Mackey for separable locally compact groups, this is a powerful technique for classifying the unitary representations of a group H when we are given a closed normal subgroup N of H; under appropriate conditions it classifies the irreducible representations of H in terms of those of N and of subgroups of the quotient group $$G=H/N$$. More generally, suppose we are given a Banach *- algebra A, together with some suitable Banach *-algebraic bundle structure $${\mathcal B}$$ for A over some locally compact group G. The “bundle generalization ” of the normal subgroup analysis will then enable us to classify the *-representations of A in terms of those of subgroups of G and those of the so-called unit fiber of $${\mathcal B}$$ (corresponding in the classical case to the normal subgroup N). The fundamental tools of the classical normal subgroup analysis are the construction of induced representations and the imprimitivity theorem.” (from authors’ preface).
The book can be useful both for the beginners and the professionals, especially that some of its parts can be read independently of the other ones. The prerequisites for reading this book consist of elementary group and ring theory, elements of general topology, general topological linear spaces and abstract measure theory. Besides the facts the reader will find here many useful comments, motivations, historical remarks and bibliographical data. Each chapter starts with an introduction and ends with a set of exercises and notes and remarks. Some activity is necessary on the side of the reader since (especially at the beginning) some proofs are omitted or shortly sketched and the details are postponed to exercises (or references). The exercises contain also useful examples and counter-examples and their difficulty varies from simple ones to fairly complicated. Besides the preface there are large introductions in both volumes (40 and 23 pages) aimed for orienting the reader towards the subject-matter.
The first chapter “Preliminaries” recalls the necessary definitions and establishes the notation on logics and set theory, algebra and elements of functional analysis. No proofs are given.
The second chapter “Integration Theory and Banach Bundles” deals with measure theory (instead of usual $$\sigma$$-rings of measurable sets the authors use here $$\delta$$-rings) and integral theory in a more general then usual setting: they consider measurability and integration of vector fields on a set X, i.e. functions on X such that for $$x\in X$$ the value f(x) is in a Banach space (or locally convex space) $$B_ x$$ depending upon x. They consider also particular cases when all $$B_ x$$ are the same or even coincide with $${\mathbb{C}}$$. The chapter contains a section on projection-valued measures and spectral integrals and ends with Banach bundles over locally compact spaces, integration of such bundles and Fubini theorem for these bundles.
Chapter three “Locally Compact Groups” starts with general facts on topological groups and fields devoting some space to the essential in the sequal concept of group extensions. The remaining part of the chapter deals with locally compact groups (Haar measure, convolution of measures and functions, group algebras, invariant and quasi-invariant measures on coset spaces).
Chapter IV “Algebraic representation theory” is an exposition of the representation theory for groups and algebras. The basic definitions and facts (irreducibility, complete irreducibility, multiplicity) are given for “operator sets”-objects more general then rep also a more rich topological version). The inducing construction is applied to (suitably defined) $${\mathcal B}$$-positivity of *-representations. It is shown that if $${\mathcal B}$$ is a saturated $$C^*$$-algebraic bundle over the locally compact group G and H is a closed subgroup of G then every *- representation of $${\mathcal B}_ H$$ is positive with respect to $${\mathcal B}$$. Also if $${\mathcal B},G,H$$ are as above then for any *-representation S of $${\mathcal B}_ H$$ the following conditions are equivalent: (i) S is $${\mathcal B}$$-positive; (ii) $$S| {\mathcal B}_ e$$ is $${\mathcal B}$$-positive (Theorems 11.10 and 11.11; here $${\mathcal B}_ H$$ is the reduction of $${\mathcal B}$$ to H. The authors also ask whether these results are true without the saturation hypothesis). Further sections in this chapter are § 12 Elementary Properties of Induced Representations of Banach *- Algebraic Bundles, § 13 Restriction and Tensor Products of Induced Representations of Banach *-Algebraic Bundles, § 14 The Imprimitivity Theorem for Banach *-Algebraic Bundles (this is the chief result of this chapter. Let $${\mathcal B}$$ be a Banach *-algebraic bundle over an l.c. group G and let H be a closed subgroup of G. The result says, roughly speaking, that any B-positive non-degenerate *-representation of $${\mathcal B}_ H$$ induces a non-degenerate system of imprimitivity for $${\mathcal B}$$ over G/H and conversely, any such a system is unitarily equivalent to a system induced by some non-degenerate $${\mathcal B}$$-positive *-representation of $${\mathcal B}_ H)$$, § 15 A Generalized Mackey-Stone-von-Neumann Theorem, § 16 Conjugation of Representations, § 17 Non-involutory Induced Representations.
The last chapter XII is entitled “The Generalized Mackey Analysis”. The authors wirte: “This final chapter... is the climax toward which the earlier chapters have been pointing. Indeed, our choice of the material to be included in earlier chapters has been determined to a considerable extent by the requirements of the present chapter. Our interest in Banach *-algebraic bundles, especially saturated ones, is largely due to the fact that they seem to form the most general natural setting for the Mackey analysis... As far as the authors know, the present chapter is the first place in which the Mackey normal subgroup analysis for saturated bundles has appeared in print.” The authors start with description of the topic of this chapter in the case of a finite group G. The procedure for classifying the structure space of G in terms of a normal subgroup N of G is divided into three steps. A similar procedure is performed further in the more general context of saturated Banach *-algebraic bundles over locally compact groups. In § 6 it is given a complete description of the structure of an arbitrary saturated $$C^*$$-algebraic bundle (over an l.c. group) whose unit fiber $$C^*$$-algebra is of compact type. Such bundles are $$C^*$$-direct sums of certain specific $$C^*$$-algebraic bundles (theorem 6.19, it is due to J. M. G. Fell and it is its first publication). The chapter is closed with a section on saturated bundles over compact groups and with two sections devoted to various examples illustrating the Mackey analysis.
The bibliography contains almost 1300 positions.
Reviewer: W.Żelazko

##### MSC:
 46K10 Representations of topological algebras with involution 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 22Dxx Locally compact groups and their algebras 46H15 Representations of topological algebras 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 55R25 Sphere bundles and vector bundles in algebraic topology 43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis 22-02 Research exposition (monographs, survey articles) pertaining to topological groups