The book gives a self-contained treatment of topics leading to special interests of the author, especially induced representations of locally compact groups. The book should be useful to those wanting an introduction to operator algebras and more so to those interested in representations of non-Abelian locally compact groups. The style is that of lecture notes and could easily be adapted for a postgraduate course. There are several definitions missing and some minor confusions, but this does not detract from the book. The index of symbols is incomplete, several concepts are not properly explained and more explanation of motivations, viz. why he is doing what he is doing, would be helpful. The reviewer does not see the point of the numbering of the bibliography as virtually no references are made to the bibliography by number.
The first few chapters present the operator algebra theory used and also the basics for G. Mackey’s theory of groups acting as transformations of Borel spaces. Chapter I is everything one may need to know about Borel spaces and mappings. Chapter II is a selection from the standard Dixmier-type introduction to von Neumann and \(C^*\)-algebras. Chapter III deals mainly with what is necessary to define a type I von Neumann algebra starting with the Boolean algebra of projections on a separable Hilbert space going on to central projections. A von Neumann algebra is defined to be of type I if every nonzero central projection contains a non-zero Abelian projection. The chapter also gives the basic results on Hilbert bundles, Borel fields of Hilbert spaces and von Neumann algebras, and on integration and disintegration of Hilbert spaces, of representations and of von Neumann algebras. These are needed later for induced representations.
In Chapter IV, “Groups and Group Actions”, the author defines invariant and quasi-invariant measures and the modular function (a continuous homomorphism of the group into the multiplicative group of non-negative reals which relates the left and right regular representations). He introduces the basic convolution algebras, viz. the measure algebra \(M(G)\) and \(L^{1}(G)\), and also the group \(C^{*}\)-algebra \(C^{*}(G)\). Chapter V, “Induced Actions and Representations” details G. Mackey’s theory including systems of imprimitivity, representations induced from subgroups and reduction of representations to subgroups (Mackey’s subgroup theorem). Chapter VI describes the various topologies for the duals (usually called the spectra) of \(C^{*}\)-algebras. It deals also with group actions in terms of spaces of orbits and gives Mackey’s characterisation of type I group algebras by smooth action of the dual group. Chapter VII, is titled “Left Hilbert Algebras”. These are algebras, originally introduced by J. Dixmier in 1951, with an antilinear adjoint operation and which are dense in the Hilbert space. The author then describes Tomita’s further development of left Hilbert algebras introducing modular automorphism groups. The chapter includes results on the theory of weights on a von Neumann algebra, viz. additive mappings from positive elements to \([0,\infty)\), extending the concept of states. The last chapter deals with the Fourier-Stieltjes algebra \(B(G)\) and the Fourier algebra \(A(G)\), analogues of the measure algebra \(M({\widehat{G}})\) and \(L^{1}({\widehat{G}})\), respectively, for the character group \({\widehat{G}}\) of an Abelian \(G\). The Fourier-Stieltjes algebra, a Banach *-algebra, is the span of the linear combinations of continuous positive definite functionals on G, or can be defined as the algebra of matrix coefficients of continuous unitary representations. The Fourier algebra is the subalgebra defined as the norm closure of elements with compact support, or as the matrix elements of the regular representation in \(L^{2}(G)\) (though the author gives another equivalent definition). The author exhibits duality in that \(B(G)\) is the dual of \(C^{*}(G)\) and \(A(G)\) is the predual of the von Neumann algebra of the regular representation. For a concrete C\(^{*}\)-algebra \(A\) he also describes the double dual \(A^{**}\) with the Arens topology which is isometrically isomorphic to the enveloping von Neumann algebra of \(A\).