\(C^*\)-algebras. Volume 4: Hilbert spaces (Q2732570)

The present fourth volume of the author's monumental five-volume work, turning to realizations of \(C^*\)-algebras, contains a large chapter 5 ``Hilbert spaces'', consisting of 6 sections. The first five of them (5.1, Pre-Hilbert spaces; 5.2, Orthogonal projections of Hilbert space; 5.3, Adjoint operators; 5.4, Representations; 5.5, Orthonormal bases) are more or less directly related to \({\mathcal L}(H)\) being a \(C^*\)-algebra. They present elements of Hilbert space theory, including the analysis of bounded operators in (between) \(H\)-spaces, mixed with a treatment of representations, rooting in the GNS-construction for \(C^*\)-algebras. Among the specialities of this part are the results around the mean ergodic theorem for contractions in subsect. 5.2.5, a detailed discussion of complexification of \(H\)-spaces in sect. 5.3 and examples of real \(C^*\)-algebras in subsect. 5.5.7. Much attention is given to compact operators and their discrete spectral resolution (subsect. 5.5.6), whereas the spectral theorem does not work here in its general form. Note that, in the adopted terminology, ``an operator on a Hilbert space is an isometry iff it is unitary'' (Cor. 5.3.1.23), though in the \(H\)-space context one usually speaks of the Wold decomposition of isometric operators. There are a few harmless misprints, like the reference to Example 5.4.21 instead of 5.4.31 on p.~157.NEWLINENEWLINEThe last section 5.6, Hilbert right \(C^*\)-modules, occupying almost half of the volume, is its main distinguishing feature, since Hilbert \(C^*\)-modules were not often treated in books [cf. \textit{C. Lance}, ``Hilbert \(C^*\)-modules. A toolkit for operator algebraists'' (Cambridge Univ. Press) (1995; Zbl 0822.46080)]. After the general definitions, this section studies properties of adjointable maps between Hilbert right \(E\)-modules over a \(C^*\)-algebra \(E\). Then, subsections on self-dual modules and von Neumann right \(W^*\)-modules follow, with an accent on summability aspects, as well as examples. However, subsect. 5.6.5, \({\mathcal K}_E\), does not invoke Kasparov's characterization of the multiplier algebra and subsect. 5.6.6, Matrices over \(C^*\)-algebras, does not tackle \(n\)-positive maps. The final subsection identifies type I \(W^*\)-algebras as those isomorphic to \(C^*\)-direct products of a family of homogeneous \(W^*\)-algebras. Thus, within chapter 5, the author's didactic principle of difficulty increasing with the page number is not discarded.