The final volume 5 of this monumental book series, covering miscellaneous topics, includes chapter 6, entitled “Selected Chapters of \(C^*\)-Algebras” and subdivided into three sections, as well as chapter 7, “\(C^*\)-Algebras Generated by Groups”, divided into two sections, with the chapters’ size ratio being 2:1. Sect. 6.1, \({\mathcal L}^p\)-spaces, studies the classes \({\mathcal L}^p(E.F)\), \(p\geq\) 1, within the set \({\mathcal K}(E,F)\) of compact linear operators \(u : E \to F\), \(E\) and \(F\) being \(H\)-spaces. The Schatten ideals \({\mathcal L}^p(E) = {\mathcal L}^p(E,E)\) of \({\mathcal L}(E)\) are considered in detail, especially \({\mathcal L}^1(E)\), i.e., trace class operators, and \({\mathcal L}^2(E)\), i.e., Hilbert-Schmidt operators. The duals of \({\mathcal L}^p\)-spaces are characterized, whence \({\mathcal L}(E)\), the bidual of \({\mathcal K}(E)\), is a \(W^*\)-algebra with the predual \({\mathcal L}^1(E)\). As a byproduct, the points of the essential spectrum of a normal \(u \in {\mathcal L}(E)\) prove to be almost eigenvalues. Subsect. 6.1.8 on the canonical projection of the tridual of \({\mathcal K}(E,F)\), in particular, identifies the dual of the Calkin algebra in the dual of \({\mathcal L}(E)\). The final subsect. 6.1.9 revises integral operators on \(L^2(\mu)\). Neither are parallels with noncommutative integration with respect to a trace of a semifinite algebra pursued, nor are the scales of \({\mathcal L}^p\)-spaces corresponding to weights of \(W^*\)-algebras mentioned.
Sect. 6.2, Selfadjoint Linear Differential Equations, continues “branch (5)” from chapter 3. Subsect. 6.2.1 is devoted to selfadjoint boundary value problems for a linear ODE on \([a,b] \subset {\mathbb R}\). The existence of the Green function determines the integral operator being an s.a. injective Hilbert-Schmidt operator on the (weighted) \(L^2[a,b]\). Subsect. 6.2.2 presents the regular Sturm-Liouville theory for ordinary differential operators of second order and 6.2.4 considers associated parabolic and hyperbolic evolution equations (in the sense of \(L^2\)-distributions) over \([a,b] \times [0,c] \subset {\mathbb R}^2\). Subsect. 6.2.3, adhering to Theorem 3.2.2.15, gives an example of a nuclear operator in \(L^2\) over a circle. The two last subsections study the elliptic second order operator \(u\) over a compact Riemannian manifold \(T\) introduced in 3.2.3. Its potential determines a positive compact operator in an appropriate \(L^2\) over \(T\), and parabolic and hyperbolic equations associated with \(u\) are also considered. Note that \(C^*\)-algebras associated to differential operators are not mentioned. Sect. 6.3, Von Neumann Algebras, develops the subject without modular theory or automorphisms and derivations. Operator topologies of \({\mathcal L}(H)\) and the strong closure of a concrete \(C^*\)-algebra are analyzed in subsect. 6.3.1, while the next section 6.3.2 deals with the universal representation and the bidual of a \(C^*\)-algebra, and 6.3.3 extends the functional calculus, in particular, to derive the Lebesgue decomposition of a spectral measure of \(x \in {\mathcal L}(H)_{sa}\). Then von Neumann algebras, introduced as \(W^*\)-subalgebras of \({\mathcal L}(H)\), are characterized by their topological, algebraic and order properties.
Other subsections consider irreducible representations and pure states of a \(C^*\)-algebra, representations of \(W^*\)-algebras, abelian von Neumann algebras, finite-dimensional \(C^*\)-algebras, and also Hilbert right modules over them. The presentation of the last theme is very instructive.
Chapter 7 opens with a resumé: many examples of \(C^*\)-algebras are constructed by projective representations of groups, which are presented in Sect. 7.1; the Clifford algebras are special classes of these representations and are treated in Sect. 7.2. The main object in Sect. 7.1 is rather an analog of the group algebra for an arbitrary group \(T\), built in \(l^p(T)\), with convolution twisted by a “Schur function” \(f : T \times T \to {\mathbb K}\) (\(f\) is a usual unimodular 2-cocycle if \({\mathbb K} = {\mathbb C}\)). Special attention is given to locally finite groups, saturated with finite subgroups, and to abelian \(T\). Subsections 7.1.3, 7.1.4 provide elaborate examples of \(C^*\)- and \(W^*\)-algebras generated by appropriate convolution operators from \({\mathcal L}(l^p(T))\), including the cases of finite \(T\). In Sect. 7.2, the general notion of a Clifford algebra with an ordered set \({\mathcal T}\) of generators is introduced and adjusted to the preceding analysis. In 7.2.2, finite-dimensional real and complex Clifford algebras and their representations are studied in detail; the final subsect. 7.2.3 deals with the case of \({\mathcal T} = {\mathbb N}\) and the corresponding complex \(C^*\)-algebra. Though sect. 7.2 is rich in content, one could add information on Shale’s analysis of Clifford algebras over a Hilbert space and on the related \(C^*\)-algebra of canonical anticommutation relations. Thus the interesting and hard enterprise undertaken by the present author, with a monograph of a remarkably original style as a result, can most certainly be continued in various directions.