The central volume 3 of this five-volume series consists of chapter 4, “\(C^*\)-algebras”, giving the essentials of the abstract \(C^*\)-theory, viewed as a “high-point of the theory of involutive Banach algebras”. The preamble stresses the importance of abelian \(C^*\)-(sub)algebras in the noncommutative setting. Sect. 4.1, The General Theory, opens with subsect. General Results, providing definitions and examples. It also treats (purely) real \(C^*\)-algebras and complexification, adjoining of a unit, semisimplicity of \(C^*\)-algebras, uniqueness of \(C^*\)-norm, due to coincidence of *-isomorphisms and isometries, etc. \(C^*\)-direct products and direct sums are introduced but neither inductive limit nor tensor products are mentioned. The remaining three subsections prove “three central results”: (i) \(C^*\)-algebras are strongly symmetrical whence the Gelfand transform is an isometry of a Gelfand algebra (say, complex abelian \(C^*\)-algebra). (ii) This is used to develop functional calculus for “self-normal” elements, i.e., normal resp. self-adjoint if the scalar field is \(\mathbb C\) resp. \(\mathbb R\). (iii) The Fuglede-Putnam theorem permits to “enlarge the toe-hold of commutativity” to commutative sets of self-normal elements.
Sect. 4.2, The Order Relations, is the largest in accord with the author’s summary, viz., “every \(C^*\)-algebra is equipped with a canonical ordering, albeit lacking a lattice structure”, playing the principal role together with the functional calculus. “Every positive element has a unique square-root, so we can define the absolute value of any element” and prove “the existence of a canonical approximate unit which is the upper section filter of an upward directed set. The ordering is used in the Segal-Schatten theorem to show that any quotient of a \(C^*\)-algebra is again a \(C^*\)-algebra...”. A special subsection deals with the ordered set of “orthogonal projections”, so the existence of \(C^*\)-algebras without nontrivial projections could be mentioned. Sect. 4.3, Supplementary Results on \(C^*\)-algebras, is mainly devoted to order complete (\(\sigma\)-complete) algebras and their lattices of projections, but the term \(AW^*\)-algebra is not used. The functional calculus for self-normal elements is extended “with Borel functions in lieu of continuous functions”. Every such element has an integral representation by means of a spectral measure. Separate subsections consider the “carriers” of an element, hereditary algebras, simple algebras and relations between the duals of a real algebra and its complexification.
The previous analysis culminates in Sect. 4.4, \(W^*\)-algebras. In particular, it demonstrates that every element of a \(W^*\)-algebra (and of its predual) has a polar decomposition and the order continuous positive linear forms are exactly the positive elements of the predual. A subsection on \(W^*\)-homomorphisms establishes also that \(C(T)\) is a \(W^*\)-algebra iff \(T\) is hyperstonian. Since the author remarks that “a great part of results in Sect. 4.4 is due to S. Sakai”, the reader could also be referred to the book by S. Sakai [“\(C^*\)-algebras and \(W^*\)-algebras” (Springer-Verlag, Berlin) (1971; Zbl 0219.46042), reprinted (1998; Zbl 1024.46001)] for further information on \(W^*\)-algebras over \(\mathbb C\). Almost all of chapter 4 belongs to the “main strand”, marked (0), and, naturally, many research directions of the last decades remain unmentioned, e.g., \(C^*\)-dynamical systems, \(K\)-theory, classification problems. Note that the monograph is tightly and accurately sewn together by the notation, terminology and numerous cross-references and the subject and symbol indices are helpful.