\(C^*\)-algebras. Volume 3: General theory of \(C^*\)-algebras (Q2732571)

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.NEWLINENEWLINESect. 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.NEWLINENEWLINEThe 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 \textit{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.