\(C^*\)-algebras. Volume 1: Banach spaces (Q2724095)

Opening the five-volume book with the common title ``\(C^*\)-algebras'', vol. 1 in its single chapter 1 presents a part of general Banach spaces theory, mostly used later to study Banach algebras, especially \(C^*\)- and \(W^*\)-algebras. The value of the entire vast project in 7 chapters can already bee seen from the complete Table of Contents of the monograph and short Introduction. Along with a detailed exposition of general background and a classical core of \(C^*\)-theory the book advantageously tackles some important modern domains of their applications. Note, in particular, Hilbert right \(C^*\)-modules (see vol. 4, ch. 5), the theory of \(L^p\)-spaces of operators and applications to selfadjoint linear differential equations (vol. 5, ch. 6) and Clifford algebras (see ch. 7). The whole theory is set forth parallel for real and complex scalars. NEWLINENEWLINENEWLINECh. 1, Banach spaces, consists of 7 sections. In sec. 1.1, Normed spaces, basic properties are given with emphasis on finite-dimensional spaces, in anticipation of compact operators appearing in vol. 2; the Minkowski theorem and the Riesz theorem are the mein results. Sec. 1.2, Operators, studies continuous linear operators between (in) \(B\)-spaces. Together with general results and standard examples, the factor spaces and complemented subspaces, the convex sets and the Alaoglu-Bourbaki theorem are also considered here. The largest sec. 1.3, the Hahn-Banach theorem, contains many related results on the transpose of an operator, the bidual, reflexive spaces and completion of a normed space as well as the Krein-Milman theorem. Sec. 1.4, Applications of Baire's theorem, is devoted to the Banach-Steinhaus principle of uniform boundedness and Banach's open mapping principle. Sec. 1.5, Banach categories, and sec. 1.6, Nuclear maps, form isolated branches, whereas sec. 1.7, Ordered Banach spaces, including analysis of order continuity, prepares a tool to study \(W^*\)-algebras. NEWLINENEWLINENEWLINEThe mentioned prerequisites for the book, addressed ``mainly to mathematicians'' and physicists, are the ``classical analysis'' and ``rudimentary knowledge of set theory, linear algebra, point set topology and integration theory''. Actually the reader of vol. 1 needs some experience in operating with (ultra) filters instead of nets, performing the Stone--Čech and Alexandroff compactifications, familiarity with cardinals and with Radon measures on locally compact spaces. A clear exposition of basic ideas accompanied with numerous examples belongs to the merits of the book. The results appear in the style ``Definition--Theorem--Proof--Example--Exercise''. They are furthermore structured, aside from the chapter--section--subsection order, into a main stem and 8 subsidiary branches, distinguished by extra marks (0), (1), \dots, (8), to facilitate reader's orientation. Additional material, including exercises to each section, is unmarked. The choice of material and its structure are rooted in its origin from the expanded notes of Lecture courses in Functional Analysis given by the author at the ETH, Zürich. NEWLINENEWLINENEWLINEAs C.~Constantinescu justly remarks, ``there is no shortage of excellent books on \(C^*\)-algebras'', but we expect that his book will successfully complement them. Unusually, no list of references is appended to this book and the Introduction mentions only 8 sources on history and perspectives of Functional Analysis, among them \textit{R.~Kadison}'s view of the first 40 years of Operator Algebras [Proc. Symp. Pure Math. 38, Part 1, Kingston/Ont. 1980, 1-18 (1982; Zbl 0506.46043)]. Throughout vol. 1, besides a reference to ``any book'' on measure theory and integration, only once N. Bourbaki's Integration, ch. V and J-I Nagah's Modern General Topology are cited in proofs and specific recent examples of \(B\)-spaces constructed resp. by R.C.~James and W.~Gowers are referred to. Notwithstanding, the text is written very accurately, with attributions of classical results, and the reader will find it interesting and useful. Undoubtedly, the book should be recommended both to students of Functional Analysis and researches interested in the beautiful \(C^*\)-theory and its applications.