Hilbert \(C^*\)-modules. A toolkit for operator algebraists. (English) Zbl 0822.46080

Lance, E. Christopher

London Mathematical Society Lecture Note Series. 210. Cambridge: Univ. Press,. ix, 130 p. (1995).

The aim of these Lecture Notes, considered by the author as a toolkit for operator algebraists, is to provide students and non specialists mathematicians, for the first time, a very clear and unified exposition of the main techniques and results that have shown themselves to be useful in a variety of contents in modern \(C^*\)-algebra theory and some of its major applications. These techniques and results centre round the concept of Hilbert \(C^*\)-module, an object like Hilbert space provided that the inner product takes its value in a general \(C^*\)-algebra instead of being complex-valued. Hilbert \(C^*\)-modules first appeared in the work of I. Kaplansky in 1953 on modules over operator algebras. The book is divided in ten chapters and contains a very selective list of references.
The first three chapters present basic definitions and concepts and the elementary theory of Hilbert \(C^*\)-modules in the model of classical Hilbert spaces. Chapter 4 presents tensor products. Chapter 5 is a treatment on completely-positive mappings between \(C^*\)-algebras and then in the context of Hilbert \(C^*\)-algebras. In the chapter 6, a universal positive response is given, under some countability condition on modules, to the following general question: Given a \(C^*\)-algebra \(A\), can we hope to classify all Hilbert-\(A\)-modules up to unitary equivalence?
Broadly, chapters 4 to 6 are motivated by applications of Hilbert \(C^*\)-modules to the \(K\)-theory and \(KK\)-theory. Chapter 7 is a short interlude on the topic of Morita equivalence, chapter 8 on slice maps and bialgebras, chapter 9 on unbounded operators and chapter 10 on the bounded transform and unbounded multipliers, are oriented towards application to \(C^*\)-algebraic quantum group theory, but quantum groups as such do not figure in this book.

Reviewer: H.Hogbe Nlend (Bordeaux)

MSC:

46L89	Other “noncommutative” mathematics based on \(C^*\)-algebra theory
46-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
46H25	Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46L05	General theory of \(C^*\)-algebras

Keywords:

Hilbert \(C^*\)-module; tensor products; completely-positive mappings between \(C^*\)-algebras; Morita equivalence; slice maps; bialgebras; bounded transform; unbounded multipliers; \(C^*\)-algebraic quantum group theory