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

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.

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 |