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Zbl 0822.46080
Lance, E.Christopher
Hilbert $C\sp*$-modules. A toolkit for operator algebraists. (English)
[B] London Mathematical Society Lecture Note Series. 210. Cambridge: Univ. Press,. ix, 130 p. \sterling\ 17.95; \$ 29.95 (1995). ISBN 0-521-47910-X/pbk

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 $C\sp*$-algebra theory and some of its major applications. These techniques and results centre round the concept of Hilbert $C\sp*$-module, an object like Hilbert space provided that the inner product takes its value in a general $C\sp*$-algebra instead of being complex-valued. Hilbert $C\sp*$-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. \par The first three chapters present basic definitions and concepts and the elementary theory of Hilbert $C\sp*$-modules in the model of classical Hilbert spaces. Chapter 4 presents tensor products. Chapter 5 is a treatment on completely-positive mappings between $C\sp*$-algebras and then in the context of Hilbert $C\sp*$-algebras. In the chapter 6, a universal positive response is given, under some countability condition on modules, to the following general question: Given a $C\sp*$-algebra $A$, can we hope to classify all Hilbert-$A$-modules up to unitary equivalence? \par Broadly, chapters 4 to 6 are motivated by applications of Hilbert $C\sp*$-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 $C\sp*$-algebraic quantum group theory, but quantum groups as such do not figure in this book.
[H.Hogbe Nlend (Bordeaux)]

MSC 2000:: *46L89 Other "noncommutative" mathematics based on C*-algebra theory
46-01 Textbooks (functional analysis)
46H25 Topological modules
46L05 General theory of C*-algebras

Keywords: Hilbert $C\sp*$-module; tensor products; completely-positive mappings between $C\sp*$-algebras; Morita equivalence; slice maps; bialgebras; bounded transform; unbounded multipliers; $C\sp*$-algebraic quantum group theory

Cited in: Zbl 1212.46085 Zbl 1165.47005 Zbl 1053.46038 Zbl 1015.46034 Zbl 1046.46002 Zbl 1015.46033 Zbl 0953.19002

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