# zbMATH — the first resource for mathematics

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.

##### 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