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 | Textbooks (functional analysis) |

46H25 | Normed modules and Banach modules, topological modules |

46L05 | General theory of ${C}^{*}$-algebras |