Similarity Problems and Completely Bounded Maps (Lecture Notes in Mathematics)

This book is mainly about three similarity problems arising in three different contexts, namely group representations, C*-algebras and uniform algebras (e.g. the disc algebra). These three problems (all still open in full generality) are studied using a common tool, completely bounded maps, which have recently emerged as a major concept in operator algebra theory. The book is devoted to the background necessary to understand these problems, to the partial solutions that are known and to numerous related concepts and results. The book is mostly self-contained and accessible to graduate students mastering basic functional and harmonic analysis. For advanced readers, it can be an invitation to the recently developed theory of "operator spaces", for which completely bounded maps are the fundamental morphisms.