Based on the author's university lecture courses, this book presents the many facets of one of the most important open problems in operator algebra theory. Central to this book is the proof of the equivalence of the various forms of the problem, including forms involving C*-algebra tensor products and free groups, ultraproducts of von Neumann algebras, and quantum information theory. The reader is guided through a number of results (some of them previously unpublished) revolving around tensor products of C*-algebras and operator spaces, which are reminiscent of Grothendieck's famous Banach space theory work. The detailed style of the book and the inclusion of background information make it easily accessible for beginning researchers, Ph.D. students, and non-specialists alike.

‘This is a very rich and detailed monograph on an enormously important subject. It is written in the crystal clear and elegant style that is the hallmark of its author, and it offers a lot of information to specialists and novices alike. The book will certainly become an authoritative guide.’

Dirk Werner Source: London Mathematical Society Student Texts

‘This book is jam packed with information, and should be an invaluable guide to anyone interested in these ideas … For the complete picture and the recent advances, Pisier’s book is the place to go.’

Source: Bulletin of the American Mathematical Society