The theory of crossed products is extremely rich and intriguing. There
are applications not only to operator algebras, but to subjects as
varied as noncommutative geometry and mathematical physics. This book
provides a detailed introduction to this vast subject suitable for
graduate students and others whose research has contact with crossed
product $C^*$-algebras. In addition to providing the basic
definitions and results, the main focus of this book is the fine ideal
structure of crossed products as revealed by the study of induced
representations via the Green-Mackey-Rieffel machine. In particular,
there is an in-depth analysis of the imprimitivity theorems on which
Rieffel's theory of induced representations and Morita equivalence of
$C^*$-algebras are based. There is also a detailed treatment
of the generalized Effros-Hahn conjecture and its proof due to
Gootman, Rosenberg, and Sauvageot.
This book is meant to be self-contained and accessible to any
graduate student coming out of a first course on operator
algebras. There are appendices that deal with ancillary subjects,
which while not central to the subject, are nevertheless crucial for a
complete understanding of the material. Some of the appendices will be
of independent interest.
Readership
Graduate students and research mathematicians interested in
$C^*$-algebras.