Preface The present memoir consists of two parts. The first part encompasses Chapters 1-4, and is concerned with the description of a class of representations of the Cuntz algebra Od, starting out with a very general description of such representations. The second part encompasses Chapters 4-18 and is a description of a class of AF- algebras with constant incidence matrices J of the special form (6.1). The two parts are thus connected by Chapter 4, where it is explained how these AF-algebras arise as the fixed-point algebras of modular automorphism groups associated to certain states on Od- Readers who are not interested in representation theory can therefore read the memoir from Chapter 5. Since the special examples we study can be understood very concretely, we hope that the memoir may serve as an invitation for graduate students who want to study isomorphism and invariants in more general settings. Oslo, Iowa City and Kiev, June 1999, January 2003, and May 2003 Ola Bratteli Palle E. T. Jorgensen Vasyl' Ostrovs'kyi
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