Preface to the Second Edition

Summary

A lot has happened in the realm of foliated spaces and their operator algebras since 1988, when this book first appeared. We are pleased that, as we had hoped, this book has served as an introduction to the subject and a reference for researchers and students.

Our colleagues have convinced us that there is merit in issuing a second edition of our work, so that a new generation of students may have access to its contents. Cambridge University Press was amenable to the idea, so we (slowly) went to work.

We have taken the opportunity of a new edition to make a number of changes and additions to the book:

• (1) We have corrected a few minor errors, filled some gaps, and made many changes to improve the exposition.

• (2) We have added updates at the end of each chapter as well as occasional footnotes in which we discuss some of the relevant mathematical developments since 1988. This discussion is understandably brief. We do try to point the reader to the papers where the results themselves appear.

• (3) We have enlarged the bibliography correspondingly.

• (4) We have added a new appendix; it is a reprint of a Mathematical Reviews Featured Review by the second author on the Gap Labeling Theorem. We felt this was appropriate since it illustrates a very interesting and important application of the Index Theorem.

• (5) We have added an index to the book.

• (6) MSRI has provided for the resetting of the book in LATEX and for the redrafting of all the art.

As originally formulated, the Connes’ Index Theorem [1979] applied to foliated manifolds. The version presented here is valid for foliated spaces, a category that is strictly larger than foliated manifolds and laminations obtained from manifolds. It turns out that this extra generality is crucial for some of the applications of the Index Theorem in the past few years. For instance, the Gap Labeling results discussed in Appendix D require this extra generality. We discuss this in some detail at the end of Chapter VIII.

We acknowledge with gratitude the help that we have received from Jean Bellissard, Alberto Candel, Larry Conlon, Steve Hurder, Jerry Kaminker, Ma-soud Khalkhali, Paul Muhly, and especially our friend and editor par excellence Silvio Levy in the preparation of this edition. The second author is grateful to Baruch Solel and the faculty of the Technion for a sabbatical year at a critical time. We are grateful to the editors of Mathematical Reviews for permission to reproduce the Featured Review on the Gap Labeling Theorem as Appendix D of this work.