© 1999

Groupoids, Inverse Semigroups, and their Operator Algebras


Part of the Progress in Mathematics book series (PM, volume 170)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Alan L. T. Paterson
    Pages 1-19
  3. Alan L. T. Paterson
    Pages 21-84
  4. Alan L. T. Paterson
    Pages 147-210
  5. Back Matter
    Pages 211-274

About this book


In recent years, it has become increasingly clear that there are important connections relating three concepts -- groupoids, inverse semigroups, and operator algebras. There has been a great deal of progress in this area over the last two decades, and this book gives a careful, up-to-date and reasonably extensive account of the subject matter. 

After an introductory first chapter, the second chapter presents a self-contained account of inverse semigroups, locally compact and r-discrete groupoids, and Lie groupoids. The section on Lie groupoids in chapter 2 contains a detailed discussion of groupoids particularly important in noncommutative geometry, including the holonomy groupoids of a foliated manifold and the tangent groupoid of a manifold. The representation theories of locally compact and r-discrete groupoids are developed in the third chapter, and it is shown that the   C*-algebras of r-discrete groupoids are the covariance   C*-algebras for inverse semigroup actions on locally compact Hausdorff spaces. A final chapter associates a universal r-discrete groupoid with any inverse semigroup. Six subsequent appendices treat topics related to those covered in the text. 

The book should appeal to a wide variety of professional mathematicians and graduate students in fields such as operator algebras, analysis on groupoids, semigroup theory, and noncommutative geometry. It will also be of interest to mathematicians interested in tilings and theoretical physicists whose focus is modeling quasicrystals with tilings. An effort has been made to make the book lucid and 'user friendly"; thus it should be accessible to any reader with a basic background in measure theory and functional analysis.


Group theory Harmonic Analysis Lie Lie groups Manifold algebra calculus differential topology function functional analysis geometry representation theory theorem topological group topology

Authors and affiliations

  1. 1.Department of MathematicsUniversity of MississippiUniversityUSA

Bibliographic information

  • Book Title Groupoids, Inverse Semigroups, and their Operator Algebras
  • Editors Alan Paterson
  • Series Title Progress in Mathematics
  • Series Abbreviated Title Progress in Mathematics(Birkhäuser)
  • DOI
  • Copyright Information Birkhäuser Boston 1999
  • Publisher Name Birkhäuser, Boston, MA
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-0-8176-4051-4
  • Softcover ISBN 978-1-4612-7276-2
  • eBook ISBN 978-1-4612-1774-9
  • Series ISSN 0743-1643
  • Series E-ISSN 2296-505X
  • Edition Number 1
  • Number of Pages XVI, 274
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Group Theory and Generalizations
    Operator Theory
    Topological Groups, Lie Groups
  • Buy this book on publisher's site


"This outstanding book brings together three mathematical objects 'which a priori seem to have nothing much in common'.…The concept of amenability is largely carried over to the present framework. In particular, Clifford semigroups are examined. An example coming from physics, quasicrystals, is discussed. Throughout the monograph explicit clear proofs are provided. The large coverage of this material makes pleasant reading."

–Zentralblatt Math