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Groupoids, Inverse Semigroups, and their Operator Algebras (Progress in Mathematics, 170),New
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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, uptodate and reasonably extensive account of the subject matter.After an introductory first chapter, the second chapter presents a selfcontained account of inverse semigroups, locally compact and rdiscrete 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 rdiscrete groupoids are developed in the third chapter, and it is shown that the C*algebras of rdiscrete groupoids are the covariance C*algebras for inverse semigroup actions on locally compact Hausdorff spaces. A final chapter associates a universal rdiscrete 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.
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