Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids / Ruy Exel, David R. Pitts.

Format:

Book

Author/Creator:

Exel, Ruy, 1956- author.

Pitts, David Ryder, author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 0075-8434 2306.

Lecture Notes in Mathematics, 0075-8434 ; Volume 2306

Language:

English

Subjects (All):

C*-algebras.

Groupoids.

Physical Description:

viii, 156 pages : illustrations ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer Nature Switzerland AG, [2022]

Summary:

This book develops tools to handle C*-algebras arising as completions of convolution algebras of sections of line bundles over possibly non-Hausdorff groupoids. A fundamental result of Gelfand describes commutative C*-algebras as continuous functions on locally compact Hausdorff spaces. Kumjian, and later Renault, showed that Gelfand's result can be extended to include non-commutative C*-algebras containing a commutative C*-algebra. In their setting, the C*-algebras in question may be described as the completion of convolution algebras of functions on twisted Hausdorff groupoids with respect to a certain norm. However, there are many natural settings in which the KumjianRenault theory does not apply, in part because the groupoids which arise are not Hausdorff. In fact, non-Hausdorff groupoids have been a source of surprising counterexamples and technical difficulties for decades. Including numerous illustrative examples, this book extends the KumjianRenault theory to a much broader class of C*-algebras. This work will be of interest to researchers and graduate students in the area of groupoid C*-algebras, the interface between dynamical systems and C*-algebras, and related fields.

Contents:

Intro

Abstract

Contents

1 Introduction

2 Inclusions

2.1 Local Modules

2.2 Regular Ideals and the Localizing Projection

2.3 Regular Inclusions

2.4 Invariant Ideals

2.5 Extended Multiplication for Normalizers

2.6 Regularity of Maximal Ideals in Regular Inclusions

2.7 Extension of Pure States, Relative Free Points and Smooth Normalizers

2.8 Free Points

2.9 Fourier Coefficients

2.10 Opaque and Gray Ideals

2.11 Topologically Free Inclusions

2.12 Pseudo-Expectations

3 Groupoids

3.1 Étale Groupoids

3.2 Twists and Line Bundles

3.3 The C*-Algebra of a Twisted Groupoid

3.4 Topologically Free Groupoids

3.5 The Essential Groupoid C*-Algebra

3.6 Kwasniewski and Meyer's Version of the Essential Groupoid C*-Algebra

3.7 The Relative Weyl Groupoid

3.8 Fell Bundles Over Inverse Semigroups

3.9 Topological Freeness of the Weyl Groupoidand the Main Theorem

3.10 Semi-Masas

3.11 Canonical States

4 Examples and Open Questions

4.1 Example: Non-Smooth Normalizers

4.2 Example: Periodic Functions on the Interval

4.3 Example: The Gray Ideal of Twisted Groupoid C*-Algebras

4.4 Some Open Questions

5 Appendix: Isotropy Projection

References

Symbol Index

Concept Index

Notes:

Includes bibliographical references (pages 151-152) and indexes.

Current copyright fee: GBP19.00 42\0.

Other Format:

Online version: Exel, Ruy, 1956- Characterizing groupoid C* -algebras of non-Hausdorff Étale groupoids.

ISBN:

9783031055126

3031055128

OCLC:

1351364571

Publisher Number:

9783031055126