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 Kumjian–Renault 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 Kumjian–Renault 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.
Author(s): Ruy Exel, David R. Pitts
Series: Lecture Notes in Mathematics, 2306
Publisher: Springer
Year: 2022
Language: English
Pages: 160
City: Cham
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
