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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 toa 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.
List of contents
- 1. Introduction. - 2. Inclusions. - 3. Groupoids. - 4. Examples and Open Questions. - 5. Appendix.
About the author
Ruy Exel is a professor of Mathematics at the Universidade Federal de Santa Catarina, in Brazil. He has published extensively in the subject of operator algebras with emphasis on its interactions with dynamical systems and mathematical physics. A pioneer in the area of partial actions of groups on C*-algebras, he has recently published the book Partial Dynamical Systems, Fell Bundles and Applications. He was an invited speaker in the 2018 ICM in Rio.
David R. Pitts is a professor of mathematics at the University of Nebraska-Lincoln. He has published work on various aspects of operator algebras, including: commutative subspace lattice algebras, free semigroup algebras, and Cartan subalgebras of von Neumann algebras and C*-algebras.
Report
"The book under review generalizes Kumjian and Renault's work to include more examples of C*-algebras. In doing this, the noncommutative space used is allowed to be non-Hausdorff. Non-Hausdorff groupoids have been the source of many exciting examples or counterexamples. As such, a better study of non-Hausdorff groupoids is welcome. ... The book ends with a section of examples and open questions. The Appendix contains details of a fundamental result in the theory of twisted groupoid C_-algebras." (Cristian Ivanescu, Mathematical Reviews, November, 2023)
"This is a nicely written monograph devoted to the new and important notion of non Hausdorff groupoids and their C*-algebras, and could be beneficial for researchers in operator algebras and mathematical physics." (Massoud Amini, zbMATH 1511.46002, 2023)
