“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)
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.
