图书简介
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.
Introduction.- Part I Inclusions.- Local modules.- Regular ideals and the localizing projection.- Regular inclusions.- Invariant ideals.- Extended multiplication for normalizers.- Regularity of maximal ideals in regular inclusions.- Extension of pure states, relative free points and smooth normalizers.- Free points.- Fourier coefficients.- Opaque and gray ideals.- Topologically free inclusions.- Pseudo-expectations.- Part II Groupoids.- E#tale groupoids.- Twists and line bundles.- The C*-algebra of a twisted groupoid.- Topologically free groupoids.- The essential groupoid C*-algebra.- Kwasniewski and Meyer’s version of the essential groupoid C*-algebra.- The relative Weyl groupoid.- Fell bundles over inverse semigroups.- Topological freeness of the Weyl groupoid and the main Theorem.- Semi-masas.- Canonical states.- Part III Examples and Open Questions.- Example: non-smooth normalizers.- Example: periodic functions on the interval.- Example: the gray ideal of twisted groupoid C*-algebras.- Some open questions.- Part IV Appendix.- Isotropy projection.- Symbol index.- Concept index.- References.