GCR and CCR Steinberg Algebras. (23rd December 2020)

• Record Type: Journal Article
• Authors: Clark, Lisa O.; Steinberg, Benjamin; van Wyk, Daniel W.
• Is Part Of: Canadian journal of mathematics. Volume 72:Number 6(2020)
• Page Start: 1581
• Page End: 1606
• Publication Date: 2020-12-23
• Subjects: 22A22, -- 18B40, -- 16D60, -- 46L89; CCR algebra, -- GCR algebra, -- groupoid, -- Steinberg algebra; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/S0008414X19000415
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: Kaplansky introduced the notions of CCR and GCR $C^{\ast }$ -algebras, because they have a tractable representation theory. Many years later, he introduced the notions of CCR and GCR rings. In this paper we characterize when the algebra of an ample groupoid over a field is CCR and GCR. The results turn out to be exact analogues of the corresponding characterization of locally compact groupoids with CCR and GCR $C^{\ast }$ -algebras. As a consequence, we classify the CCR and GCR Leavitt path algebras.