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GCR and CCR Steinberg Algebras

Part of: Topological and differentiable algebraic systems Modules, bimodules and ideals Special categories Selfadjoint operator algebras

Published online by Cambridge University Press:  23 August 2019

Lisa O. Clark ,
Benjamin Steinberg  and
Daniel W. van Wyk
Lisa O. Clark
Affiliation:
School of Mathematics and Statistics, Victoria University of Wellington, PO Box 600,Wellington 6140, New Zealand Email: lisa.clark@vuw.ac.nz
Benjamin Steinberg
Affiliation:
Department of Mathematics, City College of New York, Convent Avenue at 138th Street, New York,New York 10031, USA Email: bsteinberg@ccny.cuny.edu
Daniel W. van Wyk
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH03755-3551, USA Email: daniel.w.van.wyk@dartmouth.edu
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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.

Keywords

CCR algebraGCR algebragroupoidSteinberg algebra

MSC classification

Primary: 22A22: Topological groupoids (including differentiable and Lie groupoids)
Secondary: 18B40: Groupoids, semigroupoids, semigroups, groups (viewed as categories) 16D60: Simple and semisimple modules, primitive rings and ideals 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 6 , December 2020 , pp. 1581 - 1606
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

The second author thanks the Fulbright Commission for sponsoring his stay at the Universidade Federal de Santa Catarina in Florianopolis, Brazil where much of this work was done. The third author was supported in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

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