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GCR and CCR Steinberg algebras. (English) Zbl 1468.22010

Summary: 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.

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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