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Journal of Operator Theory

Volume 88, Issue 1, Summer 2022  pp. 85-117.

Nuclearity for partial crossed products by exact discrete groups

Authors:  Alcides Buss (1), Damian Ferraro (2), Camila F. Sehnem (3)
Author institution: (1) Departamento de Matematica, Universidade Federal de Santa Catarina, Florianopolis, 88040-900, Brazil
(2) Departamento de Matematica y Estadistica del Litoral, Universidad de la Republica, Salto, 50000, Uruguay
(3) School of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington, 6140, New Zealand

Summary:  We show that the partial crossed product of a commutative $C^*$-algebra by an exact discrete group is nuclear if the full and reduced partial crossed products coincide. This generalises a result by Matsumura for global actions. In general, we prove that a partial action of an exact discrete group on a $C^*$-algebra $A$ has Exel's approximation property if and only if the full and reduced crossed products by the diagonal partial action on $A\otimes_{\mathrm{max}} A^\mathrm{op}$ coincide. We apply our results to show that the weak containment property implies nuclearity in the case of semigroup $C^*$-algebras and $C^*$-algebras of separated graphs.

Keywords: partial action, nuclearity, approximation property, exact group

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