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.

DOI: http://dx.doi.org/10.7900/jot.2020dec01.2327
Keywords: partial action, nuclearity, approximation property, exact group

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