Nuclearity for partial crossed products by exact discrete groups

Abstract

We study partial actions of exact discrete groups on C*-algebras. We show that the partial crossed product of a commutative C*-algebra by an exact discrete group is nuclear whenever the full and reduced partial crossed products coincide. This generalises a result by Matsumura in the context of 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 partial crossed products associated to the diagonal partial action on $A{\otimes }_{\max }{A}^{\mathrm{op}}$ coincide. We apply our results to show that the reduced semigroup C*-algebra ${\mathrm{C}}_{\lambda }^{\ast }(P)$ of a submonoid of an exact discrete group is nuclear if the left regular representation on ${\ell }^2(P)$ is an isomorphism between the full and reduced C*-algebras. We also show that nuclearity is equivalent to the weak containment property in the case of C*-algebras associated to separated graphs.

arXiv comment27 pages; changes according to the referee's comments in version 2; numbering has been affected; to appear in J. Operator Theory

References in corpus (6)

• Some remarks about the weak containment property for groupoids and semigroups
• Equivalence of Fell bundles over groups
• A characterization of amenability of group actions on $C^{\ast }$-algebras
• Tensor Products of Fell Bundles over Groups
• Amenability and weak containment for actions of locally compact groups on $C^{\ast }$-algebras
• Exchange rings and real rank zero C*-algebras associated with finitely separated graphs

Cited by in corpus (1)

• Toeplitz algebras of semigroups