Nuclearity for partial crossed products by exact discrete groups

2020/11/12 by Buss, Alcides, Ferraro, Damián, Sehnem, Camila F. · 2 citations
#37B99 (Secondary) #46L55 (Primary) 46L99 #Dynamical Systems (math.DS) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2011.06686

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⊗_max A^op coincide. We apply our results to show that the reduced semigroup C*-algebra C^*_λ(P) of a submonoid of an exact discrete group is nuclear if the left regular representation on ℓ²(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.

Cited by

• Characterizations of amenability for noncommutative dynamical systems and Fell bundles
• Actions of Fell bundles

Related

• Nuclearity of semigroup C*-algebras
• Nuclearity of semigroup C*-algebras and the connection to amenability
• C*-crossed products by partial actions and actions of inverse semigroups
• Characterisations of crossed products by partial actions
• Equivariant (co)module nuclearity of C^*-crossed products