Reduced Crossed Products

Question

Given a discrete group $ G $ and a $ G $-$ C^{*} $-algebra $ A $, we can form the reduced crossed product $ A \rtimes_{\operatorname{r}} G $. I want to define it as the closure of the embedded image of $ {C_{c}}(G,A) $ inside $ \mathscr{B}({\ell^{2}}(G,A)) $, with the embedding given by the twisted left-regular representation.

Does anyone know a reference for whether this is functorial with respect to all $ G $-$ C^{*} $-algebras? It is generally not with respect to groups. Any ideas are appreciated.

Answer 1

Take a look at these notes from the Lisboa Summer School Course on Crossed Product $ C^{*} $-Algebras written by N. Christopher Phillips, available here. The following information is taken from there.

Let $ G $ be a locally compact group. A $ G $-covariant system is defined as a triple $ (G,A,\alpha) $, where $ A $ is a $ C^{*} $-algebra and $ \alpha: G \to \operatorname{Aut}(A) $ a strongly continuous action of $ G $ on $ A $ by $ * $-automorphisms.

Definition. Let $ G $ be a locally compact group. Let $ (G,A,\alpha),(G,B,\beta) $ be $ G $-covariant systems. Then a morphism from $ (G,A,\alpha) $ to $ (G,B,\beta) $ is defined as a $ * $-homomorphism $ \phi: A \to B $ that is equivariant (or $ G $-equivariant, if the group must be specified) for $ \alpha $ and $ \beta $, i.e., $$ \forall g \in G: \quad \phi \circ \alpha_{g} = \beta_{g} \circ \phi. $$ The class of $ G $-covariant systems, together with their morphisms, forms a category.

The crossed-product and reduced-crossed-product constructions are functorial by the following:

Theorem. Let $ G $ be a locally compact group. Let $ (G,A,\alpha),(G,B,\beta) $ be $ G $-covariant systems. Then for every morphism $ \phi: (G,A,\alpha) \to (G,B,\beta) $, there is a $ * $-homomorphism $$ \psi: {C_{c}}(G,A,\alpha) \to {C_{c}}(G,B,\beta) $$ given by the formula $$ \forall f \in {C_{c}}(G,A,\alpha), ~ \forall g \in G: \quad [\psi(f)](g) = \phi(f(g)). $$ This extends by continuity to a $ * $-homomorphism $ {L^{1}}(G,A,\alpha) \to {L^{1}}(G,B,\beta) $, and finally on to $ * $-homomorphisms $$ {C^{*}}(G,A,\alpha) \to {C^{*}}(G,B,\beta) \qquad \text{and} \qquad {C_{\operatorname{r}}^{*}}(G,A,\alpha) \to {C_{\operatorname{r}}^{*}}(G,B,\beta). $$

This makes both the crossed-product and reduced-crossed-product constructions functors from the category of $ G $-covariant systems, for a fixed $ G $, to the category of $ C^{*} $-algebras.