# Reduced Crossed Products

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.

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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.

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Thanks. They seem to be working with a different definition, which I wanted to avoid, and I do not yet really understand their argument and how it translates to my case. – mland May 20 '12 at 8:11