# reduced crossed products

Given a discrete group $G$ and a $G$-$C^*$-algebra $A$ we can form the reduced crossed product $A\rtimes_r G$. I want to define it by the closure of $C_c(G,A)$ in $\mathcal{B}(\ell^2(G,A)$ where this inclusion is given by the twisted left regular representation. Does anyone know a reference whether this is functorial in the $C^*$-algebra? It is in generally not in the group variable. 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 is taken from there.

Let $G$ be a locally compact group. A C$^*$-algebra $A$ equipped with an action $G \to Aut(A)$ will be called a $G$-algebra. We sometimes refer to $(G,A,\alpha)$ as a $G$-algebra.

Definition

If $(G,A,α)$ and $(G,B,β)$ are $G$-algebras, then a homomorphism $\phi: A \to B$ is said to be equivariant (or G-equivariant if the group must be specified)if for every $g\in G$, we have $$\phi \circ \alpha_g =\beta_g \circ \phi.$$ For a fixed locally compact group G, the G-algebras and equivariant homomorphisms form a category.

The crossed product construction is functorial for equivariant homomorphisms

Theorem

Let G be a locally compact group. If (G,A,α) and (G,B,β) are G-algebras and φ: A → B is an equivariant homomorphism, then there is a homomorphism $\psi: C_c(G,A,α) → C_c(G,B,β)$ given by the formula $ψ(a)(g) = φ(a(g))$ for $a \in C_c(G,A,α)$ and $g \in G$, and this homomorphism extends by continuity to a homomorphism $L_1(G,A,α) \to L1(G,B,β)$, and then to homomorphisms $$C^∗(G,A,α) → C^∗(G,B,β)\ and\ C_r^∗(G,A,α) → C_r^∗(G,B,β).$$

This construction makes the crossed product and reduced crossed product constructions functors from the category of G-algebras 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