Let $G$ be a discrete group. Consider the action of $G$ on itself 
a) by left multiplication, 
b) by conjugation. 


Under which conditions on group homomorphisms is the Group-Measure construction associated to these situations functorial? 


(Recall that the group-measure construction associated to a measurable action on a space $M$ is obtained as the weak \* closure of the operators $u\_{g}= \lambda\_{g}\otimes id $( $g \in G$) and $id\otimes \alpha(x)$( $x \in L^{\infty}(M))$, where $\alpha:L^{\infty}(M) \to B(H)$ is the GNS construction, $\lambda:G\to BL^{2}(G)$ is the regular representation and the operators $id \otimes \alpha$ and $u\_{g}$ are defined on the hilbert space $H\otimes L^{2}(G)$)



Functoriality  of  the  Group-Measure -space  construction 
I assume that the morphisms on the von Neumann algebra side are unital \*-homomorphisms.


For a group morphism $\theta:G\rightarrow H$, the algebraic morphism (on the twisted group algebra, over the ring of finitely supported functions on $G$) is given by $\psi\_{alg}(\delta\_g u\_h)=\delta\_{\theta(g)}u\_{\theta(h)}$ where $\delta\_g\in L^\infty(G)$ is the characteristic function of the singleton $\{g\}$. First of all, observe that $\theta$ has to be injective in order to make $\psi\_{alg}$ multiplicative. If this algebraic morphism is to extend to a unital morphism of the crossed product, it follows that $\psi(1)=\sup\{\psi(\delta\_g)\mid g\in G\}=\sup\{\delta\_{\theta(g)}\mid g\in G\}$ and hence $\theta$ is surjective.


So we have shown that both constructions (a) and (b) are functorial with respect to isomorphisms, but not with respect to any other morphisms.


However, in case (a) we can give a pathological functorial structure. We restrict the objects to countably infinite groups, and assume that they come with a bijection $n:N\rightarrow G$. Observe that the group-measure space construction gives $B(\ell^2(G))$ which is now canonically isomorphic to $B(\ell^2(N))$, say by the isomorphism $\psi\_G$. Mapping all morphisms $\theta:G\rightarrow H$ to the isomorphism $\psi\_\theta=\psi\_H^{-1}\circ\psi\_G$ gives us a pathologial functor.


I do not know if (b) also has such a pathological functor, but it is in any case not the one we are interested in.


