Functoriality  of  the  Group-Measure -space  construction  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 $H\otimes L^{2}(G)$)
 A: 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.
