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 $H\otimes L^{2}(G)$)

  • $\begingroup$ fun trial with respect to what morphisms? Note that the group Von Neumann algebra construction is only functorial with respect to injective homomorphisms with closed range, not all homomorphisms. I suspect that similar restrictions would be needed in the group-measure-space construction $\endgroup$ – Yemon Choi Mar 8 '12 at 17:37
  • $\begingroup$ maudite autocorrect - first word should have been "functorial" $\endgroup$ – Yemon Choi Mar 8 '12 at 17:38
  • 1
    $\begingroup$ The group-measure space construction should incorporate the group action in the definition of $u_g$. $\endgroup$ – Jesse Peterson Mar 8 '12 at 17:57

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.

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