Let $G$ be a group and form the associated Hilbert space $\ell^2(G)$ with canonical orthonormal basis $(e_g)_{g\in G}$. For $g \in G$, define the operators $u_g, v_g ∈ B(\ell^2(G))$ by $$u_g(e_h) := e_{gh}\ ,\ v_g(e_h) := e_{hg^{−1}}, \ \forall \ h\in G$$ Consider the generated von Neumann algebras $A := \overline{\{u_g : g \in G\}}^{WOT}$ and $B := \overline{\{v_g : g \in G\}}^{WOT}$.
How to prove that the commutant $A′= B$ ? From a characterization of a matrix in $A'$.
We can characterize that $a= (a_{fh})_{f,h\in G}$, an operator given by its matrix, commutes with all of the $u_g$'s if and only if $$\forall g,f,h\in G \ :\ a_{gf\ gh}= a_{f\ h}$$ The matrix of $a$ should be constant on 'diagonals' ( provided we view $h\mapsto gh$ as a translation ).
But how to show that this $a$ is a WOT-limit of $v_g$'s ?
Or is there another approach?