On the relative weak asymptotic homomorphism property for triples of group von Neumann algebras

Abstract:

A triple of finite von Neumann algebras $B\subset N\subset M$ is said to have the relative weak asymptotic homomorphism property if there exists a net of unitaries $(u_i)_{i\in I}\subset U(B)$ such that \[ \lim _{i\in I}\Vert \mathbb {E}_B(xu_iy)-\mathbb {E}_B(\mathbb {E}_N(x)u_i\mathbb {E}_N(y))\Vert _2=0 \] for all $x,y\in M$. Recently, J. Fang, M. Gao and R. Smith proved that the triple $B\subset N\subset M$ has the relative weak asymptotic homomorphism property if and only if $N$ contains the set of all $x\in M$ such that $Bx\subset \sum _{i=1}^n x_iB$ for finitely many elements $x_1,\ldots ,x_n\in M$. Furthermore, if $H<G$ is a pair of groups, they get a purely algebraic characterization of the weak asymptotic homomorphism property for the pair of von Neumann algebras $L(H)\subset L(G)$, but their proof requires a result which is very general and whose proof is rather long. We extend the result to the case of a triple of groups $H<K<G$, we present a direct and elementary proof of the above-mentioned characterization, and we introduce three more equivalent conditions on the triple $H<K<G$, one of them stating that the subspace of $H$-compact vectors of the quasi-regular representation of $H$ on $\ell ^2(G/H)$ is contained in $\ell ^2(K/H)$.