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

Abstract

for all x, y ∈ M . Then recently, J. Fang, M. Gao and R. Smith proved that the triple B ⊂ N ⊂ M has the relative weak asymptotic homomorphism property if and only if N contains the set of all x ∈ M such that Bx ⊂ ∑n i=1 xiB for finitely many elements x1, . . . , xn ∈ 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) ⊂ 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 combinatorial 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 l(G/H) is contained in l(K/H).