ON THE RELATIVE WEAK ASYMPTOTIC HOMOMORPHISM PROPERTY FOR TRIPLES OF GROUP VON NEUMANN ALGEBRAS.

A triple of finite von Neumann algebras B ⊂ N ⊂ M is said to have the relative weak asymptotic homomorphism property if there exists a net of unitaries (ui)i∈I ⊂ U(B) such that [Multiple line equation(s) cannot be represented in ASCII text] for all x, y ∈ M. 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 ⊂ Σi=1n 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 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 l²(G/H) is contained in l²(K/H). [ABSTRACT FROM AUTHOR]