On the Relative Weak Asymptotic Homomorphism Property for Triples of Groups von Neumann Algebras

TL;DR

This paper proves an equivalence related to the weak asymptotic homomorphism property for group von Neumann algebras and extends the result to triples of groups, providing a more general understanding of their algebraic structure.

Contribution

It offers a direct, elementary proof of a known equivalence and extends the result to triples of groups, broadening the theoretical framework.

Findings

Equivalence between weak asymptotic homomorphism property and embedding of quasi-normalizer.

Extension of the equivalence to triples of groups.

Elementary proof approach for the main result.

Abstract

We provide a direct and elementary proof of the equivalence between the weak asymptotic homomorphism property for the pair of group von Neumann algebras $L(H)\subset L(G)$ and the embedding into $H$ of the one sided quasi-normalizer of the pair $H<G$, as it is stated by J. Fang, M. Gao and R. R. Smith in a recent article, and we extend the result to triples of groups $H<K<G$.