Von Neumann equivalence and $M_d$ type approximation properties

TL;DR

This paper demonstrates that certain approximation and amenability properties of von Neumann algebras are preserved under von Neumann equivalence and inherited from lattices, expanding understanding of their stability.

Contribution

It establishes the stability of $M_d$-approximation, $M_d$-weak-amenability, and $M_d$-weak-Haagerup properties under von Neumann equivalence and from lattices.

Findings

Properties are stable under von Neumann equivalence.

Properties are inherited from lattices.

Results apply to measure and W*-equivalence.

Abstract

We show that $M_d$-approximation-property, $M_d$-weak-amenability, and $M_d$-weak-Haagerup-property are stable under von Neumann equivalence (hence also Measure equivalence and W*-equivalence). We also show that these properties are inherited from lattices.