The $M_d$-Approximation Property and Unitarisability

TL;DR

This paper introduces a new approximation property called the $M_d$-AP, strengthening existing concepts, and demonstrates its implications for group amenability and actions on CAT(0) cube complexes, with examples of non-weakly amenable groups satisfying it.

Contribution

It defines the $M_d$-AP property, shows its relation to amenability, and provides examples of groups satisfying this property beyond weak amenability.

Findings

Unitarisable groups with $M_d$-AP for all $d extgreater{}2$ are amenable.

Groups acting properly on finite-dimensional CAT(0) cube complexes satisfy $M_d$-AP for all $d extgreater{}2$.

Examples of non-weakly amenable groups satisfying $M_d$-AP for all $d extgreater{}2$ are provided.

Abstract

We define a strengthening of the Haagerup-Kraus approximation property by means of the subalgebras of Herz-Schur multipliers $M_d(G)$ ($d\geq 2$) introduced by Pisier. We show that unitarisable groups satisfying this property for all $d\geq 2$ are amenable. Moreover, we show that groups acting properly on finite-dimensional CAT(0) cube complexes satisfy $M_d$-AP for all $d\geq 2$. We also give examples of non-weakly amenable groups satisfying $M_d$-AP for all $d\geq 2$.