The $M_d$-Approximation Property and Unitarisability

Abstract

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

10 pages. Minor changes. Accepted for publication in Proceedings of the American Mathematical Society

References in corpus (2)

• Simple Lie groups without the Approximation Property II
• A complete characterization of connected Lie groups with the Approximation Property