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 ) M_d(G) ( d ≥ 2 d\geq 2 ) introduced by Pisier. We show that unitarisable groups satisfying this property for all d ≥ 2 d\geq 2 are amenable. Moreover, we show that groups acting properly on finite-dimensional CAT(0) cube complexes satisfy M d M_d -AP for all d ≥ 2 d\geq 2 . We also give examples of non-weakly amenable groups satisfying M d M_d -AP for all d ≥ 2 d\geq 2 .