Weak amenability for subfactors

Abstract

We define the notions of weak amenability and the Cowling–Haagerup constant for extremal finite index subfactors s of type II1. We prove that the Cowling–Haagerup constant only depends on the standard invariant of the subfactor. Hence, we define the Cowling–Haagerup constant for standard invariants. We explicitly compute the constant for Bisch–Haagerup subfactors and prove that it is equal to the constant of the group involved in the construction. Given a finite family of amenable standard invariants, we prove that their free product in the sense of Bisch–Jones is weakly amenable with constant 1. We show that the Cowling–Haagerup of the tensor product of a finite family of standard invariants is equal to the product of their Cowling–Haagerup constants.