Weak amenability for subfactors

TL;DR

This paper introduces the concepts of weak amenability and the Cowling-Haagerup constant for extremal finite index subfactors of type II_1, establishing their invariance and computing them for specific cases, thus advancing the understanding of subfactor invariants.

Contribution

It defines weak amenability and the Cowling-Haagerup constant for subfactors, proves their invariance under standard invariants, and computes these constants for certain subfactors and constructions.

Findings

Cowling-Haagerup constant depends only on the standard invariant.

Explicit computation of the constant for Bisch-Haagerup subfactors.

The Cowling-Haagerup constant of tensor products equals the product of individual constants.

Abstract

We define the notions of weak amenability and the Cowling-Haagerup constant for extremal finite index subfactors of type II_1. 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 in the sense of Popa, we prove that their free product in the sense of Bisch-Jones is weakly amenable with constant 1. We show that the Cowling-Haagerup constant of the tensor product of a finite family of standard invariants is equal to the product of their Cowling-Haagerup constants.