Approximation properties of fixed point planar algebras

TL;DR

This paper investigates how approximation properties like amenability and the Haagerup property are inherited by fixed point subfactor planar algebras under group actions, especially for bipartite graphs and trees.

Contribution

It establishes that certain approximation properties are preserved in fixed point subfactor planar algebras under group actions with approximation properties, and introduces a crossed product construction for von Neumann algebras by Hecke pairs.

Findings

Subfactor planar algebras from trees have the Haagerup property and CMAP.

Fixed point subalgebras inherit approximation properties from the acting group.

Provides examples of non-amenable subfactor planar algebras with approximation properties.

Abstract

Let $(\Gamma,\mu)$ be a bipartite graph together with a weight on its vertices. Assume that $\mu$ is an eigenvector for the adjacency matrix of $\Gamma$. Let Aut$(\Gamma, \mu)$ be the automorphism group of the bipartite graph $\Gamma$ that scales the weight $\mu$. It is a locally compact totally disconnected group that acts on the bipartite graph planar algebra $P$ associated to $(\Gamma,\mu)$. Consider a subgroup G < Aut$(\Gamma, \mu)$ and the set of fixed points $P^G \subset P$ that we assume to be a subfactor planar algebra. If the closure of G inside Aut$(\Gamma, \mu)$ satisfies an approximation property such as amenability, the Haagerup property, weak amenability, or not having property (T), then the subfactor planar algebra $P^G$ inherits this property respectively. As a corollary we show that if $\Gamma$ is a tree, then the subfactor planar algebra $P^G$ has the Haagerup property and has the complete metric approximation property (CMAP). This provides an infinite family of subfactor planar algebras that have non-integer index, are non-amenable, have the Haagerup property, and have CMAP. We define the crossed product of a (finite) von Neumann algebra by a Hecke pair of groups. We show that a large class of symmetric enveloping inclusions of subfactor planar algebras are described by such a crossed product including Bisch-Haagerup subfactors.