Approximation properties of fixed point planar algebras

Let $(\mathrm{\Gamma },\mu )$ be a bipartite graph together with a weight on its vertices. Assume that $\mu$ is an eigenvector for the adjacency matrix of $\mathrm{\Gamma }$. Let Aut$(\mathrm{\Gamma },\mu )$ be the automorphism group of the bipartite graph $\mathrm{\Gamma }$ that scales the weight $\mu$. It is a locally compact totally disconnected group that acts on the bipartite graph planar algebra $P$ associated to $(\mathrm{\Gamma },\mu )$. Consider a subgroup G < Aut$(\mathrm{\Gamma },\mu )$ 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$(\mathrm{\Gamma },\mu )$ 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 $\mathrm{\Gamma }$ 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.