On fixed point planar algebras

Abstract

To a weighted graph can be associated a bipartite graph planar algebra P. We construct and study the symmetric enveloping inclusion of P. We show that this construction is equivariant with respect to the automorphism group of P. The automorphism group of the weighted graph acts on P. We consider subgroups G of the automorphism group of the weighted graph such that the G-fixed point space P^G is a subfactor planar algebra. As an application we show that if G is amenable, then P^G is amenable as a subfactor planar algebra. We define the notions of a cocycle action of a Hecke pair on a tracial von Neumann algebra and the corresponding cross product. We show that a large class of symmetric enveloping inclusions of subfactor planar algebras can be described by such a cross product.

This paper replaces a previous version arXiv:1509.06654 entitled "Approximation properties for fixed point planar algebras" that contained a mistake

References in corpus (6)

• On co-amenability for groups and von Neumann algebras
• The planar algebra of diagonal subfactors
• C*-tensor categories and subfactors for totally disconnected groups
• Crossed products by Hecke pairs I: *-algebraic theory
• Crossed products by Hecke pairs II: C*-completions
• Automorphisms of the bipartite graph planar algebra