On fixed point planar algebras

TL;DR

This paper explores the structure of fixed point subalgebras in bipartite graph planar algebras under automorphism groups, establishing conditions for amenability and describing their construction via cocycle actions and cross products.

Contribution

It introduces a framework for analyzing fixed point subalgebras of bipartite graph planar algebras using symmetric enveloping inclusions and cocycle actions, extending understanding of their automorphism groups.

Findings

Fixed point spaces under amenable groups are themselves amenable.

Construction of symmetric enveloping inclusions is equivariant under automorphisms.

Large classes of inclusions can be described via cocycle actions and cross products.

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.