Weights on bimodules

TL;DR

This paper characterizes weights on planar algebras via vertex functions on principal graphs and demonstrates that trivial perturbation class property is preserved under Connes fusion, providing a constructive perturbation method.

Contribution

It offers an alternative characterization of weights on planar algebras and proves closure of trivial perturbation class under Connes fusion with a constructive perturbation approach.

Findings

Weights characterized by vertex functions on principal graphs

Trivial perturbation class property is closed under Connes fusion

Constructive method for perturbing bimodules with positive weights

Abstract

The concept of a {\em weight} on a planar algebra was introduced in \cite{DGG}. In this article we give an alternate characterization of weights on a planar algebra in terms of `weight functions' on the vertices of the principal graphs. Using this characterization we show that the property of bifinite bimodules of having a `trivial perturbation class' is closed under Connes fusion. We give a direct and constructive method of perturbing a bifinite bimodule by a positive weight in such a way that the bimodule planar algebra of the perturbed bimodule is isomorphic to the perturbation of the one associated to the initial bimodule by the given weight.