Rigid C^*-tensor categories of bimodules over interpolated free group factors

TL;DR

This paper constructs a planar algebra from a given rigid C^*-tensor category and demonstrates its equivalence to a category of bifinite bimodules over an interpolated free group factor, linking abstract categories to operator algebra structures.

Contribution

It introduces a method to realize any countably generated rigid C^*-tensor category as a category of bifinite bimodules over L(F_infty), connecting categorical and operator algebra frameworks.

Findings

Constructed a planar algebra P from C

Established equivalence between Pro and Bim categories

Linked abstract categories to bimodules over free group factors

Abstract

Given a countably generated rigid C^*-tensor category C, we construct a planar algebra P whose category of projections Pro is equivalent to C. From P, we use methods of Guionnet-Jones-Shlyakhtenko-Walker to construct a rigid C^*-tensor category Bim whose objects are bifinite bimodules over an interpolated free group factor, and we show Bim is equivalent to Pro. We use these constructions to show C is equivalent to a category of bifinite bimodules over L(F_infty).