$C^*$-algebras from planar algebras II: the Guionnet-Jones-Shlyakhtenko $C^*$-algebras

TL;DR

This paper investigates the properties of $C^*$-algebras constructed from planar algebras via the GJS method, revealing their Morita equivalence, $K$-theory, and structural features like simplicity and stable rank.

Contribution

It demonstrates that these $C^*$-algebras are pairwise strongly Morita equivalent, computes their $K$-groups, and establishes key properties such as simplicity and unique trace.

Findings

The $C^*$-algebras are pairwise strongly Morita equivalent.

The $K$-groups of these algebras are computed.

They exhibit properties like simplicity, unique trace, and stable rank 1.

Abstract

We study the $C^*$-algebras arising in the construction of Guionnet-Jones-Shlyakhtenko (GJS) for a planar algebra. In particular, we show they are pairwise strongly Morita equivalent, we compute their $K$-groups, and we prove many properties, such as simplicity, unique trace, and stable rank 1. Interestingly, we see a $K$-theoretic obstruction to the GJS $C^*$-algebra analog of Goldman-type theorems for II$_1$-subfactors. This is the second article in a series studying canonical $C^*$-algebras associated to a planar algebra.