$C^{\ast }$-algebras from planar algebras I: canonical $C^{\ast }$-algebras associated to a planar algebra

Abstract

From a planar algebra, we give a functorial construction to produce numerous associated $C^{\ast }$-algebras. Our main construction is a Hilbert $C^{\ast }$-bimodule with a canonical real subspace which produces Pimsner-Toeplitz, Cuntz-Pimsner, and generalized free semicircular $C^{\ast }$-algebras. By compressing this system, we obtain various canonical $C^{\ast }$-algebras, including Doplicher-Roberts algebras, Guionnet-Jones-Shlyakhtenko algebras, universal (Toeplitz-)Cuntz-Krieger algebras, and the newly introduced free graph algebras. This is the first article in a series studying canonical $C^{\ast }$-algebras associated to a planar algebra.

47 pages, many figures

References in corpus (5)

• Random matrices, free probability, planar algebras and subfactors
• Rigid C^*-tensor categories of bimodules over interpolated free group factors
• Representations of multicategories of planar diagrams and tensor categories
• KMS weights on groupoid and graph C*-algebras
• KMS states on the C*-algebras of finite graphs

Cited by in corpus (1)

• Free transport for finite depth subfactor planar algebras