C*-algebras from planar algebras I: Canonical C*-algebras associated to a planar algebra

Hartglass, Michael; Penneys, David

doi:10.1090/tran/6781

C$^$-algebras from planar algebras I: Canonical C$^$-algebras associated to a planar algebra
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by Michael Hartglass and David Penneys PDF

Trans. Amer. Math. Soc. 369 (2017), 3977-4019

Abstract:

From a planar algebra, we give a functorial construction to produce numerous associated C$^*$-algebras. Our main construction is a Hilbert C$^*$-bimodule with a canonical real subspace which produces Pimsner-Toeplitz, Cuntz-Pimsner, and generalized free semicircular C$^*$-algebras. By compressing this system, we obtain various canonical C$^*$-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$^*$-algebras associated to a planar algebra.

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