$C^*$-algebras from planar algebras I: canonical $C^*$-algebras associated to a planar algebra
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2014
- DOI:
- 10.48550/arXiv.1401.2485
- arXiv:
- arXiv:1401.2485
- Bibcode:
- 2014arXiv1401.2485H
- Keywords:
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- Mathematics - Operator Algebras;
- Mathematics - Quantum Algebra;
- 46L37;
- 46L05 (Primary);
- 46L54 (Secondary)
- E-Print:
- 47 pages, many figures