$C^*$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 PimsnerToeplitz, CuntzPimsner, and generalized free semicircular $C^*$algebras. By compressing this system, we obtain various canonical $C^*$algebras, including DoplicherRoberts algebras, GuionnetJonesShlyakhtenko algebras, universal (Toeplitz)CuntzKrieger 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 eprints
 Pub Date:
 January 2014
 arXiv:
 arXiv:1401.2485
 Bibcode:
 2014arXiv1401.2485H
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Quantum Algebra;
 46L37;
 46L05 (Primary);
 46L54 (Secondary)
 EPrint:
 47 pages, many figures