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


Authors: Michael Hartglass and David Penneys
Journal: Trans. Amer. Math. Soc. 369 (2017), 3977-4019
MSC (2010): Primary 46L05, 46L37; Secondary 46L54
DOI: https://doi.org/10.1090/tran/6781
Published electronically: October 27, 2016
MathSciNet review: 3624399
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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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Additional Information

Michael Hartglass
Affiliation: Department of Mathematics, University of California, Riverside, Riverside, California 92521

David Penneys
Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90045-1555
Address at time of publication: Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, Ohio 43210-1174
Email: penneys.2@osu.edu

DOI: https://doi.org/10.1090/tran/6781
Received by editor(s): June 9, 2014
Received by editor(s) in revised form: May 31, 2015
Published electronically: October 27, 2016
Article copyright: © Copyright 2016 by the authors

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