Braided tensor C*-categories, Hecke symmetries

and actions on extended Cuntz algebras

Author:
Anna Paolucci

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2249-2258

MSC (1991):
Primary 46M05, 16W30, 81R50

DOI:
https://doi.org/10.1090/S0002-9939-99-04693-6

Published electronically:
April 16, 1999

MathSciNet review:
1476384

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we deal with braided tensor C*-categories. For every object of the category we associate a C*-algebra denoted by . An analysis of the braiding is carried out by using the conjugate equations. If the braiding is a Hecke symmetry and the -dimension is appropriately chosen, we characterize the C*-algebra as the one generated by the representation given by the Markov trace. This analysis leads to the existence of an action of on . Such actions (Theorem 1) correspond to *-monomorphisms of on which generalize the ones obtained earlier by the author (1997).

**1.**T. Banica, (1996), The quantization of and some related amenability questions from the fusion semiring viewpoint, preprint, Marseille.**2.**J. Cuntz, (1977) Simple C*-algebras generated by isometries, Comm. Math. Phys. 57, p. 173-185. MR**57:7189****3.**S. Doplicher, J.E. Roberts, (1989), A new duality theory for compact groups, Invent. math., 98, 157-218. MR**90k:22005****4.**S. Doplicher, J.E. Roberts, (1987), Duals of compact Lie groups realized in the Cuntz algebras, J. Functional Analysis, 74, 90-120. MR**98a:22011****5.**V.F.R. Jones, (1987), Hecke algebra representations of braid groups and link polynomials, Annals of Mathematics, 126, p. 335-388. MR**89c:46092****6.**A. Joyal, R. Street, (1986) Braided monoidal categories, Macquarie Mathematics report n. 860081.**7.**C. Kassel, (1994), Quantum Groups, Springer Verlag. MR**96:17041****8.**R. Longo, J. E. Roberts, (1995), Theory of dimension, preprint, Rome.**9.**A. Paolucci, (1996), Remarks on braided C*-categories and endomorphisms of C*-algebras, J. Operator Theory, 36, p. 157-177. MR**97m:46113****10.**A. Paolucci, (1997), Coactions of Hopf Algebras on Cuntz algebras and their fixed point algebras, Proc. Amer. Math. Soc., 125, p. 1033-1042. MR**97f:46106****11.**S.L. Woronowicz, (1987), Compact matrix pseudogroups, Comm. Math. Phys. 111, 613-665. MR**88m:46079****12.**S.L. Woronowicz, (1987), Twisted SU(2) group. An example of a non commutative differential calculus, Publ. RIMS 23, 117-181.**13.**S.L. Woronowicz, (1988), Tannaka-Krein duality for compact matrix pseudogroups, twisted groups, Inventiones Math., 93, p. 35-76. MR**90e:22033****14.**S.L. Woronowicz, (1993), Compact Quantum Group, preprint, Warsaw.**15.**D.N. Yetter, (1990), Quantum groups and representations of monoidal categories, Math. Proceedings Camb. Phil. Soc. 108, p. 261-290. MR**91k:16028****16.**H. Wenzl, (1988), Hecke algebras of type and subfactors, Invent. Math. 92, p. 349-383. MR**90b:46118**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46M05,
16W30,
81R50

Retrieve articles in all journals with MSC (1991): 46M05, 16W30, 81R50

Additional Information

**Anna Paolucci**

Affiliation:
School of Mathematics, University of Leeds, LS2 9JT, United Kingdom

Address at time of publication:
Dipartimento di Matematica, Università di Torino, via Carlo Alberto, 10, 10124 Torino, Italy

Email:
paolucci@dm.unito.it

DOI:
https://doi.org/10.1090/S0002-9939-99-04693-6

Keywords:
C*-algebras,
Hilbert spaces,
Hecke symmetries,
braided C*-categories.

Received by editor(s):
July 22, 1997

Received by editor(s) in revised form:
August 11, 1997, August 18, 1997, and September 11, 1997

Published electronically:
April 16, 1999

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1999
American Mathematical Society