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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Actions of compact quantum groups
on $C^*$-algebras

Author: Marcin Marciniak
Journal: Proc. Amer. Math. Soc. 126 (1998), 607-616
MSC (1991): Primary 22D25; Secondary 46L60, 81R50
MathSciNet review: 1415332
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Abstract: In this paper we investigate a structure of the fixed point algebra under an action of compact matrix quantum group on a $C^*$-algebra $\mathcal{B}$. We also show that the categories of $\mathcal C$-comodules in $\mathcal B$ and inner endomorphisms restricted to the fixed point algebra coincide when the relative commutant of the fixed point algebra is trivial. Next we show a version of the Tannaka duality theorem for twisted unitary groups.

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  • [1] Araki H., Carey A.L., Evans D.E., On $O_{n+1}$, J. Operator Theory 12 (1984), 247-264 MR 86g:46082
  • [2] Bratteli O., Evans D.E., Derivations tangential to compact groups: the non-abelian case, Proc. London Math. Soc. 52 (1986), 369-384 MR 87f:46123
  • [3] Cuntz J., Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), 173-185 MR 57:7189
  • [4] Doplicher S., Roberts J.E., Duals of Compact Lie Groups Realized in the Cuntz Algebras and Their Actions on $C^*$-Algebras, J. Funct. Anal. 74 (1987), 96-120 MR 89a:22011
  • [5] Doplicher S., Roberts J.E., Compact group actions on $C^*$-algebras, J. Operator Theory 19 (1988), 283-305 MR 90f:46104
  • [6] Doplicher S., Roberts J.E., A new duality theory for compact groups, Invent. Math. 98 (1989), 157-218 MR 90k:22005
  • [7] Doplicher S., Roberts J.E., Why There is a Field Algebra with a Compact Gauge Group Describing the Superselection Structure in Particle Physics, Comm. Math. Phys. 131 (1990), 51-107 MR 91k:81082
  • [8] Fannes M., Nachtergaele B., Werner R.F., Quantum Spin Chains with Quantum Group Symmetry, Comm. Math. Phys. 174 (1996), 477-507 CMP 96:07
  • [9] Fredenhagen K., Rehren K.-H., Schroer B., Superselection Sectors with Braid Group Statistics and Exchange Algebras I, Comm. Math. Phys. 125 (1989), 201-226 MR 91c:81047
  • [10] Fredenhagen K., Rehren K.-H., Schroer B., Superselection Sectors with Braid Group Statistics and Exchange Algebras II, Rev. Math. Phys. - Special Issue (1992), 113-157 MR 94g:81119
  • [11] Haag R., Local Quantum Physics, Springer-Verlag, Berlin Heidelberg 1992 MR 94d:81001
  • [12] Konishi Y., Nagisa M., Watatani Y., Some remarks on actions of compact matrix quantum groups on $C^*$-algebras, Pacific J. Math. 153 (1992), 119-127 MR 93c:46121
  • [13] Mack G., Schomerus V., Conformal Field Algebras with Quantum Symmetry from the Theory of Superselection Sectors, Comm. Math. Phys. 134 (1990), 139-196 MR 92i:81298
  • [14] Mack G., Schomerus V., Quasi Hopf quantum symmetry in quantum theory, Nucl. Phys. B 370 (1992), 185-230 MR 93c:81081
  • [15] Paschke W.L., The crossed product of a $C^*$-algebra by an endomorphism, Proc. Amer. Math. Soc. 80 (1980), 113-118 MR 81m:46081
  • [16] Powers R.T., Price G., Derivations Vanishing on $S(\infty )$, Comm. Math. Phys. 84 (1982), 439-447 MR 83k:46056
  • [17] Price G., Extremal Traces on Some Group-Invariant $C^*$-Algebras, J. Funct. Anal. 49 (1982), 145-151 MR 84e:46075
  • [18] Størmer E., Symmetric States of Infinite Tensor Products of $C^*$-algebras, J. Funct. Anal. 3 (1969), 48-68 MR 39:3327
  • [19] Wang S., Free Products of Compact Quantum Groups, Comm. Math. Phys. 167 (1995), 671-692 MR 95k:46104
  • [20] Wang S., Tensor products and crossed products of compact quantum groups, Proc. London Math. Soc. 71 (1995), 695-720 MR 96i:46074
  • [21] Wenzl H., Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92 (1988), 349-383 MR 90b:46118
  • [22] Woronowicz S.L., Twisted $SU(2)$ group. An example of a non-commutative differential calculus, R.I.M.S. Publ. Kyoto University 23 (1987), 117-181 MR 88h:46130
  • [23] Woronowicz S.L., Compact Matrix Pseudogroups, Comm. Math. Phys. 111 (1987), 613-665 MR 88m:46079
  • [24] Woronowicz S.L., Tannaka-Krein duality for compact matrix pseudogroups. Twisted $SU(N)$ groups, Invent. Math. 93 (1988), 35-76 MR 90e:22033
  • [25] Woronowicz S.L., Compact quantum groups, preprint, University of Warsaw.

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Additional Information

Marcin Marciniak
Affiliation: Institute of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952 Gdańsk, Poland

Received by editor(s): May 7, 1996
Received by editor(s) in revised form: July 16, 1996, and August 8, 1996
Additional Notes: The author was supported by KBN grant 2 P301 020 07
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

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