Compact quantum group actions on C*-algebras and invariant derivations

Dumitru, Raluca; Peligrad, Costel

doi:10.1090/S0002-9939-07-09011-9

Compact quantum group actions on C*-algebras and invariant derivations
HTML articles powered by AMS MathViewer

by Raluca Dumitru and Costel Peligrad

Proc. Amer. Math. Soc. 135 (2007), 3977-3984

DOI: https://doi.org/10.1090/S0002-9939-07-09011-9

Published electronically: July 27, 2007

PDF | Request permission

Abstract:

We define the notion of invariant derivation of a C*-algebra under a compact quantum group action and prove that in certain conditions, such derivations are generators of one parameter automorphism groups.

References

Saad Baaj and Georges Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de $C^*$-algèbres, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 4, 425–488 (French, with English summary). MR 1235438, DOI 10.24033/asens.1677
Julien Bichon, An De Rijdt, and Stefaan Vaes, Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys. 262 (2006), no. 3, 703–728. MR 2202309, DOI 10.1007/s00220-005-1442-2
Florin P. Boca, Ergodic actions of compact matrix pseudogroups on $C^*$-algebras, Astérisque 232 (1995), 93–109. Recent advances in operator algebras (Orléans, 1992). MR 1372527
Ola Bratteli and Derek W. Robinson, Operator algebras and quantum-statistical mechanics. II, Texts and Monographs in Physics, Springer-Verlag, New York-Berlin, 1981. Equilibrium states. Models in quantum-statistical mechanics. MR 611508, DOI 10.1007/978-3-662-09089-3
Ola Bratteli, Derivations, dissipations and group actions on $C^*$-algebras, Lecture Notes in Mathematics, vol. 1229, Springer-Verlag, Berlin, 1986. MR 871870, DOI 10.1007/BFb0098817
F. Goodman and P. E. T. Jørgensen, Unbounded derivations commuting with compact group actions, Comm. Math. Phys. 82 (1981/82), no. 3, 399–405. MR 641770, DOI 10.1007/BF01237047
C. Peligrad, Derivations of $C^{\ast }$-algebras which are invariant under an automorphism group, Topics in modern operator theory (Timişoara/Herculane, 1980), Operator Theory: Advances and Applications, vol. 2, Birkhäuser, Basel-Boston, Mass., 1981, pp. 259–268. MR 672827
Piotr Podleś, Symmetries of quantum spaces. Subgroups and quotient spaces of quantum $\textrm {SU}(2)$ and $\textrm {SO}(3)$ groups, Comm. Math. Phys. 170 (1995), no. 1, 1–20. MR 1331688, DOI 10.1007/BF02099436
Shôichirô Sakai, On one-parameter subgroups of $^*$-automorphisms on operator algebras and the corresponding unbounded derivations, Amer. J. Math. 98 (1976), no. 2, 427–440. MR 412824, DOI 10.2307/2373896
Shuzhou Wang, Ergodic actions of universal quantum groups on operator algebras, Comm. Math. Phys. 203 (1999), no. 2, 481–498. MR 1697607, DOI 10.1007/s002200050622
S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613–665. MR 901157, DOI 10.1007/BF01219077
S. L. Woronowicz, Compact quantum groups, Symétries quantiques (Les Houches, 1995) North-Holland, Amsterdam, 1998, pp. 845–884. MR 1616348