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Operator biprojectivity of compact quantum groups

Author: Matthew Daws
Journal: Proc. Amer. Math. Soc. 138 (2010), 1349-1359
MSC (2010): Primary 46L89, 46M10; Secondary 22D25, 46L07, 46L65, 47L25, 47L50, 81R15
Published electronically: November 25, 2009
MathSciNet review: 2578527
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Abstract: Given a (reduced) locally compact quantum group $ A$, we can consider the convolution algebra $ L^1(A)$ (which can be identified as the predual of the von Neumann algebra form of $ A$). It is conjectured that $ L^1(A)$ is operator biprojective if and only if $ A$ is compact. The ``only if'' part always holds, and the ``if'' part holds for Kac algebras. We show that if the splitting morphism associated with $ L^1(A)$ being biprojective can be chosen to be completely positive, or just contractive, then we already have a Kac algebra. We give another proof of the converse, indicating how modular properties of the Haar state seem to be important.

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  • 1. O. Yu. Aristov, Amenability and compact type for Hopf-von Neumann algebras from the homological point of view. In Banach algebras and their applications, Contemp. Math., 363, 15-37. Amer. Math. Soc., Providence, RI, 2004. MR 2097947 (2006f:46055)
  • 2. O. Yu. Aristov, Biprojective algebras and operator spaces, J. Math. Sci. (New York) 111 (2002), 3339-3386. MR 1847551 (2003f:46090)
  • 3. E. Bédos, G. J. Murphy, and L. Tuset, Co-amenability of compact quantum groups, J. Geom. Phys. 40 (2001), 130-153. MR 1862084 (2002m:46100)
  • 4. E. Bédos, G. J. Murphy, and L. Tuset, Amenability and co-amenability of algebraic quantum groups. II, J. Funct. Anal. 201 (2003), 303-340. MR 1986692 (2004e:46085)
  • 5. E. Christensen and A. M. Sinclair, Module mappings into von Neumann algebras and injectivity, Proc. London Math. Soc. (3) 71 (1995), 618-640. MR 1347407 (96m:46107)
  • 6. E. G. Effros and Z.-J. Ruan, Operator spaces, London Math. Society Monographs, New Series, 23. The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753 (2002a:46082)
  • 7. M. Enock and J.-M. Schwartz, Kac algebras and duality of locally compact groups, Springer-Verlag, Berlin, 1992. MR 1215933 (94e:46001)
  • 8. U. Haagerup and M. Musat, Classification of hyperfinite factors up to completely bounded isomorphism of their preduals, J. Reine Angew. Math. 630 (2009), 141-176. MR 2526788
  • 9. A. Ya. Helemskii, The homology of Banach and topological algebras, translated from the Russian by Alan West. Mathematics and Its Applications (Soviet Series), 41. Kluwer Academic Publishers Group, Dordrecht, 1989. MR 1093462 (92d:46178)
  • 10. J. Kustermans, Locally compact quantum groups. In Quantum independent increment processes. I, Lecture Notes in Math., 1865. Springer, Berlin, 2005, 99-180. MR 2132094 (2006f:46073)
  • 11. J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), 837-934. MR 1832993 (2002f:46108)
  • 12. J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), 68-92. MR 1951446 (2003k:46081)
  • 13. A. Maes and A. Van Daele, Notes on compact quantum groups, Nieuw Arch. Wisk. (4) 16 (1998), 73-112. MR 1645264 (99g:46105)
  • 14. V. I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, 146. Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472 (88h:46111)
  • 15. Z.-J. Ruan and G. Xu, Splitting properties of operator bimodules and operator amenability of Kac algebras. In Operator theory, operator algebras and related topics (Timişoara, 1996), 193-216, Theta Found., Bucharest, 1997. MR 1728421 (2000j:46113)
  • 16. V. Runde, Biflatness and biprojectivity of the Fourier algebra, Arch. Math. (Basel) 92 (2009), 525-530. MR 2506954
  • 17. V. Runde, Lectures on amenability, Lecture Notes in Math., 1774. Springer-Verlag, Berlin, 2002. MR 1874893 (2003h:46001)
  • 18. P. M. Sołtan, Quantum Bohr compactification, Illinois J. Math. 49 (2005), 1245-1270. MR 2210362 (2007a:46079)
  • 19. M. Takesaki, Theory of operator algebras. I, reprint of the first (1979) edition. Encyclopaedia of Mathematical Sciences, 124. Operator Algebras and Non-commutative Geometry, 5. Springer-Verlag, Berlin, 2002. MR 1873025 (2002m:46083)
  • 20. M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, 125. Operator Algebras and Non-commutative Geometry, 6. Springer-Verlag, Berlin, 2003. MR 1943006 (2004g:46079)
  • 21. T. Timmermann, An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitaries and beyond. European Mathematical Society (EMS), Zürich, 2008. MR 2397671 (2009f:46079)
  • 22. J. Tomiyama, On the projection of norm one in $ W^*$-algebras, Proc. Japan Acad. 33 (1957), 608-612. MR 0096140 (20:2635)
  • 23. P. J. Wood, The operator biprojectivity of the Fourier algebra, Canad. J. Math. 54 (2002), 1100-1120. MR 1924715 (2003j:46085)
  • 24. S. L. Woronowicz, Compact quantum groups. In Symétries quantiques (Les Houches, 1995), 845-884. North-Holland, Amsterdam, 1998. MR 1616348 (99m:46164)
  • 25. S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665. MR 901157 (88m:46079)
  • 26. S. L. Woronowicz, Twisted $ {SU}(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), 117-181. MR 890482 (88h:46130)

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Additional Information

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

Keywords: Compact quantum group, biprojective, Kac algebra, modular automorphism group.
Received by editor(s): May 16, 2009
Received by editor(s) in revised form: July 12, 2009
Published electronically: November 25, 2009
Communicated by: Marius Junge
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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