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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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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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