## Operator biprojectivity of compact quantum groups

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- by Matthew Daws
- Proc. Amer. Math. Soc.
**138**(2010), 1349-1359 - DOI: https://doi.org/10.1090/S0002-9939-09-10220-4
- Published electronically: November 25, 2009
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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.## References

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## Bibliographic Information

**Matthew Daws**- Affiliation: School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom
- Email: matt.daws@cantab.net
- 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
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**138**(2010), 1349-1359 - MSC (2010): Primary 46L89, 46M10; Secondary 22D25, 46L07, 46L65, 47L25, 47L50, 81R15
- DOI: https://doi.org/10.1090/S0002-9939-09-10220-4
- MathSciNet review: 2578527