Operator biprojectivity of compact quantum groups

Daws, Matthew

doi:10.1090/S0002-9939-09-10220-4

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.

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