A Note on Operator Biprojectivity of Compact Quantum Groups

Details A Note on Operator Biprojectivity of Compact Quantum Groups A Note on Operator Biprojectivity of Compact Quantum Groups

2009-05-12

arxiv.org/abs/0905.1935v1

Proc. Amer. Math. Soc. 138 (2010), 1349-1359.

Mathematics

Operator Algebras

11 pages

Scientific paper

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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