OPERATOR BIPROJECTIVITY OF COMPACT QUANTUM GROUPS

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.