A HILBERT SPACE APPROACH TO APPROXIMATE DIAGONALS FOR LOCALLY COMPACT QUANTUM GROUPS

For a locally compact quantum group G, the quantum group algebra L-1(G) is operator amenable if and only if it has an operator bounded approximate diagonal. It is known that if L-1(G) is operator biflat and has a bounded approximate identity then it is operator amenable. In this paper, we consider nets in L-2(G) which suffice to show these two conditions and combine them to make an approximate diagonal of the form omega W'*xi circle times eta where W is the multiplicative unitary and xi circle times eta are simple tensors in L-2(G) circle times L-2(G). Indeed, if L-1(G) and L-1((G) over cap) both have a bounded approximate identity and either of the corresponding nets in L-2(G) satisfies a condition generalizing quasicentrality then this construction generates an operator bounded approximate diagonal. In the classical group case, this provides a new method for constructing approximate diagonals emphasizing the relation between the operator amenability of the group algebra L-1(G) and the Fourier algebra A(G).