On Haar weight on von Neumann algebra of locally compact group (Q1333673)

We continue the investigation of two special classes of normal weights on von Neumann algebras: locally finite and regular weights. They had been introduced by one of the authors in connection with problems of the non- commutative integration. In the first section we obtain certain necessary conditions of local finiteness and regularity which are sufficient for the weight on semi-finite algebra. In the second section we show by means of these conditions that the canonical Haar weight of the left von Neumann algebra of a locally compact group is locally finite or regular if and only if the proper group is unimodular. In particular, this result implies that the problem of taking the positively defined convolutional square root of every positively defined function belonging to the Fourier algebra is solvable for the unimodular groups, only.