This paper is concerned with
finding necessary and sufficient conditions for the von Neumann algebra ℳ(G)
generated by the left regular representation λG of a locally compact, separable,
non-unimodular group G to be type I, semifinite, or to have a central summand of
type III. In the case where the modular function δG of G has closed range, we
are able to give a complete solution in terms of the orbit structure of the
natural action of G on the reduced quasi-dual (ΓH,μB) of the maximal
unimodular subgroup H =kernelδG. Thus ℳ(G) is semifinite if and only if the
action is smooth with isotropy subgroup H, and of type III0 if and only if
the action is completely nonsmooth. Conditions of a similar type are given
which are necessary and sufficient for ℳ(G) to have a summand of type
IIIλ, λ ∈ (0,1].
