Vol. 79, No. 1, 1978

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Type analysis of the regular representation of a nonunimodular group

Colin Eric Sutherland

Vol. 79 (1978), No. 1, 225–250

This paper is concerned with finding necessary and sufficient conditions for the von Neumann algebra (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 HB) of the maximal unimodular subgroup H = kernel δG. Thus (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 (G) to have a summand of type IIIλ, λ (0,1].

Mathematical Subject Classification 2000
Primary: 22D25
Secondary: 46L45
Received: 22 April 1977
Published: 1 November 1978
Colin Eric Sutherland