Vol. 87, No. 2, 1980

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Subobjects of virtual groups

Arlan Bruce Ramsay

Vol. 87 (1980), No. 2, 389–454

Suppose a locally compact group G (always second countable) has a Borel action on an analytic Borel space S so that each element of G transforms a given measure μ into an equivalent measure. If S0 is the coset space for a closed subgroup H, then there is a natural action of G on S0 which comes from translations of G on itself and there is such a quasi-invariant measure. Thus it is reasonable to think of such a space (S,μ), for some purposes, as a generalized sort of subgroup, or a virtual subgroup of G. In fact, the set S × G can be given algebraic and measure-theoretic structure so that many of the procedures used with subgroups can be carried over to this general setting. There is a general notion of virtual group, not necessarily “contained in” a group, which can be derived from this, and it turns out to include equivalence relations with suitable measures as a special case. These virtual groups appear in studying group representations, operator algebras, foliations, etc. Since there is a general setting for virtual groups, it seems desirable to see whether the intuitive idea of an action of a group as representing a subobject fits into this framework in a compatible way. The purpose of this paper is to show that “images” under homomorphisms, “kernels”, etc. do fit together properly.

Mathematical Subject Classification 2000
Primary: 22D40
Secondary: 28C10
Received: 13 October 1978
Revised: 8 January 1979
Published: 1 April 1980
Arlan Bruce Ramsay