Infinite Dimensional Systems of Imprimitivity
To generalize the finite dimensional definition given in the preceding section, a suitable replacement for the set X of vector subspaces of H which is permuted by the representation U is needed. As it turns out, a naïve approach base on subspaces of H will not work; for example the translation representation of R on L2(R) has no system of imprimitivity in this sense. The right formulation of direct sum decomposition is formulated in terms of projection-valued measures.
Mackey's original formulation was expressed in terms of a locally compact second countable (lcsc) group G, a standard Borel space X and a Borel group action
We will refer to this as a standard Borel G-space.
The definitions can be given in a much more general context, but the original setup used by Mackey is still quite general and requires fewer technicalities.
Definition. Let G be a lcsc group acting on a standard Borel space X. A system of imprimitivity based on (G, X) consists of a separable Hilbert space H and a pair consisting of
- A strongly-continuous unitary representation U: g → Ug of G on H.
- A projection-valued measure π on the Borel sets of X with values in the projections of H;
which satisfy
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