System of Imprimitivity - Infinite Dimensional Systems of Imprimitivity

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 → U_g of G on H.
• A projection-valued measure π on the Borel sets of X with values in the projections of H;

which satisfy