Editing System of imprimitivity (section)

== 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 based on subspaces of ''H'' will not work; for example the translation representation of '''R''' on ''L''<sup>2</sup>('''R''') has no system of imprimitivity in this sense. The right formulation of direct sum decomposition is formulated in terms of [[projection-valued measure]]s.

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 (mathematics)|group action]]

:<math> G \times X \rightarrow X, \quad (g,x) \mapsto g \cdot x. </math>

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''<sub>''g''</sub> of ''G'' on ''H''.
* A [[projection-valued measure]] π on the Borel sets of ''X'' with values in the projections of ''H'';
which satisfy

:<math> U_g \pi(A) U_{g^{-1}} = \pi(g \cdot A). </math>

===Example ===

Let ''X'' be a standard ''G'' space and μ a σ-finite countably additive ''invariant'' measure on ''X''. This means

:<math> \mu(g^{-1} A) = \mu(A) \quad </math>

for all ''g'' ∈ ''G'' and Borel subsets ''A'' of ''G''.

Let π(''A'') be multiplication by the indicator function of ''A'' and ''U''<sub>''g''</sub> be the operator

:<math> [U_g \psi] (x) =\psi(g^{-1} x).\quad </math>

Then (''U'', π) is a system of imprimitivity of (''G'', ''X'') on ''L''<sup>2</sup><sub>μ</sub>(''X'').

This system of imprimitivity is sometimes called the ''Koopman system of imprimitivity''.