System of Imprimitivity - Homogeneous Systems of Imprimitivity

Homogeneous Systems of Imprimitivity

A system of imprimitivity is homogeneous of multiplicity n, where 1 ≤ n ≤ ω if and only if the corresponding projection-valued measure π on X is homogeneous of multiplicity n. In fact, X breaks up into a countable disjoint family {X_n} _{1 ≤ n ≤ ω} of Borel sets such that π is homogeneous of multiplicity n on X_n. It is also easy to show X_n is G invariant.

Lemma. Any system of imprimitivity is an orthogonal direct sum of homogeneous ones.

It can be shown that if the action of G on X is transitive, then any system of imprimitivity on X is homogeneous. More generally, if the action of G on X is ergodic (meaning that X cannot be reduced by invariant proper Borel sets of X) then any system of imprimitivity on X is homogeneous.

We now discuss how the structure of homogeneous systems of imprimitivity can be expressed in a form which generalizes the Koopman representation given in the example above.

In the following, we assume that μ is a σ-finite measure on a standard Borel G-space X such that the action of G respects the measure class of μ. This condition is weaker than invariance, but it suffices to construct a unitary translation operator similar to the Koopman operator in the example above. G respects the measure class of μ means that the Radon-Nikodym derivative

is well-defined for every g ∈ G, where

It can be shown that there is a version of s which is jointly Borel measurable, that is

is Borel measurable and satisfies

for almost all values of (g, x) ∈ G × X.

Suppose H is a separable Hilbert space, U(H) the unitary operators on H. A unitary cocycle is a Borel mapping

such that

for almost all x ∈ X

for almost all (g, h, x). A unitary cocycle is strict if and only if the above relations hold for all (g, h, x). It can be shown that for any unitary cocycle there is a strict unitary cocycle which is equal almost everywhere to it (Varadarajan, 1985).

Theorem. Define

Then U is a unitary representation of G on the Hilbert space

Moreover, if for any Borel set A, π(A) is the projection operator

then (U, π) is a system of imprimitivity of (G,X).

Conversely, any homogeneous system of imprimitivity is of this form, for some measure σ-finite measure μ. This measure is unique up to measure equivalence, that is to say, two such measures have the same sets of measure 0.

Indeed much more can be said about the correspondence between homogeneous systems of imprimitivity and cocycles.

When the action of G on X is transitive however, the correspondence takes a particularly explicit form based on the representation obtained by restricting the cocycle Φ to a fixed point subgroup of the action. We consider this case in the next section.