  We study generalized filters that are associated to multiplicity functions
and homomorphisms of the dual of an abelian group. These notions are based on
the structure of generalized multiresolution analyses. We investigate when the
Ruelle operator corresponding to such a filter is a pure isometry, and then use
that characterization to study the problem of when a collection of closed
subspaces, which satisfies all the conditions of a GMRA except the trivial
intersection condition, must in fact have a trivial intersection. In this
context, we obtain a generalization of a theorem of Bownik and Rzeszotnik.
