  The interplay between the invariant subspace theory and spectral synthesis
for locally compact abelian group discovered by Arveson is extended to include
other topics as harmonic analysis for Varopoulos algebras and approximation by
projection-valued measures. We propose a ''coordinate'' approach which
nevertheless does not use the technique of pseudo-integral operators, as well
as a coordinate free one which allows to extend to non-separable spaces some
important results and constructions of [W.Arveson, Operator Alegebras and
Invariant subspaces, Ann. of Math. (2) 100 (1974)] and solve some problems
posed there.
