A generalized spectral duality theorem

Abstract

We establish a version of the spectral duality theorem relating the point spectrum of a family of*-representations of a certain covariance algebra to the continuous spectrum of an associated family of*-representations. Using that version, we prove that almost all the images of any element of a certain space of fixed points of some*-automorphism of an irrational rotation algebra via standard*-representations of the algebra inl 2ℤ do not have pure point spectrum over any non-empty open subset of the common spectrum of those images. As another application of the spectral duality theorem, we prove that if almost all the Bloch operators associated with a real almost periodic function on ℝ have pure point spectrum over a Borel subset of ℝ, then almost all the Schrödinger operators with potentials belonging to the compact hull of the translates of this function have, over the same set, purely continuous spectrum.