Interpolation operators associated with sub-frame sets

Interpolation operators associated with wavelets sets introduced by Dai and Larson play an important role in their operator algebraic approach to wavelet theory. These operators are also related to the von Neumann subalgebras in the ``local commutant' space, which provides the parametrizations of wavelets. It is a particularly interesting question of how to construct operators which are in the local commutant but not in the commutant. Motivated by some questions about interpolation family and C*-algebras in the local commutant, we investigate the interpolation partial isometry operators induced by sub-frame sets. In particular we introduce the -congruence domain of the associated mapping between two sub-frame sets and use it to characterize these partial isometries in the local commutant. As an application, we obtain that if two wavelet sets have the same -congruence domain, then one is a multiresolution analysis (MRA) wavelet set which implies that the other is also an MRA wavelet set.