Von Neumann Algebras and Wavelets

Abstract

Orthonormal wavelets can be regarded as complete wandering vectors for a system of bilateral shifts acting on a separable infinite dimensional Hilbert space. The local (or “point”) commutant of a system at a vector ψ is the set of all bounded linear operators which commute with each element of the system locally at ψ. In the theory we shall develop, we will show that in the standard one-dimensional dyadic orthonormal wavelet theory the local commutant at certain (perhaps all) wavelets ψ contains non-commutative von Neumann algebras. The unitary group of such a locally-commuting von Neumann algebra parameterizes in a natural way a connected family of orthonormal wavelets. We will outline, as the simplest nontrivial special case, how Meyer’s classical class of dyadic orthonormal wavelets with compactly supported Fourier transform can be derived in this way beginning with two wavelets (an interpolation pair) of a much more elementary nature. From this pair one computes an interpolation von Neumann algebra. Wavelets in Meyer’s class then correspond to elements of its unitary group. Extensions of these results and ideas are also discussed.