Wavelets and Hilbert Modules

Abstract

A Hilbert C*-module is a generalization of a Hilbert space for which the inner product takes its values in a C*-algebra instead of the complex numbers. We use the bracket product to construct some Hilbert C*-modules over a group C*-algebra which is generated by the group of translations associated with a wavelet. We shall investigate bracket products and their Fourier transform in the space of square integrable functions in Euclidean space. We will also show that some wavelets are associated with Hilbert C*-modules over the space of essentially bounded functions over higher dimensional tori.