Wavelets and Hilbert Modules

A Hilbert C∗-module is a generalisation 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. We shall examine in detail a construction employing Hilbert C-modules that relates to wavelet theory. A Hilbert C-module (also known as a Hilbert module or a B-rigged space) is in some ways similar to a Hilbert space except that the inner product takes its value in a C-algebra instead of the complex numbers. The C-algebra valued inner product that we shall use is sometimes known as the bracket product. The Hilbert Cmodule described here has its linear space contained in L(R). Much of the work in this paper was done during the author’s PhD [Wo]. The bracket product has been used in wavelet theory before, see for example [BDR], [Fi] and [BCMO]. The connection between Hilbert C-module theory and wavelet theory that is being investigated here was described in a talk given by M. A. Rieffel in 1997 [Ri4]. The material in [Ri4] has more recently been elaborated on in two papers by J. A. Packer and M. A. Rieffel [PR1, PR2]. In [PR1], a Hilbert C-module is constructed with a linear space contained in C(Z), and is used to study the properties of continuous filters. The Hilbert C-modules described in [PR2] are similar to the ones described here. The paper [PR2] contains some