February Fourier Talks 2015

In January 1997, Marc Rieffel gave a talk at a special session of the Joint Annual Meetings entitled "Multiwavelets and operator algebras", which related the multiresolution analysis theory of wavelets theory to the K-theory of the (commutative) torus. In his talk, he discussed a way to construct nested sequences of Hilbert modules over continuous functions on the torus. In 2003 and 2004 Rieffel and I developed the notion of projective multiresolution analyses further; some of our results were related to function spaces first studied by G. Zimmermann. Since then, there have been a variety of attempts to generalize the theory of projective multiresolution analysis to Hilbert modules over noncommutative C*-algebras. This talk will discuss recent developments along these lines, including the construction of B. Purkis of projective multiresolution analyses over irrational rotation algebras, and the construction of projective multiresolution structures for noncommutative solenoids. In this construction, we let {C_n} be a nested sequence of unital C*-algebras, with direct limit algebra C that preserves the unit, and we let X be a finitely generated projective C-module. We then give the definition of a projective multiresolution structure for the pair (C, X) in terms of a nested sequence of C_n modules satisfying appropriate conditions. We will discuss applications to abstract frame theory by giving examples from the theory of noncommutative solenoids. This latter construction is joint work with F. Latrémoliére of the University of Denver.