Operator Algebras and Representations

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Abstract Jorgensen Spectpal pairs were studied first by Palle E.T. Jorgensen and Steen Pedersen in connection with their work on the Fuglede conjecture, but they turned out later to also have connections to wavelet theory, fractal iterations, Julia sets, and quasi-crystals. The harmonic analysis which is needed will be project 1 of the proposal, and it is based on analysis of certain representations of the Cuntz algebras, and on endomorphisms of von Neumann algebras. This representation theory is further expected to be useful in connection with the study of concrete geometric tiling questions. But the representations needed are type III (generically), while some of the methods from earlier work are type I. Nonetheless they have already proved amenable for adaptation to the infinite case. Jorgensen also plans to collaborate with his colleague Florin Radulescu on dynamical systems of endomorphisms which arise in Berezin quantization. The index theory which is available for the time-independent systems is likely to help us understand the time-dependent case which is the result of the operator algebraic approach to Berezin quantization. The proposal is in the interface of basic mathematics with its technology applications. The latter refer to mathematical tools from harmonic analysis and wavelet theory which are applied in current data-compression algorithms. These application spin-offs include "fingerprint electronic storage" and high-resolution television; but the proposal is on the theoretical side of the equation. There are two basic operations from mathematics (translation and scaling) which go into the filters that govern algorithms, and the repetition in the recursive process refers to the preservation of the same minimal data from one step to the next. Hence the term "filter bank". Surprisingly, and noted by the proposer in recent research papers, these filter-constructions may be understood as representations of the kinds of algebras which form the basis for the proposed researc h.