Operator Algebras, Operator Theory and Applications

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Abstract Larson This project has three areas of emphasis. The first focuses on problems related to the structural theory of non-selfadjoint operator algebras. Prior work by the investigator on quasitriangularity considerations within von Neumann algebras has impacted some work of others outside the area in applied control theory and in function theory. He will continue this direction of research. The second ongoing project concerns a new functional- analytic approach to some basic issues in wavelet theory. The PI has shown that certain classical orthonormal wavelets could be derived using operator-algebraic techniques, and he also proved the existence of single-function dyadic orthonormal wavelets in all dimensions greater than one. This was a surprise to some other researchers because it contradicted a wavelet folklore which indicated that such wavelets were impossible. He has also shown that the unitary group of a von Neumann algebra can in some cases be used to parameterize a norm-path-connected family of wavelets. This suggests the possibility of perturbation techniques. The third thrust extends prior work of this investigator and collaborators in which counterexamples were obtained to some old open problems concerning operator-algebraic reflexivity and related properties of single operators. In prior supported work this investigator solved, with collaborators, a conjecture in the area of non-selfadjoint operator algebras that had been posed about ten years earlier by another researcher. This led to an unsuspected development in the applied area of control theory which impacted work of others outside of mathematics. He will continue pursuing this line of research. In a second ongoing direction, this investigator and a former student have shown that some aspects of wavelet theory are amenable to operator algebraic computations of a nature that were previously unsuspected. Wavelet analysis and wavelet oriented technology has been the scene of a tremendous research drive in mathematics and engineering during the past few years. Primary applications have been to signal processing and data compression. They showed that certain classical wavelets could be derived using their techniques, and they also proved the existence of certain wavelets in higher dimensions which were previously thought to be impossible by many specialists. In a third direction the PI has recently extended some prior work in which counterexamples were obtained to some old open problems concerning properties of single operators. Several doctoral students have been involved in all of this work.