Analytic and Geometric Aspects of Operator Algebras

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Abstract Anderson The principal investigator will continue to develop a geometric spectral theory for n-tuples of self-adjoint operators in finite von Neumann algebras. One goal is to use this new approach to try to develop a coherent theory of the joint spectrum of such operators. Another aim is to use it to deepen our understanding of the structure of such operator algebras as irrational rotation algebras and von Neumann algebras determined by free groups. This project is concerned with the study of certain aspects of operator algebras. These objects have proved fruitful in the study of a wide variety of questions, including problems in such fields as mathematical physics, quantum mechanics and knot theory. The principal investigator and his coworkers have discovered a new way to view certain analytic data (eigenvalues of matrices and their infinite dimensional analogues - spectra of operators). A major goal of the project is to solve a difficult problem (joint spectrum for non-commuting operators), which thus far has withstood repeated vigorous attacks. It is well known that eigenvalues and other spectral data are essential in understanding many real world problems in engineering and science. Thus, it is possible that this project could produce methods for attacking questions in applied mathematics that have not been previously accessible.