Mathematical Sciences: Analytic and Geometric Aspects of Operator Algebras

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This research is in the area of operator algebras and consists of three projects. First, is a study of the problem of uniqueness of extension of complex homomorphisms on maximal abelian diagonal subalgebras to pure states of the algebra. Second, the research will involve work on the Index theory of inclusions of arbitrary factors and the study of the 'chi' invariant for finite factors. Third, the research will continue previous work on the inclusions of von Neumann algebras which involve group-like objects called 'quantized groups'. This research concerns the general theory of operators on a space. One of the fundamental concepts in physics is the notion of the symmetry groups of operators on a space. The most important groups arise as the symmetries of structured spaces, such as the symmetric, linear, and unitary groups from a set, linear space, and Hilbert space, respectively. In quantum physics, the space is replaced by a noncommutative algebra of operators on a Hilbert space. The quantum groups, and among them the deformations of classical groups, have proved important in several areas of physics and mathematics.