Mathematical Sciences: Crossed Product Algebras, Non-Commutatuve Duality, and Applications to Group Representations

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This project is mathematical research in the theory of operator algebras. These objects turn up in some formulations of quantum mechanics, where they (or rather their selfadjoint part) represent the observables of the system being modeled. One of the basic constructions in this area is the formation of the crossed product from a group action on a given algebra. (In physics, the group would most likely be the real numbers, acting on the algebra of observables by time translation to give the dynamics of the system.) Recently, a dual construction called the crossed coproduct has turned out to be very useful, especially when the group in question is nonabelian. Professor Gootman will continue his investigations of crossed products and coproducts, particularly with an eye to studying various properties of these algebras (ideal structure, dual topology, type, etc.) by interrelating the use of non- commutative duality theory, non-abelian harmonic analysis, and representation theory.