Mathematical Sciences: Studies in Semigroups of Endomorphisms, Operator Algebras, and Numerical Quantum Mechanics

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Arveson will investigate several areas of non-commutative analysis involving von Neumann algebras and C*-algebras, quantized index theory, and the role of the non-commutative spheres of Bratteli, Elliott, Evans and Kishimoto in numerical quantum mechanics. The research will involve the index theory and classification theory of semigroups of endomorphisms of factors, the structure of the C*-algebras associated to such semigroups, and the properties of C*-algebras associated with the discretized canonical commutation relations. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.