Mathematical Sciences: Semigroup of Endomorphisms of Operator Algebras

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9500291 Arveson The research proposed here addresses problems related to the structure and classification of semigroups of automorphisms of operator algebras and their relationship to several areas of modern analysis, quantum physics, and probability theory. A principal component of the project is to establish a connection between certain semigroups and continuous tensor products of Hilbert spaces. more precisely, the classification of E_0-semigroups up to cocycle conjugacy was shown to be equivalent to the problem of classifying these continuous tensor product systems. In turn, the latter structures give rise to computable invariants of various types: numerical (an index invariant), topological (a non-commutative topological space, the spectral C*-algebra). Moreover, such continuous tensor products occur naturally in probability theory; for example, they can be associated with random distributions having stationary independent increments, and with other martingale-type processes as well. A secondary project is to continue the proposer's recent work on the role of operator algebras in the problem of computing the spectra of infinite dimensional self-adjoint operators. The general area of mathematics of this project has its basis in the theory of algebras of operators on Hilbert space. 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 seemingly 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. ***