Studies in noncommutative dynamics and multivariable operator theory

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AbstractAversonThis work relates to two loosely connected areas, a) the theory of one-parameter groups of automorphisms of the algebra of all operators on a Hilbert space which posses a certain causal structure, and b) the theory of commuting n-tuples of operators acting on a Hilbert space. We have been attempting to understand the nature of noncommutative dynamics for many years, and our approach recently has been based on the theory of E_0-semigroups. There has been exciting recent progress on several fronts, the full implications of which are still being sorted out. In multivariable operator theory, we have introduced a "curvature" invariant which is somewhat analogous to the integral of curvature of a Riemannian manifold. We have shown that it is an integer in many cases by relating it to another invariant, the Euler characteristic of a certain finitely generated module. However, the key formula is valid only in certain cases, and we are now in the process of relating the curvature invariant to a more subtle integer invariant, essentially the index of a "Dirac operator" that can be associated with the given n-tuple of operators.In quantum theory the observable quantities are represented by operators. The algebra of operators differs sharply from the algebra of numbers because the result of multiplying two operators A and B depends on the order in which they are multiplied: AB is not the same as BA. This failure of the commutativity law has profound consequences, the most basic one being the uncertainty principle. The flow of time in quantum theory is represented by certain transformations, each of which moves operators in subtle ways, and the dynamics of quantum theory is the study of such groups of transformations. This "noncommutative dynamics" is very different from the commutative dynamics of classical physics. Our approach is based on a certain notion of causality, which involves the technical idea of semigroups of endomorphisms. Recent progress has been very encouraging - with the discovery of new connections with probability theory, in which one may now pass back and forth between noncommutative flows of time and certain random processes which can be thought of as "off-white" noise, in that they are close to white noise, but not exactly white noise. In a different but related direction, our work on sets of (commuting) operators establishes significant connections between sets of operators and fundamental geometric ideas such as curvature. At issue is a numerical invariant for sets of operators. While this invariant appears to be a real number capable of taking on any value, it is in fact an integer. What integer? The answer that appears to be emerging now is that this number is the numerical index of a certain Dirac operator. There appear to be significant connections with other parts of mathematics, including Riemannian geometry and algebraic geometry.