RUI: Quantum Dynamical Systems and Marginal States

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This proposal is directed towards the investigation of some problems on dynamical systems on von Neumann algebras, and is related to recent work by a number of authors on the subject of semigroups of endomorphisms on von Neumann algebra factors. This research project involves a number of areas of mathematics, including operator algebras, finite fields, number theory, and combinatorics. In contrast to semigroups of automorphisms, a semigroup of endomorphisms may be viewed as a dynamical system that may proceed forward but not backward in time. It is remarkable how challenging this subject has proven to be in light of the relatively simpler theory of one-parameter semigroups of automorphisms on a type I factor. A principal goal of this proposal is to make additional progress in the classification of one-parameter semigroups of unital endomorphisms, analogous to Wigner's characterization of groups of automorphisms acting on factors of type I. The study of operator algebras traces its origins back to the work of von Neumann and others. Their goal was to construct mathematical models that capture the behavior of quantum mechanical systems, and to use these models to make predictions about the time evolution of such systems. As knowledge has grown and techniques in the field have been refined, connections have been established between operator algebras and a number of other areas of mathematics and science. This proposal involves connections among the fields of operator algebras, commutative algebra, number theory, and combinatorics. The principal objects of study in this project are known as binary shifts on a certain type of operator algebra and are defined using bitstreams of 0's and 1's such as one studies in the theory of linear recurring sequences. A major goal of this project is to complete the classification of the binary shifts and to relate this classification to the analysis of linear recurring sequences. Binary shifts will also be studied for their potential applications to the theory of quantum dynamical systems, specifically those systems that may proceed forward, but not backward, in time.