Operator Algebras and Dynamics

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In the 1980's, A. Connes introduced noncommutative spaces, and began to develop noncommutative geometry, a rapidly growing new area of mathematics which contributes to many disciplines in mathematics and physics. A large source of examples of noncommutative spaces is given by the quotient of equivalence relations. With I.F. Putnam, C.F. Skau and more recently H. Matui, I have extensively studied the noncommutative spaces associated to minimal homeomorphisms on a Cantor set. We plan to pursue the study of orbit equivalence for larger class of minimal actions. Connes and Woods introduced a new property of ergodic theory called approximate transitivity (AT) to characterize a class of von Neumann algebras. They noted also that the Poisson boundary of some random walks is AT. Recently D.E. Handelman and I defined dimension spaces, a measure theoretic version of dimension groups. Using this new notion, we will continue the study of AT(k)_actions, a generalized verson of approximate transitivity and of the Poisson boundary of matrix-valued random walks. One of most important recent development in the theory of C*-algebras is Elliott's classification scheme of amenable C*-algebras. His starting point was an example by Blackadar of a non approximately finite (AF) sub C*-algebra of an AF algebra. I want to study the real C*-algebra equivalent to both Blackadar's example and Elliott's program. Several properties of von Neumann algebras and C*-algebras translate in properties of their unitary groups. I plan to continue to study and extend this correspondence for both C*-algebras and von Neumann algebras. The notion of sofic groups, introduced by Gromov and Weiss, was extended to measurable actions of groups and to measurable equivalence relations. In this research I will study properties of the full group asociated to sofic equivalence relations