Topological orbit equivalence, amenability, approximate transitivity, antiautomorphisms, unitary groups and operator algebras

点击翻译按钮获取中文摘要

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.  I.F. Putnam, C.F. Skau and I have extensively studied the noncommutative spaces associated with minimal homeomorphisms on a Cantor set. In collaboration with H. Matui, we plan to pursue the study of orbit equivalence for larger classes of minimal actions.Connes and Woods defined a new property of ergodic theory called approximate transitivity (AT) to characterize a class of von Neumann algebras. They also noted that the Poisson boundary of some random walks is AT. Recently D.E. Handelman and I developed dimension spaces, a measure theoretic version of dimension groups. We will continue the study of the Poisson boundary of random walks using dimension spaces and will apply it to von Neumann algebras.One of most important recent developments in the classification of C*-algebras is Elliott's classification program of amenable C*-algebras. Its 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.Many properties of von Neumann algebras and C*-algebras translate in properties of their unitary groups, in some cases considered as discrete or as topological groups. I plan to continue the study of such correspondences for classes of both C*-algebras and with Ping Ng of von Neumann algebras.