Grant: Discovery Grants Program - Individual

Operator Algebras originated in the work of von Neumann, in his search for a mathematical framework for quantum mechanics. They are the noncommutative versions of measure theory, topology and differential geometry. The deep connections between Ergodic Theory and von Neumann algebras have recently found their counterpart in equally deep connections linking Topological Dynamics and C*-algebras. This proposal will deepen these connections by studying important classes of dynamical systems, will allow new applications of operator algebras to physics by classifying natural classes of real C*-algebras, and will study specific questions in the theory of C*-algebras. it will increase our fundamental knowledge in mathematics and will enable the development of applications to other sciences. Last but not least, the core of this research proposal is the training of young mathematicians, whose research will strengthen Canada's scientific capacity. Among the four main themes of this proposal, let me for lack of space present below two of them. The equivalence of amenability and hyperfiniteness for measured equivalence relations due to Connes, Feldman, and Weiss was inspired by Connes' celebrated work on the classification of injective factors. TTo study for minimal topological equivalence relations on the Cantor set the relation between amenability and affability, the topological counterpart of hyperfiniteness, is one objective of this proposal. In an earlier work with Matui, Putnam and Skau, we proved the affability of free minimal actions on the Cantor set of finitely generated free abelian groups. The Zd-odometers are an important class of topological dynamical systems. With Putnam and Skau, we recently proved that the first cohomology group of such Zd-action is a fundamental algebraic invariant. With tools developed for the classification of the class of approximately finite C*-algebras and used by Herman, Putnam and Skau in their construction of Bratteli-Vershik models, we will study the first cohomology group of a general free minimal action of Z2 on the Cantor set. Our long-term objective is to extend the Bratteli-Vershik construction to free minimal actions of other groups of homeomorphisms. The theory of operator algebras was mainly developed as algebras over the complex numbers, but to fully enable the applications of C*-algebras to physics, it is essential that we consider real (and graded) structures in many areas of C*-algebra theory. These applications in physics have become very important and a catalyst to the development of the classification of important classes of real C*-algebras. The remarkable classification of an important class of complex nuclear C*-algebras, the so-called Elliott program, has been obtained. To complete the classification of the nuclear real C*-algebras, satisfying the real version of Elliott program is a main objective of this proposal.