The Eulerian Bratteli Diagram and Traces on Its Associated Dimension Group

Abstract

In this thesis we present two important closely related examples of Bratteli diagrams: the Pascal triangle and the Eulerian Bratteli diagram. The former is well-known and related to binomial coefficients. The latter, which is the main object of the thesis, is related to the Eulerian numbers. Bratteli diagrams were introduced in 1972 by Ola Bratteli in his study of approximately finite dimensional (AF) C*-algebras. In 1976, George Arthur Elliott associated to an AF C*-algebra or to a corresponding Bratteli diagram an ordered group, he called dimension group.

In the first part of the thesis we study the space of infinite paths of the Eulerian diagram, and we realize it as a projective limit of finite permutation groups. In the second part, we study the state space of the dimension group associated to the Eulerian Bratteli diagram. It is a compact convex set and we describe its extremal points. Finally, we use this description to give a necessary and sufficient condition for an element of this dimension group to be positive.