9400899 Spielberg It is proposed to study C*-algebras obtained from boundary actions of Fuchsian groups. These C*-algebras have interesting relations with certain dynamical systems known as Smale spaces. Smale spaces are a ubiquitous feature in dynamics. The relation with Fuchsian groups arises from the hyperbolicity of the dynamics, and promises to yield fine topological structure not accessible by techniques of ergodic theory. It is also proposed to continue the study of upper and lower spectral multiplicity in the spectrum of a C*-algebra. These notions were introduced by Archbold as tools to understand trace functionals on C*-algebras,and to analyze the structure of liminal ideals. Applications are invisioned to the study of C*-algebras of groups and groupoids. An operator is an infinite dimensional version of a matrix. These operators can be added and multiplied in a natural way and consequently collections of operators can be consider as an algebra of operators. This project deals with operator algebras that arise in an interesting way from dynamical systems. The important branch of mathematics called dynamical systems attempts to understand the repeated action of transformations. The successful completion of this project will be to the mutual benefit of the abstract theory of operator algebras and the important applied area of dynamical systems. ***
