Classification of Nuclear C*-algebras and (Noncommutative) Dynamical Systems

University Of Puerto Rico, San Juan PR

Investigators

Abstract

Abstract Pasnicu The ASH algebras (respectively AH algebras) are C*-algebras arising as inductive limits of finite direct sums of subalgebras (respectively corner subalgebras) of matrix algebras over unital,commutative C*-algebras.A C*-algebra is said to have the ideal property if each ideal is generated by projections.The investigator proposes to classify a large class of nuclear ASH algebras with the ideal property and also to classify the AH algebras with the ideal property.He also proposes to work on a conjecture which states that "many" nuclear,separable C*-algebras with the ideal property which are the crossed product of a unital,commutative C*-algebra or of an AF algebra by the integers is an ASH algebra in the above class.This project is related to Elliott's program of the classification of the separable,nuclear C*-algebras and to a problem of Effros and could have an impact in operator algebras but also in ergodic theory,in the study of the (noncommutative) dynamical systems and in geometry. C*-algebras could be thought as collections of infinite matrices of numbers endowed with an interesting algebraic and topological structure. The C*-algebras have significant applications to other parts of mathematics (geometry,topology,ergodic theory),to parts of physics (quantum mechanics and statistical mechanics) or to other sciences (the structure of DNA and other molecules).A complete classification ("enumeration") of a special class of operator algebras,called amenable von Neumann algebras,was given by Connes in his Fields Medal winning work. This project has two main goals.One is to classify ("enumerate") large classes of amenable (nuclear) C*-algebras with the ideal property (an interesting technical condition) which are defined by a particular construction ("inductive limits").The other one is to show that many amenable C*-algebras with the ideal property arising from a completely different and natural construction ("crossed products") belong in fact to one of the above classes (of "inductive limits") that the investigator proposes to classify ("enumerate").This project could have an important impact in several mathematical fields including operator algebras, dynamical systems,geometry and also in some domains outside mathematics (e.g. in quantum physics).