Mathematical Sciences: On the Classification of the Simple, Stably Finite, Separable, Amenable C-Algebras

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DMS-9622250 PI: Gong The ASH algebras (or AH algebras, respectively) are the C*-algebras arising as inductive limits of direct sums of subalgebras (or corner subalgebras, respectively) of matrix algebras over commutative C*-algebras. These algebras play an essential role in the Elliott program for the classification of separable amenable C*-algebras. The investigators propose to construct ASH (or AH) inductive limit decompositions for C*-algebras with certain general properties --- and, in particular, to prove the conjecture that any simple separable amenable C*-algebra of real rank zero and stable rank one is an AH algebra. They also propose to classify arbitrary simple ASH algebras in terms of the K-theory and tracial state space invariants (used, in special cases, by Elliott and others including the investigators). The passage from a finite to an infinite number of degrees of freedom in quantum physics led to the mathematical theory of certain infinite dimensional algebras, called operator algebras. A complete enumeration (or classification) of a special class of operator algebras, called (amenable) von Neumann algebras, was eiven by A. Connes in his Fields Medal winning work of 1975. (The Fields Medal is the equivalent of the Nobel Prize in mathematics.) More recently, V. F. R. Jones has applied the theory of von Neumann algebras to study ordinary knots. This work, which also won a Fields Medal, has shed light on quantum physics, and on the structure of DNA. The investigators propose to work on a complete enumeration (or classification) of another important class of operator algebras, called amenable C*-algebras. The successful classification of such C*-algebras will be essential for us to understand the infinite dimensional world of quantum physics.