Topological and Related Aspects of the Structure of C* - Algebras

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Abstract Phillips The Principal Investigator, N. Christopher Phillips, proposes three lines of research: (1) work on the classification and structure of simple C*-algebras; (2) a functional analytic characterization of the algebra of smooth functions on a smooth manifold; and (3) a continuation of his previous work on exponential rank. In (1), he proposes to work with Qing Lin on understanding the structure of transformation group C*-algebras of minimal diffeomorphisms. The eventual goal is to show that they are direct limits of sub-homogeneous C*-algebras, which would put them close to the classifiable class of stably finite simple C*-algebras. Several already interesting intermediate results, such as cancellation results in the K-theory of these algebras, are closer to realization. He also proposes to follow up recent work on the purely infinite case of the classification in several ways: a possibility of interesting invariants in the non-nuclear case, a long shot possibility for proving that nuclearity implies the Universal Coefficient Theorem, and a very plausible approach to the real case of the known classification theorem. In (2), he proposes to try to prove a functional analytic characterization of the algebra of smooth functions on a smooth manifold, in an effort to gain a better understanding of what a noncommutative manifold should be. In (3), he proposes to search for a simple C*-algebra with large exponential rank, and to try to understand better the exponential rank of stable and homogeneous C*-algebras. The purpose of this project is to improve the understanding of he "simple" C*-algebras. A C*-algebra is a kind of algebraic system (somewhat like the set of real numbers, with its operations of addition, subtraction, multiplication, and division, but somewhat more complicated). It has additional structure which, roughly speaking, describes when something is "large" or "small" (again, somewhat like the set of real numbers). C*-algebras turn out to be one of the more impor tant kinds of structures in mathematics. They have significant applications to other parts of mathematics which at first sight seem rather unrelated (geometry, for example), and they are also one of the kinds of structure that is important in quantum mechanics, the (rather counterintuitive) physical theory needed to deal properly with atoms and other very small objects. The simple C*-algebras are those that cannot be broken into smaller pieces, and in some sense all C*-algebras are built out of them. The project has two main goals. One is to advance the understanding of the "internal structure" of simple C*-algebras, more or less to know about each one all that it is possible to know. This is relevant when they are used in other subjects. The other is to improve the knowledge of the classification of C*-algebras: one wants a complete list of all of them, with a recipe for deciding when two simple C*-algebras, obtained in different ways, are actually the same.