Mathematical Sciences: Operator Algebras and Noncommutative Topology

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Abstract Blackadar The investigators propose to continue work on a range of problems concerning the structure of operator algebras and noncommutative algebraic topology. Specific topics to be studied are: structure of inductive limits of C*-algebras, structure of groupoids and their C*-algebras, nonstable K-theory and the fundamental comparability question, and existence and properties of cohomology theories on C*-algebras. If one takes the point of view that a space may be studied via the commutative C*-algebra of complex-valued continuous functions on the space, then the study of noncommutative C*-algebras is a natural generalization. Furthermore, there are numerous situations in ordinary geometry and topology where the natural object of study is a "singular space" which often cannot be studied directly in purely topological terms, but where there is a noncommutative C*-algebra which plays the exact role of the algebra of functions, and where C*-algebra theory has been a crucial tool in advancing the understanding of the topological situation. There has been extensive work in recent years in generalizing notions of algebraic topology, and more recently related notions of differential geometry, to C*-algebras. This work has already paid off spectacularly with fundamental advances in the understanding of the structure of C*-algebras, as well as the beautiful and deep applications in geometry, topology, and mathematical physics. Perhaps the most important result, though, is simply the bridge which has been constructed between operator algebras and topology, and the broad new level of understanding of the interconnections. Virtually all of the research the proposers have done in recent years fits into this general picture of noncommutative topology. The specific projects proposed are continuations or natural outgrowths of previous work, taking into account the way the subject as a whole has developed.