Index Theory and K-Theory for Operator Algebras

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9704001 Baum This research project centers on a conjecture formulated by P. Baum and A. Connes. If true, this conjecture gives an answer to the problem of understanding and calculating the K-theory of group C*-algebras. Validity of the conjecture implies validity for the Novikov conjecture (homotopy invariance of higher signatures), the stable Gromov-Lawson-Rosenberg conjecture (necessary and sufficient conditions for a closed Spin manifold to admit a Riemannian metric of positive scalar curvature), and the Kadison-Kaplansky conjecture (non-existence of idempotents in the reduced C*-algebra of a torsion-free discrete group). When applied to Lie groups, the conjecture makes precise the Mackey-Wigner analogy between the tempered representation theory of a semi-simple Lie group G and the representation theory of the semi-direct product Lie group that Mackey and Wigner associate to G. Thus the conjecture is unusual in that it cuts across several different areas of mathematics and unifies a number of problems and issues that previously appeared to be unrelated. A theme in nineteenth and twentieth century mathematics has been that certain features of mathematical systems that at first glance seem to be analytical (i.e., based on methods of calculus such as differentiation and integration) in fact are topological (i.e., depend only on elementary continuous geometry). This research project develops this theme within the new context of non-commutative geometry. A conjecture (formulated by P. Baum and A. Connes) will be studied that relates analytic and topological invariants of locally compact groups. If true, the conjecture will be an underlying principle revealing an unexpected unity in a number of mathematical problems and issues. ***