This research is a continuation of the previous project titled ``Operator K-theory methods in geometry, topology, harmonic analysis''. The main objectives of the present project are applications of operator K-theory to a number of well known problems in geometry, topology, representation theory and index theory of elliptic operators. These problems are the Baum-Connes conjecture in K-theory of group C*-algebras, the positive Ricci curvature conjecture in relation to the Dirac operator on a loop space and the problem of constructing a useful index theory of elliptic operators on infinite-dimensional manifolds. The Baum-Connes conjecture part of the project consists of two different problems: the Baum-Connes conjecture for discrete groups which act properly and affine-isometrically on Lp spaces and the Baum-Connes conjecture for discrete arithmetic groups. Concerning index theory on infinite-dimensional manifolds, some non-rigorous rudiments of it already exist in theoretical physics, but providing a solid basis for this theory will be another problem of the present project. The positive Ricci curvature conjecture is a well-known topological conjecture on the vanishing of the so called Witten genus. The main approach to it is by the use of index theory on infinite-dimensional manifolds.
The development of quantum mechanics in the 20th century has led to the creation of the theory of operators in Hilbert space and later to the creation of the theory of C*-algebras. The theory of C*-algebras is closely related with the contemporary field theory in physics, as well as with geometry, topology, group representation theory. Operator K-theory came to the C*-algebra theory from topology and grew out into a highly powerful tool. Nowadays, operator K-theory is part of the non-commutative geometry which is a synthesis of analysis and geometry with wide applications in many areas of mathematics and theoretical physics. The emphasis of the present project is on the applications of operator K-theory to several well known problems in topology, geometry, and infinite-dimensional analysis. For instance, constructing index theory in infinite dimensions, which is part of our project, will provide the basis for numerous applications in mathematics and theoretical physics. Operator K-theory methods will be the main tool for this research.
The K-theory of Banach algebras is a general technical tool which allows to obtain qualitative and quantitative information in analysis, geometry and topology. For example, the Novikov conjecture, which has its origins in the classification theory of compact smooth manifolds has been interpreted in the language of K-theory of Banach algebras and successfully solved in many cases. A different, but very much related example is the Baum-Connes conjecture, which essentially originates in the group representation theory. It suggests the way to describe unitary group representations using topological tools. Most of the results on these conjectures were obtained so far using Hilbert space methods. The research conducted in this project gives a new approach: we suggest to extend the class of Banach algebras from algebras acting on Hilbert spaces to algebras acting on L^p Banach spaces. A significant step in this direction was previously made by V. Lafforgue about 15 years ago. But our approach constitutes a new step, introducing a new K-theory tool, Banach E-theory, and extending the statement of the Novikov and the Baum-Connes conjectures to cover algebras of operators on L^p Banach spaces. The result on the L^p Baum-Connes conjecture obtained in this research will certainly have applications in the representation theory of discrete groups, where very few results are known so far.
