The Baum-Connes Conjecture and Coarse Geometry

 Roughly speaking coarse space is a set X together with a family of subsets of the product of X with itself, called enourages, which play the role of the sets of pairs of points in a metric space of distance R or less apart, where R is a natural number. In a coarse space one can speak of a family of pairs of points being uniformly boundedly separated, but one does not have available a notion of convergence: there is no small-scale structure to a course space, only large-scale structure. It is possible to associate to a coarse geometric space a C*-algebra - this construction is due mostly to John Roe. The Baum-Connes conjecture for coarse geometric spaces proposes a formula for the K-theory of this C*-algebra. The formula is inspired by closely related conjectures in controlled topology. A key feature of the coarse Baum-Connes conjecture is that in many cases the Baum-Connes conjecture for the coarse space underlying a discrete group implies the injectivity of the Baum-Connes assembly map for the group itself. This in turn has powerful topological consequences like the Novikov conjecture. This has led to a great deal of progress on the Novikov conjecture, including most notably Yu's theorem that the Novikov conjecture holds for groups which admit a uniform embedding into Hilbert space

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