Mm Obius Transformations in Noncommutative Conformal Geometry

We study the projective linear group PGL 2 (A) associated with an arbitrary algebra A and its subgroups from the point of view of their action on the space of involutions in A. This action formally resembles M obius transformations known from complex geometry. By specifying A to be an algebra of bounded operators in a Hilbert space H, we rediscover the Mobius group ev (M) de ned by Connes and study its action on the space of Fredholm modules over the algebra A. There is an induced action on the K-homology of A, which turns out to be trivial. Moreover, this action leads naturally to a simpler object, the polarized module underlying a given Fredholm module, and we discuss this relation in detail. Any polarized module can be lifted to a Fredholm module, and the set of di erent lifts forms a category, whose morphisms are given by generalized Mobius tranformations. We present an example of a polarized module canonically associated with the di erentiable structure of a smooth manifold V . Using our lifting procedure we obtain a class of Fredholm modules characterizing the conformal structures on V . Fredholm modules obtained in this way are a special case of those constructed by Connes, Sullivan and Teleman. Supported in part by the Visitor Fund of the Department of Mathematics, University of Exeter. y Supported in part by an LMS Scheme 4 grant and a grant from the Exeter University Research Fund.