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 Moebius
transformations known from complex geometry. By specifying A to be an algebra
of bounded operators in a Hilbert space H, we rediscover the Moebius group
defined 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 different lifts forms a category, whose morphisms are given by
generalized Moebius tranformations. We present an example of a polarized module
canonically associated with the differentiable 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.