PROJECTIVE SPACE OF A C^*-MODULE

Let X be a right Hilbert C^*-module over A. We study the geometry and the topology of the projective space of X, consisting of the orthocomplemented submodules of X which are generated by a single element. We also study the geometry of the p-sphere S_p(X) and the natural fibration , where S_p(X) = {x ∈ X: <x, x> = p}, for p ∈ A a projection. The projective space and the p-sphere are shown to be homogeneous differentiable spaces of the unitary group of the algebra ℒ_A(X) of adjointable operators of X. The homotopy theory of these spaces is examined.