Projective space of a C*-module

Let X be a right Hilbert C*-module over A. We study the geometry and the to pology of the projective space P(X) of X, consisting of the orthocomplement ed submodules of X which are generated by a single element. We also study t he geometry of the P-sphere Sp(X) and the natural fibration Sp(X) --> P(X), where Sp(X) = {x is an element of X: [x, x] = p}, for p is an element of A a projection. The projective space and the p-sphere axe shown to be homoge neous 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 .