Projective space of a C*-module

Abstract

Let X be a right Hilbert C^*-module over A. We study the geometry and the topology of the projective space P(X) 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 S_p(X) →P(X), 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.