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 [Formula: see text] 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 Sp(X) and the natural fibration [Formula: see text], where Sp(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 ℒA(X) of adjointable operators of X. The homotopy theory of these spaces is examined.