Hopf fibration in a C*-module

Abstract

Let X be a right C*-module over a unital C*-algebra A. We study the Hopf fibration of X: h:X_p→ P1(X) =projective space of X, where the projective space of X is the set of singly generated orthocomplemented submodules of X, X_p is the set of elements of X, which generate such submodules, and h(x)= module generated by X. The group of unitary operators of the module X acts on both spaces. We introduce a Finsler metric in X_p, which is invariant under the unitary action. Our main results establish that the map h is distance decreasing (when the projective space of X is considered with its natural unitary invariant metric), and a minimality result in X_p, characterizing metric geodesics in this space.