Hopf fibration in a C*‐module†

Abstract

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