Hopf fibration in a C^*‐module^†

Abstract

Let X be a right C*‐module over a unital C*‐algebra A$ {\cal A}$. We study the Hopf fibration of X: frakturh:XP→P1(X)goodbreak=projective space of0.33emboldX,$$\begin{equation*} \mathfrak {h}: {\bf X} _ {\cal P} \rightarrow P1({\bf X}) = \hbox{projective space of } {\bf X} , \end{equation*}$$where the projective space of X is the set of singly generated orthocomplemented submodules of X, boldXscriptP$ {\bf X} _ {\cal P}$ is the set of elements of X, which generate such submodules, and frakturhfalse(boldxfalse)=$\mathfrak {h}({\bf x} )=$module generated by boldx∈XP$ {\bf x} \in {\bf X} _ {\cal P}$. The group of unitary operators of the module X acts on both spaces. We introduce a Finsler metric in boldXscriptP$ {\bf X} _ {\cal P}$, which is invariant under the unitary action. Our main results establish that the map h$\mathfrak {h}$ is distance decreasing (when the projective space of X is considered with its natural unitary invariant metric), and a minimality result in boldXscriptP$ {\bf X} _ {\cal P}$, characterizing metric geodesics in this space.