Differential geometry of partial isometries and partial unitaries

Abstract

Let $\a$ be a C$^*$-algebra. In this paper the sets $\ii$ of partial isometries and $\ii_\Delta\subset\ii$ of partial unitaries (i.e., partial isometries which commute with their adjoints) are studied from a differential geometric point of view. These sets are complemented submanifolds of $\a$. Special attention is paid to geodesic curves. The space $\ii$ is a homogeneous reductive space of the group $U_\a \times U_\a$, where $U_\a$ denotes the unitary group of $\a$, and geodesics are computed in a standard fashion. Here we study the problem of the existence and uniqueness of geodesics joining two given endpoints. The space $\ii_\Delta$ is \emph{not} homogeneous, and therefore a completely different treatment is given. A principal bundle with base space $\ii_\Delta$ is introduced, and a natural connection in it defined. Additional data, namely certain translating maps, enable one to produce a \emph{linear} connection in $\ii_\Delta$, whose geodesics are characterized.