Differential geometry of partial isometries and partial unitaries (Q1425771)

For a \(C^*\)-algebra \(A\), the sets \(I\) of partial isometries and \(I_\Delta\subset I\) of partial unitaries (i.e., partial isometries that commute with their adjoints) are submanifolds of \(A\). The space \(I\) is a homogeneous reductive space of the group \(U_A\times U_A\), where \(U_A\) is the unitary group of \(A\), and geodesics are computed in a standard fashion. A condition is given for existence of geodesics joining two endpoints. A principal bundle with base \(I_\Delta\) is introduced and a natural connection is defined. Additional data enable the authors to construct a linear connection on \(I_\Delta\) and to characterize its geodesics.