Differential geometry of partial isometries and partial unitaries

Abstract

Let a be a C^*-algebra. In this paper the sets I of partial isometries and I_Δ ⊂ I 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 I is a homogeneous reductive space of the group U_A x 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 I_Δ is not homogeneous, and therefore a completely different treatment is given. A principal bundle with base space I_Δ is introduced, and a natural connection in it defined. Additional data, namely certain translating maps, enable one to produce a linear connection in I_Δ, whose geodesics are characterized.