Geometry of positive operators and uhlmannʹs approach to the geometric phase

In Uhlmannʹs description of the differential geometry of the space Ω of density operators, a relevant role is played by the parallel condition ω*ω = ω*ω, where ω is a lifting of a curve γ in Ω, i.e. ω(t)ω(t)* = γ(t) for all t. In this paper we get a principal bundle with a natural connection over the space G+ of all positive invertible elements of a C*-algebra such that the parallel transport is ruled by Uhlmannʹs parallel equation.