Geometry of positive operators and Uhlmann's approach to the geometric phase

In Uhlmann's description of the differential geometry of the space Omega of density operators, a relevant role is played by the parallel condition ome ga*omega = (omega) over dot *omega, where omega is a lifting of a curve gam ma in Omega, i.e. omega (t)omega (t)* = gamma (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 tra nsport is ruled by Uhlmann's parallel equation.