GEODESICS AND OPERATOR MEANS IN THE SPACE OF POSITIVE OPERATORS

The set A+ of positive invertible elements of a C*-algebra has a natural structure of reductive homogeneous manifold with a Finsler metric. Because pairs of points can be joined by uniquely determined geodesics and geodesics are "short" curves, there is a natural notion of convexity: C ⊂ A+ is convex if the geodesic segment joining a, b ∈ C is contained in C. We show that this notion is related to the classical convexity of real and operator valued functions. Several results about convexity are proved in this paper. The expressions of these results are closely related to the operator means of Kubo and Ando, in particular to the geometric mean of Pusz and Woronowicz, and they produce several norm estimations and operator inequalities.