Sphere and projective space of a C*-algebra with a faithful state

Abstract

LetAbe a unitalC*-algebra with a faithful state?. We study the geometry of the unit sphereS?={x?A:?(x*x) = 1}and the projective spaceP?=S?/T. These spaces are shown to be smooth manifoldsand homogeneous spaces of the groupU?(A)of isomorphisms acting inAwhich preserve the inner productinduced by?, which is a smooth Banach-Lie group. An important role is played by the theory of operators inBanach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric inP?, and provethe existence of minimal geodesics, both with given initial data, and given endpoints.