THE GRASSMANN MANIFOLD OF A C-STAR ALGEBRA

A study is made of the geometry of the Grassmann manifold of a C*-algebra. The curvature of the canonical connection on this Grassmann manifold is calculated. It is shown that the Grassmann manifold admits a natural complex structure which is parallel with respect to the canonical connection. A study is made of the properties of geodesics of that canonical connection. It is shown that a connected component of the Grassmann manifold has the property that every length-minimising curve is a geodesic if and only if the restriction of the norm of the C*-algebra to the tangent spaces of that connected component gives rise to a Riemannian metric on that component, and if this condition is satisfied then that connected component is holomorphically isometric to the complex projective space of some Hilbert space.