Completeness of Inner Product Spaces Induced by States on Jordan and C au-Algebras

We show that a state phi on a Jordan algebra oe" induces complete inner product space if and only if phi is a convex combination of pure states. Inner product spaces generated by Type I (n) factor states and states on spin factors are described. We initiate study of completely positive maps in this connection by showing that pure completely positive map on a C (au)-algebra gives always complete inner product space in the Stinespring construction.