The Normal State Space of ß (H)

Abstract

In this chapter we will study the normal state space K of the algebra ß(H) of all bounded operators on a Hilbert space H. In the first section we will investigate the facial structure of K and explain how it is related to the Grassmannian. In the second section we will introduce the geodesic metric for ∂ _eK,which is related to the transition probability in physics. In the third section we will prove some basic facts about *-isomorphisms and *anti-isomorphisms, which are important for our later study of orientation of state spaces. The fourth section will be our first encounter with the concept of orientation, here only for the state space of the 2 * 2-matrix algebra. But this case already holds the key to understanding the general case, as we shall see later on.