The theory of pure operations

Abstract

This paper continues the study started in [13] where classes of operations were investigated in the partially ordered vector space approach to the theory of statistical physical systems. In this approach the set of states is represented by a norm closed generating coneK in a complete base norm space (V, K, B) and the set of operations is represented by the setP of positive norm non-increasing linear operators onV. In actual physical experiments it is usually the case that only certain subsets ofK are available and it is supposed that the set Γ(K) of such subsets is the set of split faces ofK. The properties of two important classes of operation are examined. The first classP of strong operations has the property that each member leaves every element of Γ(K) invariant and therefore can be measured in every restricted situation. The second classPP of pure operations has the property above and also sends pure states into pure states. A study is made, in terms of the structure of Γ(K), of when such operations are physically relevant. The paper ends with an examination of Γ(K),P, PP in the Von Neumann algebra model.