Existence and Density Theorems for Stochastic Maps on Commutative C*-algebras

This paper presents theoremes on the structure of stochastic and normalized positive linear maps over commutative C*-algebras. We show how strongly the solution of the n-tupel problem for stochastic maps relates to the fact that stochastic maps of finite rank are weakly dense within stochastic maps in case of a commutative C*-algebra. We give a new proof of the density theorem and derive (besides the solution of the n-tupel problem) results concerning the extremal maps of certain convex subsets which are weakly dense. All stated facts suggest application in Statistical Physics (algebraic approach), especially concerning questions around evolution of classical systems.