Positive semigroups in ordered Banach spaces

Abstract

In this chapter, we deal with one-parameter positive semigroups in ordered Banach spaces. Firstly, we discuss the notion of ideally ordered Banach spaces and uniformly order convex Banach spaces. Both classes include L^p-spaces (1 ≤ p < ∞) as well as preduals of von Neumann algebras. We prove several theorems about positive semigroups in such Banach spaces. Then we consider positive semigroups in Banach lattices and investigate several types of asymptotic regularity of these semigroups. In the last section of this chapter, we deal with relations between the geometry of Banach lattices and mean ergodicity of bounded positive semigroups in them.