Strongly positive semigroups and faithful invariant states

Abstract

Let (Jf9 τ, ω) denote a FF*-algebra Jί, a semigroup ί>0κτ ( of linear maps of Jί into Jί, and a faithful τ-invariant normal state ω over Jί. We assume that τ is strongly positive in the sense that τt(A*A)^τt{A)*τt(A) for all AeJί and ί>0. Therefore one can define a contraction semigroup T on = τt(A)Ω, AeJί, where Ω is the cyclic and separating vector associated with ω. We prove 1. the fixed points Ji(τ) of τ are given by Jί(τ) = Jί'r\T' = Jί'c\E where E is the orthogonal projection onto the subspace of T-invariant vectors, 2. the state ω has a unique decomposition into τ-ergodic states if, and only if, Jί(τ) or {JiyjE}' is abelian or, equivalently, if (Jί, τ, ω) is R+-abelian, 3. the state ω is τ-ergodic if, and only if, MKJE is irreducible or if inf ||ω-ω||=0 ωeCoω'oτ