Strongly positive semigroups and faithful invariant states

Abstract

Let (ℳ, τ, ω) denote aW*-algebra ℳ, a semigroupt>0↦τ _t of linear maps of ℳ into ℳ, and a faithful τ-invariant normal state ω over ℳ. We assume that τ is strongly positive in the sense that

$$\tau _t (A^ * A) \geqq \tau _t (A)^ * \tau _t (A)$$

for allA∈ℳ andt>0. Therefore one can define a contraction semigroupT on ℋ=$\overline {\mathcal{M}\Omega }$ by

$$T_t A\Omega = \tau _t (A)\Omega ,{\rm A} \in \mathcal{M},$$

where Ω is the cyclic and separating vector associated with ω. We prove

1. the fixed points ℳ(τ) of τ are given by ℳ(τ)=ℳ∩T′=ℳ∩E′, whereE is the orthogonal projection onto the subspace ofT-invariant vectors,

2. the state ω has a unique decomposition into τ-ergodic states if, and only if, ℳ(τ) or {ℳυE}′ is abelian or, equivalently, if (ℳ, τ, ω) is ℝ-abelian,

3. the state ω is τ-ergodic if, and only if, ℳυE is irreducible or if

$$\mathop {\inf }\limits_{\omega '' \in Co\omega 'o\tau } \left\| {\omega '' - \omega '} \right\| = 0$$

for all normal states ω′ where Coω′°τ denotes the convex hull of {ω′°τ _t } _t>0.

Subsequently we assume that τ is 2-positive,T is normal, andT* _t ℳ₊Ω$\subseteqq \overline {\mathcal{M}_ + \Omega }$, and then prove

4. there exists a strongly positive semigroup |τ| which commutes with τ and is determined by

$$\left| \tau \right|_t \left( A \right)\Omega = \left| {T_t } \right|A\Omega ,$$

5. results similar to 1 and 2 apply to |τ| but the τ-invariant state ω is |τ|-ergodic if, and only if,

$$\mathop {\lim }\limits_{t \to \infty } \left\| {\omega 'o\tau _t - \omega } \right\| = 0$$

for all normal states ω′.