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
for allA∈ℳ andt>0. Therefore one can define a contraction semigroupT on ℋ=\(\overline {\mathcal{M}\Omega } \) by
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
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
5. results similar to 1 and 2 apply to |τ| but the τ-invariant state ω is |τ|-ergodic if, and only if,
for all normal states ω′.
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Communicated by R. Jost
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Robinson, D.W. Strongly positive semigroups and faithful invariant states. Commun.Math. Phys. 85, 129–142 (1982). https://doi.org/10.1007/BF02029138
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DOI: https://doi.org/10.1007/BF02029138