ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
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′.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02029138
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