## Weak ergodicity of nonhomogeneous Markov chains on noncommutative $$L^1$$-spaces.(English)Zbl 1276.47012

Let $$M$$ be a von Neumann algebra with faithful normal finite trace $$\tau$$. Let $$L^1 (M,\tau )$$ be an $$L^1$$-space associated with $$M$$. The set of all self-adjoint elements in $$M$$ is denoted by $$M_sa$$. Let $$T: L^1 (M,\tau ) \rightarrow L^1 (M,\tau )$$ be a linear bounded operator, $$X=\{ x\in L^1 (M_sa ,\tau ) : \tau (x)=0 \}$$, $$\delta (T)= \sup_{x\in X, x\neq 0} \frac {||Tx||_1}{||x||_1}$$. $$\delta (T)$$ is called the Dobrushin ergodicity coefficient of $$T$$. The author establishes several properties of the Dobrushin ergodicity coefficient in a noncommutative setting. These results allow to investigate the weak ergodicity of nonhomogeneous discrete Markov processes (NDMP) by means of the ergodicity coefficient. Sufficient conditions are given for such processes to satisfy the weak ergodicity. A necessary and sufficient conditions are given for the satisfaction of the $$L^1$$-weak ergodicity of NDMP. The author gives an example showing that $$L^1$$-weak ergodicity is weaker that weak ergodicity. He also gives several concrete examples of noncommutative NDMP.

### MSC:

 47A35 Ergodic theory of linear operators 28D05 Measure-preserving transformations
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### References:

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