Pure inductive limit state and Kolmogorov's property-II

Details Pure inductive limit state and Kolmogorov's property-II Pure inductive limit state and Kolmogorov's property-II

2011-01-31

arxiv.org/abs/1101.5961v1

Mathematics

Operator Algebras

Scientific paper

In this paper we continue our investigation for an useful necessary and sufficient condition for purity of an inductive limit state of a $C^*$-dynamical systems of injective endomorphisms. In our earlier paper [Mo2], we have proved that Kolmogorov's property of the state is a sufficient condition for purity of the state and found it's relation with asymptotic behavior of the associated Markov map on the corner algebra determined by the support projection of the state in the $C^*$- dynamical system of endomorphisms. However such a criteria is not necessary. Here we will refine our investigation and prove that a weaker Kolmogorov's type of asymptotic criteria is not only sufficient but also necessary for purity. Further such a result has a ready generalization to inductive limit states associated with a $C^*$-dynamical systems of a commuting family of endomorphisms.

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