Abstract
In this paper we study unique ergodicity of C ∗ -dynamical system ( A , T ) , consisting of a unital C ∗ -algebra A and a Markov operator T : A ↦ A , relative to its fixed point subspace, in terms of Riesz summation which is weaker than Cesaro one. Namely, it is proven that ( A , T ) is uniquely ergodic relative to its fixed point subspace if and only if its Riesz means 1 p 1 + ⋯ + p n ∑ k = 1 n p k T k x converge to E T ( x ) in A for any x ∈ A , as n → ∞ , here E T is an projection of A to the fixed point subspace of T. It is also constructed a uniquely ergodic entangled Markov operator relative to its fixed point subspace, which is not ergodic.
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