A note on noncommutative unique ergodicity and weighted means

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 {mapping} 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 meansfrac(1, p₁ + ⋯ + p_n) underover(∑, k = 1, n) p_k T^k xconverge 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.