A note on noncommutative unique ergodicity and weighted means

2008/03/03 by Luigi Accardi, Farrukh Mukhamedov, Accardi, Luigi +1
Mathematics · #46L35 #46L55 #47A35 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.0803.0073

openalex publication_date 2008/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study unique ergodicity of C^*-dynamical system (\ga,T), consisting of a unital C^*-algebra \ga and a Markov operator T:\ga↦\ga, relative to its fixed point subspace, in terms of Riesz summation which is weaker than Cesaro one. Namely, it is proven that (\ga,T) is uniquely ergodic relative to its fixed point subspace if and only if its Riesz means equation* (1)/(p₁+...+p_n)∑_k=1ⁿp_kT^kx equation* converge to E_T(x) in \ga for any x∈\ga, as n→∞, here E_T is an projection of \ga 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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