Asymptotic behavior of Markov semigroups on preduals of von Neumann algebras

Abstract

We develop a new approach for investigation of asymptotic behavior of Markov semigroup on preduals of von Neumann algebras. With using of our technique we establish several results about mean ergodicity, statistical stability, and constrictiviness of Markov semigroups.  2005 Elsevier Inc. All rights reserved.

A Survey on Invariance and Ergodicity of Quantum Markov Semigroups

“…The material presented in this chapter is largely based on the works done by Fagnola, Rebolledo or both (see [32][33][34]) and recent results obtained by Umanita ([68,69]) and Emel'syanov and Wolff [25].…”

“…While these equivalent conditions serve as characterization of mean ergodicity, they are very difficult to verify especially for infinite dimensional cases (see Umanita [68], [69]). To overcome this shortcoming, Emel'syanov and Wolff [25] introduced the quantum version of mean lower bound for positive quantum state and prove that (Theorem (6.25)) if the distance between Cesaro mean of any normal state can be made asymptotically closed to the order interval of a mean lower bouond element, then the QMS {T * t , t ≥ 0} is mean ergodic. Furthermore, if A is atomic, then the space of fixed points F(T * ) is finite-dimensional.…”

“…Theorem 6.25 (Emel'yanov and Wolff [25]) If there exists η 0 ∈ (A * ) + and δ ∈ R, 0 ≤ δ < 1 such that…”

“…Recent contributions in this area include works done by Carbone [8], Luczak [50], Fagnola [31], Rebolledo [56] or both (see [32][33][34][35][36]) and recent results obtained by Choi [10], and Umanita [68,69] and Emel'syanov and Wolff [25]. The main purpose of this article is to explore and survey recent results on invariances, mean ergodicity and mean ergodicity of the semigroup {T * t , t ≥ 0}.…”

“…The main purpose of this article is to explore and survey recent results on invariances, mean ergodicity and mean ergodicity of the semigroup {T * t , t ≥ 0}. The material presented in this survey is largely based on the works done by (see [32][33][34]) and recent results obtained by Umanita [68,69] and Emel'syanov and Wolff [25].…”