tailieunhanh - Báo cáo toán học: "Uniform ergodic theorems for identity preserving Schwarz maps on W*-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Thống nhất định lý ergodic nhận dạng bảo quản Schwarz bản đồ trên W *- đại số. | J. OPERATOR THEORY J 1 1984 395-404 Copyright by INCREST 1984 UNIFORM ERGODIC THEOREMS FOR IDENTITY PRESERVING SCHWARZ MAPS ON IT -ALGEBRAS ULRICH GROH 1. A bounded linear operator T on a Banach space E is called uniformly ergodic 1 1 resp. strongly ergodic if the averages M T - y Tk converge in the uniform n k 0 operator topology resp. strong operator topology . The limit p of the sequence Af T satisfies p PT TP p2 and is called the ergodic projection associated with T. It follows that p is a projection of E onto the fixed space F T x G E Tx x of T. Obviously every uniformly ergodic operator is strongly ergodic with the same ergodic projection. But in general there is no further connection between these two concepts. For example every contraction on a Hilbert space is strongly ergodic 13 Theorem whereas a contraction is uniformly ergodic iff 1 is a pole of the resolvent R - T 5 VII 9 Proposition on p. 223 . An important class of uniformly ergodic operators is formed by the quasi-compact operators . those T for which there exists a sequence Kn of compact operators such that lim T x 0 5 . But in general even a uniformly ergodic n operator with finite-dimensional fixed space is not quasi-compact for an example see 9 p. 224 . In this paper we study identity preserving Schwarz maps on unital C -algebras sd . those T6 j jaZ such that 7T 1 and T xxS 5s T x T x . In 7 it is proved that such a map is uniformly ergodic with finite-dimensional fixed space iff it is quasi-compact. This surprising result has great influence on the peripheral spectrum of T see 7 for details . In this paper we prove uniform ergodic theorems for identity preserving Schwarz maps T on Jf -algebras y z. We give a short outline of our results In Section two we study the ultrapower of the predual z of yZZ and the structure of the fixed space F Tf if T possesses a preadjoint T 6 -Z . The key for our uniform ergodic theorems is Lemma . 396 ULRICH GROH We show that an identity .