Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên phổ ngoại vi của các nhà khai thác tích cực ergodic thống nhất trên C *- đại số. | J. OPERATOR THEORY 10 1983 31 -37 Copyright by INCREST 1983 ON THE PERIPHERAL SPECTRUM OF UNIFORMLY ERGODIC POSITIVE OPERATORS ON C -ALGEBRAS ULRICH GROH 1. Let 21 be a unital c -algebra and let T e 2l . T is called uniformly er-II-1 godic if the averages T n 1 X T - converge in the uniform operator topology to an operator p e 2I . Pisa projection onto the fixed space F T x 6 21 Tx x and TP PT p. We call p the ergodic projection associated with T. If Te ố 2l is positive . T 2I 2l where 21 is the positive cone of 2Í then the following holds see 6 Theorem 5 . Proposition. If T is a positive operator on a c -algebra the following assertions are equivalent a T is uniformly ergodic. b r T 1 and 1 is a pole of the resolvent R Ẳ T of order 1. c r T 1 and lim z 1 R 2 T exists in the uniform operator topology. If a positive operator is uniformly ergodic then the associated ergodic projection the residue of the resolvent at 1 and the limit in c are equal. For the notations concerning the spectrum a T spectral radius r T resolvent R Ấ T pole of the resolvent etc. we refer to the book of Dunford-Schwartz 3 . Suppose r r T is a pole of the resolvent for a positive operator on a c -al-gebra. Then r is a pole of maximal order on the peripheral spectrum ơ T n rT of T where r 2eC A 1 10 App. . Moreover if T is irreducible . leaves no non trivial closed face of 2l invariant see . 4 Definition then r has order one App. 10 App. i . In our main theorem we will show that for an uniformly ergodic irreducible and completely positive operator the peripheral spectrum consists entirely of first order poles of the resolvent. For the definition and properties of completely positive maps we refer to 13 . Since for an irreducible positive operator r T 0 4 we may assume without loss of generality r T 1 in the theorem below. 32 ULRICH GROH . Main Theorem. Let T be an irreducible and completely positive operator with spectral radius r T 1 on a C - -algebra. IfT .
