Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một số các loại bất bình đẳng cho các bản đồ tích cực tuyến tính trên C *- đại số. | Copyright by INCREST 1980 J. OPERATOR THEORY 4 1980 271-285 SOME ASSORTED INEQUALITIES FOR POSITIVE LINEAR MAPS ON C -ALGEBRAS MAN-DUEN CHOI 1. INTRODUCTION The general structure of positive linear maps on c -algebras is highly combinatorial see Appendix B yet its subtlety should surpass its complexity. In this paper we try to exhibit some special features of positive linear maps by means of 2x2 operator-matrix manipulations. The main results here concerning some assorted inequalities of the Schwarz type appear non-trivial even if all c -algebras involved are limited to the finite-dimensional matrix algebras. It is noteworthy that some preliminary results presented here are well known but we include the proofs for completeness this article should thus be considered as a partly expository paper. The readers are also referred to 1 4 16 for some recent research on the subject. Throughout the paper c -algebras possess a unit I and are written in German type 91 . á 3 denotes the algebra of all bounded operators on the Hilbert space 3 . denotes the algebra of all n X n complex matrices. - 91 is the algebra of n X n matrices over 91. For an operator T 6 SS 3 we denote by C T for the unital c -algebra generated by T. T is said to be invertible if T has a bounded inverse T l. T is said to be positive in notation T 0 if T is hermitian and the spectrum of T is included in 0 oo in the matrix case T is also said to be positive semi-definite the unique positive square root of T is denoted by T1 2. A linear map 91 - between two c -algebras is said to be unital if P I r Ị is said to be positive if Ĩ A is positive for every positive A in 91 Ị is said to be 2-positive if 0 A1 p i12 1 All A12 is positive for every positive _ y421 22 . -Aịi A22. in M2 9l . We present in 2 some basic 2x2 operator-matrix manipulations for preliminaries. 272 MAN-DUEN CHOI In 3 we are concerned with some Schwarz-type inequalities and the cognate conjectures for every unital positive linear map i 9I- . .
