Positive Linear Maps on C*-Algebras

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators. Herein, C*-algebras possess an identity and are written in German type 21, 93, S. Capital letters A, B, C stand for operators, script l e t t e r s ^ , J ^ for vector spaces, small letters x, y, z for vectors. Capital Greek letters $>, ^ stand for linear maps on C*-algebras, small Greek letters a, /3, y for complex numbers. We denote by Wln the collection of all n X n complex matrices. SDIw(2l) = SI (x) 9Jtn is the C*-algebra of n X n matrices over 31. A linear map $: 21 —» 93 is positive if $(A) is positive for all positive A in 21. We define $ (x) ln: aWn(8)-*9W»(»)by