Decompositions of Linear Maps Into Non-Separable $C^{\ast }$-Algebras

Abstract

We study positive decompositions of bounded linear maps between $C^{\ast }$-algebras. A characterization of commutative injective $C^{\ast }$-algebras is given in terms of positive decompositions with certain norm condition of linear maps. We also provide under the Continuum Hypothesis a completely bounded map into the Calkin algebra which admits no positive decomposition.