Decomposition of linear maps into non-separable \(C^*\)-algebras (Q1076291)

The main result is that a commutative \(C^*\)-algebra B is injective if every self-adjoint bounded linear map \(\phi\) from a \(C^*\)-algebra into B can be expressed in the form \(\phi =\phi^+-\phi^-\) with \(\phi^+,\phi^-\) positive and \(\| \phi \| =\| \phi^+\| +\| \phi^-\|\). A completely bounded map with no positive decomposition into the Calkin algebra is constructed, assuming the Continuum Hypothesis.