On a generalization of decomposable maps on C*-algebras

We propose the notion of countable decomposability of maps on C*-algebras: a bounded linear map $\phi :\mathcal{A}\to B(\mathcal{H})$, where $\mathcal{A}$ is a C*-algebra and $\mathcal{H}$ a Hilbert space, will be called countably decomposable if it admits a representation $\phi ={\sum }_{k=1}^{\mathrm{\infty }}{\psi }_k\circ {\phi }_k$ for completely positive maps ${\psi }_k:\mathcal{A}\to B(\mathcal{H})$ and bounded *-maps ${\phi }_k:\mathcal{A}\to \mathcal{A}$. A characterization of countable decomposability is given in certain cases with various assumptions imposed on maps ${\phi }_k$. Our findings provide extensions of a classical result of Størmer from Proc. Amer. Math. Soc. 86 (1982), 402-404, originally formulated for decomposable positive maps.