On a generalization of decomposable maps on C*-algebras
Krzysztof Szczygielski
TL;DR
This paper generalizes decomposable maps on C*-algebras by introducing countable decomposability: a map $\varphi: \mathscr{A}\to B(\mathcal{H})$ admits a representation $\varphi=\sum_{k=1}^{\infty} \psi_k\circ\phi_k$ with CP maps $\psi_k$ and *-maps $\phi_k$. It develops finite and infinite decomposition theories, linking them to dilation theorems (Stinespring) and extension results (Arveson), and provides characterizations via the cones $\Gamma_n^{+}(\phi)$ in terms of positivity of ampliations $\,[a_{ij}]$ and $\,[\varphi(a_{ij})]$. The main results, including finite-decomposition equivalences, infinite-decomposition criteria under *-homomorphism/vanishing-sequence assumptions, and nonunital extensions, show that $\mathsf{D}_{\phi}(\mathscr{A},\mathcal{H})$ forms a BW-closed left mapping cone, thereby generalizing Stormer's 1982 characterization to a countable setting. Overall, the work broadens the structural understanding of positive maps on C*-algebras and their dilations, offering a framework for further study of positivity, topology, and nonunital cases in operator algebra theory.
Abstract
We propose the notion of countable decomposability of maps on C*-algebras: a bounded linear map $\varphi : \mathscr{A}\to B(\mathcal{H})$, where $\mathscr{A}$ is a C*-algebra and $\mathcal{H}$ a Hilbert space, will be called countably decomposable if it admits a representation $\varphi = \sum_{k=1}^{\infty} ψ_k \circ φ_k$ for completely positive maps $ψ_k : \mathscr{A}\to B(\mathcal{H})$ and bounded *-maps $φ_k : \mathscr{A}\to\mathscr{A}$. A characterization of countable decomposability is given in certain cases with various assumptions imposed on maps $φ_k$. 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.