On a generalization of decomposable maps on C*-algebras

TL;DR

This paper introduces the concept of countable decomposability for maps on C*-algebras, extending classical results on decomposable positive maps by providing characterizations under various assumptions.

Contribution

It generalizes the notion of decomposable maps to a countable setting and offers new characterizations, broadening the understanding of map structures on C*-algebras.

Findings

Provides a new definition of countable decomposability for maps on C*-algebras.

Extends classical results of Størmer to the countable case.

Offers characterizations of countable decomposability under different assumptions.

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 $\varphi = \sum_{k=1}^{\infty} \psi_k \circ \phi_k$ for completely positive maps $\psi_k : \mathscr{A}\to B(\mathcal{H})$ and bounded *-maps $\phi_k : \mathscr{A}\to\mathscr{A}$. 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 St{\o}rmer from Proc. Amer. Math. Soc. 86 (1982), 402-404, originally formulated for decomposable positive maps.