Approximately order zero maps between C*-algebras

TL;DR

This paper studies linear maps between C*-algebras that nearly preserve involution and orthogonality, providing structural insights and approximation results especially in finite-dimensional cases.

Contribution

It introduces the concept of approximately order zero maps and shows they can be approximated by approximate Jordan *-homomorphisms under certain conditions.

Findings

Structural properties of approximately order zero maps

Approximation of such maps by Jordan *-homomorphisms in finite dimensions

Dependence of approximation errors on map norms and epsilon

Abstract

We investigate linear operators between C$^\ast$-algebras which approximately preserve involution and orthogonality, the latter meaning that for some $\varepsilon>0$ we have $\|\phi(x)\phi(y)\|\leq\varepsilon\|x\|\|y\|$ for all positive $x,y$ with $xy=0$. We establish some structural properties of such maps concerning approximate Jordan-like equations and almost commutation relations. In some situations (e.g. when the codomain is finite-dimensional), we show that $\phi$ can be approximated by an approximate Jordan $^\ast$-homomorphism, with both errors depending only on $\|\phi\|$ and $\varepsilon$.