Approximately order zero maps between C*-algebras

Abstract

We investigate linear operators between C${}^{\ast }$-algebras which approximately preserve involution and orthogonality, the latter meaning that for some $\epsilon >0$ we have $\Vert \varphi (x)\varphi (y)\Vert \le \epsilon \Vert x\Vert \Vert y\Vert$ 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 $\varphi$ can be approximated by an approximate Jordan ${}^{\ast }$-homomorphism, with both errors depending only on $\Vert \varphi \Vert$ and $\epsilon$.

43 pp