Approximately order zero maps between C*-algebras

We investigate linear operators between C${}^{\ast }$-algebras which approximately preserve involution and orthogonality, the latter meaning that for some $\epsilon >0$ we have $\mathrm{\parallel }\varphi (x)\varphi (y)\mathrm{\parallel }\le \epsilon \mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$ 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 $\mathrm{\parallel }\varphi \mathrm{\parallel }$ and $\epsilon$.