Approximately order zero maps between C*-algebras

We investigate linear operators between C*-algebras which approximately preserve involution and orthogonality, the latter meaning that for some epsilon > 0 we have parallel to phi(x)phi(y)parallel to <= epsilon parallel to x parallel to parallel to y parallel to 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 *-homomorphism, with both errors depending only on parallel to phi parallel to and epsilon. (C) 2021 Elsevier Inc. All rights reserved.