2-positive almost order zero maps and decomposition rank

Abstract

We consider 2-positive almost order zero (disjointness preserving) maps on C∗-algebras. Generalizing the argument of M. Choi for multiplicative domains, we provide an internal characterization of almost order zero for 2-positive maps. In addition, it is shown that complete positivity can be reduced to 2-positivity in the definition of decomposition rank for unital separable C∗-algebras.