Disjointness preservers of AW⁎-algebras

In this paper, we study bijective linear maps θ:A→B between AW⁎-algebras preserving either zero products or range orthogonality. Such a map is automatically continuous, and provides an algebra or a ⁎-algebra isomorphism π(⋅)=θ(⋅)θ⁎⁎(1)−1 from A onto B. Our results extend previous works for the case of W⁎-algebras, and also cover such maps between C⁎-algebras satisfying an extra assumption on preserving abelian C⁎-subalgebras.