Structure of preservers of range orthogonality on *-rings and C⁎-algebras

Abstract

The topic of this paper lies between algebraic theory of *-rings and *-algebras on one side, and analytic theory of C⁎-algebras on the other side. A map θ:A→B between unital *-rings is called range orthogonal isomorphism if it is bijective and preserves range orthogonality in both directions. We show that any additive (resp. linear) range orthogonal isomorphism is canonical, that is, it is a *-isomorphism followed by multiplication from the right by an invertible element, provided that A is generated by projections as a *-ring. In case of general * rings and *-algebras we show that direct summands generated by projections are well behaved with respect to range orthogonal morphisms. In particular, we show that additive range orthogonality isomorphisms are canonical on proper nonabelian parts of Baer *-algebras. We apply algebraic results to matrix C⁎-algebras to show that any range orthogonal isomorphisms between them is canonical. The same holds for C⁎-algebras having proper nonabelian part generated by projections.