Weighted homomorphisms between ${\mathrm{C}}^{\ast }$-algebras

Abstract

We show that a bounded, linear map between ${\mathrm{C}}^{\ast }$-algebras is a weighted $\ast$-homomorphism (the central compression of a $\ast$-homomorphism) if and only if it preserves zero-products, range-orthogonality, and domain-orthogonality. It follows that a self-adjoint, bounded, linear map is a weighted $\ast$-homomorphism if and only if it preserves zero-products. As an application we show that a linear map between ${\mathrm{C}}^{\ast }$-algebras is completely positive, order zero in the sense of Winter–Zacharias if and only if it is positive and preserves zero-products.