Weighted homomorphisms between C*-algebras

TL;DR

This paper characterizes weighted *-homomorphisms between C*-algebras through preservation of zero-products and orthogonality, providing new insights into their structure and applications to positive maps.

Contribution

It offers a characterization of weighted *-homomorphisms via zero-product and orthogonality preservation, extending understanding of linear maps between C*-algebras.

Findings

Weighted *-homomorphisms characterized by zero-product preservation

Self-adjoint maps are weighted *-homomorphisms if they preserve zero-products

Linear maps that are positive and preserve zero-products are exactly order zero maps

Abstract

We show that a bounded, linear map between C*-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 C*-algebras is completely positive, order zero in the sense of Winter-Zacharias if and only if it is positive and preserves zero-products.