Linear maps between C*-algebras that are *-homomorphisms at a fixed point

Abstract

Let $A$ and $B$ be C${}^{\ast }$-algebras. A linear map $T:A\to B$ is said to be a ${}^{\ast }$-homomorphism at an element $z\in A$ if $a{b}^{\ast }=z$ in $A$ implies $T(a{b}^{\ast })=T(a)T(b{)}^{\ast }=T(z)$, and $c^{\ast }d=z$ in $A$ gives $T(c^{\ast }d)=T(c{)}^{\ast }T(d)=T(z).$ Assuming that $A$ is unital, we prove that every linear map $T:A\to B$ which is a ${}^{\ast }$-homomorphism at the unit of $A$ is a Jordan ${}^{\ast }$-homomorphism. If $A$ is simple and infinite, then we establish that a linear map $T:A\to B$ is a ${}^{\ast }$-homomorphism if and only if $T$ is a ${}^{\ast }$-homomorphism at the unit of $A$. For a general unital C${}^{\ast }$-algebra $A$ and a linear map $T:A\to B$, we prove that $T$ is a ${}^{\ast }$-homomorphism if, and only if, $T$ is a ${}^{\ast }$-homomorphism at $0$ and at $1$. Actually if $p$ is a non-zero projection in $A$, and $T$ is a ${}^{\ast }$-homomorphism at $p$ and at $1-p$, then we prove that $T$ is a Jordan ${}^{\ast }$-homomorphism. We also study bounded linear maps that are ${}^{\ast }$-homomorphisms at a unitary element in $A$.