Isometries between C*-algebras

Abstract

Let $A$ and $B$ be C*-algebras and let $T$ be a linear isometry from $A$ into $B$. We show that there is a largest projection $p$ in $B^{\ast \ast }$ such that $T(\cdot )p:A⟶B^{\ast \ast }$ is a Jordan triple homomorphism and

for all $a$, $b$, $c$ in $A$. When $A$ is abelian, we have $\Vert T(a)p\Vert =\Vert a\Vert$ for all $a$ in $A$. It follows that a (possibly non-surjective) linear isometry between any C*-algebras reduces locally to a Jordan triple isomorphism, by a projection.