Isometries between C*-algebras

Abstract

Let $A$ and $B$ be C*-algebras and let $T$ be a linear isometry from $A$ \emph{into} $B$. We show that there is a largest projection $p$ in $B^{**}$ such that $T(\cdot)p : A \longrightarrow B^{**}$ is a Jordan triple homomorphism and $$ T(a b^* c + c b^* a) p= T(a) T(b)^* T(c) p + T(c) T(b)^* T(a) p $$ for all $a$, $b$, $c$ in $A$. When $A$ is abelian, we have $\|T(a)p\|=\|a\|$ for all $a$ in $A$. It follows that a (possibly non-surjective) linear isometry between any C*-algebras reduces {\it locally} to a Jordan triple isomorphism, by a projection.