Inner products and module maps of Hilbert C*-modules

Abstract

Let $E$ and $F$ be two Hilbert $C^*$-modules over $C^*$-algebras $A$ and $B$, respectively. Let $T$ be a surjective linear isometry from $E$ onto $F$ and $\varphi$ a map from $A$ into $B$. We will prove in this paper that if the $C^*$-algebras $A$ and $B$ are commutative, then $T$ preserves the inner products and $T$ is a module map, i.e., there exists a $*$-isomorphism $\varphi$ between the $C^*$-algebras such that $$ \langle Tx,Ty\rangle=\varphi(\langle x,y\rangle), $$ and $$ T(xa)=T(x)\varphi(a). $$ In case $A$ or $B$ is noncommutative $C^*$-algebra, $T$ may not satisfy the equations above in general. We will also give some condition such that $T$ preserves the inner products and $T$ is a module map.