Inner products and module maps of Hilbert C*-modules

Abstract

Let $E$ and $F$ be two Hilbert $C^{\ast }$-modules over $C^{\ast }$-algebras $A$ and $B$, respectively. Let $T$ be a surjective linear isometry from $E$ onto $F$ and $\phi$ a map from $A$ into $B$. We will prove in this paper that if the $C^{\ast }$-algebras $A$ and $B$ are commutative, then $T$ preserves the inner products and $T$ is a module map, i.e., there exists a $\ast$-isomorphism $\phi$ between the $C^{\ast }$-algebras such that $$⟨Tx,Ty⟩=\phi (⟨x,y⟩),$$ and $$T(xa)=T(x)\phi (a).$$ In case $A$ or $B$ is noncommutative $C^{\ast }$-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.