Inner products and module maps of Hilbert C*-modules

TL;DR

This paper investigates the structure of surjective isometries between Hilbert C*-modules over commutative C*-algebras, showing they preserve inner products and module actions, and explores conditions for similar properties in noncommutative cases.

Contribution

It proves that surjective linear isometries between Hilbert C*-modules over commutative C*-algebras preserve inner products and are module maps, extending understanding of their structure.

Findings

In commutative case, isometries preserve inner products.

In commutative case, isometries are module maps via *-isomorphisms.

Conditions are provided for preservation in noncommutative cases.

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.