A characterization of Hilbert C*-modules over commutative C*-algebras

Abstract

Abstract Let M be a Hilbert C∗-module. Set |x|=⟨x,x⟩12 for every x∈M and C={⟨x,y⟩:x,y∈M}¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯. We prove that |xa|=|x||a| for every x∈M and a∈C if and only if C is a commutative C∗-algebra.