On extendability of functionals on Hilbert $C^*$-modules

TL;DR

This paper investigates the extendability of functionals on Hilbert $C^*$-modules, demonstrating conditions under which non-zero functionals can vanish on submodules, with results varying across different algebra types.

Contribution

It extends previous results by showing that non-zero functionals can vanish on submodules even in monotone complete $C^*$-algebras, and identifies cases where this cannot occur.

Findings

Existence of non-zero $A$-valued functionals vanishing on submodules in monotone complete $C^*$-algebras.

Non-extendability of such functionals in certain type I $W^*$-algebras.

Abstract

Let $M\subset N$ be Hilbert $C^*$-modules over a $C^*$-algebra $A$ with $M^\perp=0$. It was shown recently by J. Kaad and M. Skeide that there exists a non-zero $A$-valued functional on $N$ such that its restriction onto $M$ is zero. Here we show that this may happen even if $A$ is monotone complete. On the other hand, we show that for certain type I $W^*$-algebras this cannot happen.