Derivations on ideals in commutative $AW^*$-algebras

TL;DR

This paper characterizes when non-zero band preserving derivations exist from ideals in commutative $AW^*$-algebras to certain subspaces, showing triviality under specific conditions and linking existence to the Boolean algebra's $\sigma$-distributivity.

Contribution

It provides necessary and sufficient conditions for the existence of non-zero band preserving derivations in commutative $AW^*$-algebras, highlighting the role of the Boolean algebra structure.

Findings

Non-zero derivations exist iff the Boolean algebra of projections is not $\sigma$-distributive.

Derivations into $ ext{a quasi-normed solid space}$ are always trivial.

Derivations into $ ext{the algebra of measurable operators}$ can be non-zero under certain conditions.

Abstract

Let $\mathcal{A}$ be a commutative $AW^*$-algebra, let $S(\mathcal{A})$ be the *-algebra of all measurable operators affiliated with $\mathcal{A}$, let $\mathcal{I}$ be an ideal in $\mathcal{A}$, let $s(\mathcal{I})$ be the support of the ideal $\mathcal{I}$ and let $\mathbb{Y}$ be a solid subspace in $S(\mathcal{A})$. The necessary and sufficient conditions of existence of non-zero band preserving derivations from $\mathcal{I}$ to $\mathbb{Y}$ are given. We show that, in case when $\mathbb{Y}\subset\mathcal{A}$, or $\mathbb{Y}$ is a quasi-normed solid space, any band preserving derivation from $\mathcal{I}$ into $\mathbb{Y}$ is always trivial. At the same time, there exist non-zero band preserving derivations from $\mathcal{I}$ with values in $S(\mathcal{A})$, if and only if the Boolean algebra of all projections from the $AW^*$-algebra $s(\mathcal{I})\mathcal{A}$ is not $\sigma$-distributive.