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

Let $\mathcal A$ be a commutative $AW^*$-algebra.We denote by $S(\mathcal A)$ the $*$-algebra of measurable operators that are affiliated with $\mathcal A$. For an ideal $\mathcal I$ in $\mathcal A$, let $s(\mathcal I)$ denote the support of $\mathcal I$. Let $\mathbb Y$ be a solid linear subspace in $S(\mathcal A)$. We find necessary and sufficient conditions for existence of nonzero band preserving derivations from $\mathcal I$ to $\mathbb Y$. We prove that no nonzero band preserving derivation from $\mathcal I$ to $\mathbb Y$ exists if either $\mathbb Y\subset\mathcal A$ or $\mathbb Y$ is a quasi-normed solid space. We also show that a nonzero band preserving derivation from $\mathcal I$ to $S(\mathcal A)$ exists if and only if the boolean algebra of projections in the $AW^*$-algebra $s(\mathcal I)\mathcal A$ is not $\sigma$-distributive.