Derivations in disjointly complete commutative regular algebras

Abstract

We show that any nonexpansive derivation on a subalgebra of a dis-jointly complete commutative regular algebra extends up to a derivation on . For an algebra of functions , continuous on a dense open subset of Stone compact X, we establish that the lack of nontrivial derivation is equivalent to σ-distributivity of the Boolean algebra of clopen subsets of X. The field is an arbitrary normed field of charachteristic zero containing a complete non-discrete subfield. Our work is motivated by two seemingly unrelated problems due to Ayupov [2] and Wickstead [32].