Derivations in disjointly complete commutative regular algebras

Abstract

We show that any nonexpansive derivation on a subalgebra of a disjointly complete commutative regular algebra A extends up to a derivation on A. For an algebra C∞(X, K) of functions X → K, 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 K 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].