Non-trivial derivations on commutative regular algebras

Abstract

Necessary and suffcient conditions are given for a (complete) commuta- tive algebra that is regular in the sense of von Neumann to have a non-zero derivation. In particular, it is shown that there exist non-zero derivations on the algebra L(M) of all measurable operators affliated with a commutative von Neumann algebra M, whose Boolean algebra of projections is not atomic. Such derivations are not continuous with respect to measure convergence. In the classical setting of the algebra S[0; 1] of all Lebesgue measurable func- tions on [0; 1], our results imply that the first (Hochschild) cohomology group H¹(S[0; 1]; S[0; 1]) is non-trivial.