Local Derivations on Operator Algebras

When attempting to find sufficient conditions for a linear mapping to be a derivation, an obvious candidate is the concept of a local derivation. Local derivations on operator algebras have been investigated in recent papers of Kadison (J. Algebra130(1990), 494–509) and Larson and Sourour (Proc. Symp. Pure Math.51(1990), 187–194). A local derivationηis a (norm continuous) linear map from an operator algebra A into an A-bimodule M which agrees with some derivation at each point in the algebra. We show that if A is the direct limit of finite dimensional CSL algebras via *-extendable embeddings (e.g., a triangular AF algebra), then a local derivation on A must be a derivation. Further, we show that for many finite dimensional operator algebras, any inner local derivation must be an inner derivation.