Local Derivations on Operator Algebras

✍ Scribed by Randall L. Crist

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 553 KB
• Volume: 135
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0004

✦ Synopsis

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. Algebra 130 (1990), 494 509) and Larson and Sourour (Proc. Symp. Pure Math. 51 (1990), 187 194). A local derivation ' is 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.