Spatiality of Derivations of Operator Algebras in Banach Spaces

Suppose that A is a transitive subalgebra of B(X) and its norm closure A¯ contains a nonzero minimal left ideal I. It is shown that if δ is a bounded reflexive transitive derivation from A into B(X), then δ is spatial and implemented uniquely; that is, there exists T∈B(X) such that δ(A)=TA−AT for each A∈A, and the implementation T of δ is unique only up to an additive constant. This extends a result of E. Kissin that “if A¯ contains the ideal C(H) of all compact operators in B(H), then a bounded reflexive transitive derivation from A into B(H) is spatial and implemented uniquely.” in an algebraic direction and provides an alternative proof of it. It is also shown that a bounded reflexive transitive derivation from A into B(X) is spatial and implemented uniquely, if X is a reflexive Banach space and A¯ contains a nonzero minimal right ideal I