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.