Spatiality of Derivations of Operator Algebras in Banach Spaces

Abstract

Suppose thatAis a transitive subalgebra ofB(X)and its norm closureA¯contains a nonzero minimal left idealI. It is shown that ifδis a bounded reflexive transitive derivation fromAintoB(X), thenδis spatial and implemented uniquely; that is, there existsT∈B(X)such thatδ(A)=TA−ATfor eachA∈A, and the implementationTofδis unique only up to an additive constant. This extends a result of E. Kissin that “ifA¯contains the idealC(H)of all compact operators inB(H), then a bounded reflexive transitive derivation fromAintoB(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 fromAintoB(X)is spatial and implemented uniquely, ifXis a reflexive Banach space andA¯contains a nonzero minimal right idealI.