Spatiality of (α, β)-Derivations of Operator Algebras in Banach Spaces

Abstract:

SThe topic of the present paper is the spatiality of (α, β)-derivations of operator algebras. Suppose that X is a Banach space, A is a subalgebra of B(X) and α, β are automorphisms on B(X). It is shown that any reflexive transitive (α, β)-derivation is quasi-spatial. If the norm closure of A contains a nonzero minimal left ideal, then a bounded reflexive transitive (α, β)-derivation δ from A into B(X) is spatial and the implementation T of δ is unique only up to an additive constant.