Spatiality of Derivations of Operator Algebras in Banach Spaces

and Applied Analysis 3 The role of “compact operators” is replaced by that of “minimal one-sided ideals”. The proof of our results relies on the quasispatiality of the derivation and Banach algebra techniques. This paper is a continuation of 5 . Some definitions and notations can be found in 5 . 2. Preliminaries Throughout this paper, X is a complex Banach space, and X∗ is the topological dual space of X, the Banach space of all continuous linear functionals on X. We denote by F X the algebra of all finite-rank operators on X. If a subalgebra A contains F X , then A is called a standard operator algebra. For a bounded operator A on X, denote by LatA the lattice of all closed invariant subspaces ofA andA∗ the adjoint operator ofA. For a subalgebraA of B X , denote by Lat A the lattice of all closed subspaces invariant under every operator inA. For a setL of subspaces ofX, denote by AlgL the algebra of all operators in B X which leave all subspaces in L invariant. An operator algebraA is transitive if LatA {{0}, X};A is reflexive if A AlgLatA, where AlgLatA {T ∈ B X : LatA ⊂ Lat T}. 2.1 For 0/ x ∈ X and 0/ f ∈ X∗, the rank-one operator x⊗f acts onX by x⊗f y f y x for y ∈ X. Let A be an operator on X with Dom A ⊆ X. If x ∈ Dom A and f ∈ Dom A∗ , thenA x ⊗ f Ax ⊗ f and x ⊗ f A x ⊗ A∗f . LetM be a nonempty subset of X andN a nonempty subset ofX∗. The annihilatorM⊥ ofM and the preannihilator ⊥N ofN are defined as follows 15 : M⊥ {f ∈ X∗ : f x 0 for all x ∈ M}, ⊥N {x ∈ X : f x 0 for all f ∈ N}. It is obvious thatM⊥ is a weak∗-closed subspace ofX∗ and ⊥N is a norm-closed subspace of X. For a subalgebra A and a closed, densely defined operator T with domain Dom T , we say that Tcommutes with A, if A Dom T ⊆ Dom T and TAξ ATξ for any A ∈ A and any ξ ∈ Dom T . For a subalgebra A of B X , let A∗ {A∗ : A ∈ A} in notation. A subset I of an algebra A is a left ideal of A if AI ⊆ I, a right ideal if IA ⊆ I, and a two-sided ideal if it is both a left and a right ideal. A left ideal I of A is minimal if every left ideal of A included in I is either I or {0}, similarly for minimal right ideals. A derivation δ is bounded resp., closed if the map Dom δ A → δ A ∈ B X is bounded resp., closed in the operator norm topology. The derivation δ is transitive if its domain Dom δ is a transitive operator algebra; δ is reflexive if Aδ { Â ( A δ A 0 A ) : A ∈ Dom δ } 2.2 is a reflexive operator algebra onX⊕X. Denote by Imp δ the set of all closed, densely defined operators implementing the derivation δ as in 1.3 . For a densely defined, closed operator T with domain Dom T , we can define the derivation ΔT with domain Dom ΔT {A ∈ B X : A Dom T ⊆ Dom T , TA −AT is bounded on Dom T } 2.3 4 Abstract and Applied Analysis