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


Introduction
Throughout this paper, X is a Banach space and X will be replaced by H if it is a Hilbert space and A is a subalgebra of B X , the Banach algebra of all bounded operators on X. Suppose that A ⊆ B X and M ⊆ B X is an A-bimodule.A linear map δ from A into M is called a derivation if δ AB δ A B Aδ B for any A, B ∈ A.

1.1
Then A is called the domain of δ and denoted by Dom δ .The derivation δ is called inner resp., spatial if there exists an operator T ∈ M resp., T ∈ B X such that δ A TA − AT for any A ∈ A.

2 Abstract and Applied Analysis
If the operator T is not bounded, then δ is said to be quasispatial.More precisely, if there exists a densely defined, closed operator T : Dom T → X such that A Dom T ⊆ Dom T , δ A x TA − AT x for any A ∈ A, x ∈ Dom T , 1.3 then the derivation δ is called quasispatial, and the operator T is an implementation of δ.
Compared to the spatiality, the quasispatiality is a slightly weaker notion.
Given a bounded derivation δ on an operator algebra, the natural question is whether the derivation δ is inner or spatial .The spatiality of derivations is a classical problem when formulated for self-adjoint algebras and non-self-adjoint reflexive operator algebras.And it has been extensively studied in the literature in a large variety of situations, and some interesting results have been obtained 1-13 .For example, every derivation of a C * -algebra is spatial 12 , every derivation of a von Neumann algebra is inner 13 , and so is the derivation of a nest algebra 14 .Every derivation from an atom Boolean subspace lattice algebra into its ideal is quasispatial 7 .A necessary and sufficient condition is given for a derivation on CDC algebras to be quasispatial 6 .In 10 , the quasispatiality of derivations on CSL algebras is studied.
As to general operator algebras, it is well known that every derivation of B X is inner 9 and that every derivation of a standard operator subalgebra on a normed space X is spatial 4 .Since these operator algebras are transitive, the above question for a reflexive transitive derivation in a Banach space X is raised naturally as follows.
Problem 1. Suppose that A is a transitive subalgebra of B X .Let δ be a bounded reflexive transitive derivation from A into B X .Does there always exist T ∈ B X such that δ A TA − AT for each A ∈ A? Is the implementation T unique only up to an additive constant if there is any?
In the case when X is a Hilbert space, it is Problem 2.12 of 5 .Although the problem is still open, some strong conditions have been found by Kissin to imply that such derivations are spatial and implemented uniquely Proposition 2.11, 5 .In particular, Kissin has proved that the answer to Problem 1 is affirmative under the conditions that A is a transitive subalgebra of B H , and A contains the ideal C H of all compact operators in B H .As far as we know, there are no other solution to Problem 1.
The purpose of this paper is to investigate the quasispatiality of derivations and to address the above question in a Banach space X.The paper is organized as follows.
In Section 2, we give some preliminaries.In Section 3, we investigate the quasispatiality and spatiality of derivations.The main result Theorem 3.1 shows that if A is a transitive operator algebra on a Banach space X and A contains a nonzero minimal left ideal I, then a bounded reflexive transitive derivation δ from A into B X is spatial and implemented uniquely.As an application, the quasispatiality of the adjoint of a derivation is discussed in Section 4. The main result Theorem 4.2 in Section 4 shows that under the conditions that X is a reflexive Banach space and A contains a nonzero minimal right ideal I, δ is also spatial and implemented uniquely if δ is a bounded reflexive transitive derivation from A into B X .As another application, Proposition 2.11 ii of 5 can be proved by using Theorem 3.1 of this paper, which is Corollary 4.3.Since Theorem 2.5 and Proposition 2.11 of 5 hold in a Hilbert space and they are not valid in a Banach space without the approximation property, Theorems 3.1 and 4.2 of this paper extend the result of Kissin to Banach spaces in an algebraic direction.
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 .

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 Lat A the lattice of all closed invariant subspaces of A and A * the adjoint operator of A. For a subalgebra A of B X , denote by Lat A the lattice of all closed subspaces invariant under every operator in A. For a set L of subspaces of X, denote by Alg L the algebra of all operators in B X which leave all subspaces in L invariant.An operator algebra A is transitive if Lat A {{0}, X}; A is reflexive if For 0 / x ∈ X and 0 / f ∈ X * , the rank-one operator x ⊗ f acts on X by x ⊗ f y f y x for y ∈ X.Let A be an operator on X with Dom A ⊆ X.
Let M be a nonempty subset of X and N a nonempty subset of X * .The annihilator M ⊥ of M and the preannihilator ⊥ N of N 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 that M ⊥ is a weak * -closed subspace of X * 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 T commutes with A, if A Dom T ⊆ Dom T and TAξ AT ξ 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 is a reflexive operator algebra on X⊕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 Δ T with domain If a densely defined, closed operator T implements a derivation δ, then the derivation Δ T is an extension of the derivation δ.In fact, A ∈ Dom δ implies A ∈ Dom Δ T and Δ T A δ A for any A ∈ Dom δ by 1.3 and 2.4 .For any set Λ of derivations

2.5
In particular, Remark 2.1.Δ T is a reflexive transitive derivation for any densely defined, closed operator T .Indeed, Dom Δ T is a subalgebra of B X , and Let M ∈ Lat Dom Δ T with 0 / x 0 ∈ M. Then x ⊗ h x 0 h x 0 x ∈ M for any x ∈ Dom T and h ∈ Dom T * .Since Dom T * is dense, there exists h 0 ∈ Dom T * such that h 0 x 0 / 0 by Hahn-Banach Theorem.It follows that x ∈ M for any x ∈ Dom T so that Dom T ⊆ M. Thus X Dom T ⊆ M M. Therefore Dom Δ T is transitive.
For the derivation

Main Results
In this section, we discuss the quasispatiality of bounded transitive derivations on operator algebras in a Banach space.The main result is as follows.
Theorem 3.1.Suppose that A is a transitive subalgebra of B X and A, the norm closure of A, contains a nonzero minimal left ideal I.If δ is a bounded reflexive transitive derivation from A into B X , then δ is spatial and implemented uniquely, or, more precisely, there exists T ∈ B X such that δ A TA − AT for each A ∈ A and the implementation T is unique only up to an additive constant.
The proof of Theorem 3.1 will proceed through several lemmas, in each of which we maintain the same notation.The results similar to the following two lemmas Lemmas 3.2 and 3.3 can be found in 5 .For the sake of completeness, we outline the proof.

Lemma 3.2. Let δ be a reflexive transitive derivation from
Proof.If Imp δ is nonempty, then the lemma is trivially true.Therefore, for the rest of the argument, we assume that Imp δ is empty.
First, we have that if δ is a transitive derivation from Dom δ into B X , then x ∈ Dom T } is the graph of T .Indeed, it is easy to see that {0}⊕{0}, X⊕{0}, X⊕X, and all G T are invariant subspaces of A δ .For the converse, suppose that M ∈ Lat A δ such that M / {0} ⊕ {0}, M / X ⊕ {0}, and M / X ⊕ X.If there is a vector of form x 0 ∈ M with x / 0, then Ax 0 ∈ M for any A ∈ A. It follows that X ⊕ {0} ⊆ M by the transitivity of Dom δ .If M contains no vector of form 0 x with x / 0, then M X ⊕ {0}; otherwise, there is a vector of form 0 x ∈ M with x / 0, then we have that M X ⊕ X. Therefore x 0 ∈ M implies that x 0. It follows that there is a closed operator T such that

3.3
It is a contradiction to the assumption that δ is reflexive, which shows that Imp δ / ∅.

3.4
Therefore δ is quasispatial.Proof.We complete the proof step by step.
Step 1.We have that Indeed, suppose that I 2 {0}.Let 0 / A 0 ∈ I. Then AA 0 ⊆ I.It follows that AA 0 2 ⊆ I 2 and AA 0 2 {0}.Set X 0 {x ∈ X : A x 0 for all A ∈ A}.It is obvious that X 0 is an Ainvariant closed subspace of X. Hence X 0 {0} since A is a transitive operator algebra.Since {0} so that A 0 AA 0 {0}.We can choose x 0 ∈ X x 0 / 0 such that A 0 x 0 / 0. It is obvious that AA 0 x 0 is an A-invariant linear manifold of X and AA 0 x 0 / {0}.It follows that AA 0 x 0 is dense in X.However, AA 0 x 0 is contained in the null space of A 0 , A 0 0, which is a contradiction.
Step 2. Since I 2 / {0}, it follows from Lemma 2.1.5and Corollary 2.1.6 of 16 that there exists an idempotent P in A such that I AP and P AP is a division algebra consisting of scalar multiples of P with identity P , that is, P AP μP : μ is a complex scalar .

3.9
Then P is a rank-one operator; that is, there exist x 1 ∈ X and f ∈ X * such that

3.10
Indeed, let x 1 : 0 / x 1 ∈ X such that Px 1 x 1 and let M 1 A x 1 {A x 1 : A ∈ A}.It is obvious that M 1 / {0} and M 1 is an A-invariant linear manifold of X. Hence M 1 is dense in X.Since PA x 1 PAP x 1 μP x 1 μx 1 for any A ∈ A, the restriction P | M 1 of P on M 1 has one-dimensional range.As P is bounded, it also has one-dimensional range.Hence P x 1 ⊗ f, where f is a continuous functional.
Step 3. We have P x 1 ⊗ f ∈ I ⊆ A. As A is an algebra, the set of rank-one operators lies in A ∩ F X .For each 0 / y ∈ X, the linear manifold is dense in X.Indeed, if M y is not dense in X, by Hahn-Banach theorem, there exists 0 / g ∈ X * such that g z 0 for all z ∈ M y .Hence

3.13
If f By 0 for all B ∈ A, then f vanishes on the linear manifold Ay {By : B ∈ A}.However, as A is transitive, the manifold Ay is dense in X, so that f 0. This contradiction shows that there is B 1 ∈ A such that f B 1 y / 0. Then 3.13 implies that g Ax 1 0 for all A ∈ A. Repeating the above argument, we obtain that g 0. This contradiction shows that any manifold M y is dense in X. Therefore the algebra A ∩ F X is transitive.Lemma 3.5.Suppose that A is a transitive subalgebra of B X and A contains a nonzero minimal left ideal I. Then the operators commuting with A are scalars.More precisely, if T is a closed, densely defined operator such that A Dom T ⊆ Dom T and AT ξ TAξ for any A ∈ A and any ξ ∈ Dom T , then T μI for some complex scalar μ.
Proof.For any y ∈ X, there exists a net of operators {A λ } ⊆ A such that lim λ A λ x 1 y by the transitivity of A.

3.14
Suppose that T is a closed, densely defined operator commuting with A. As S y ⊗ f ∈ A for any y ∈ X, then y⊗f ξ ∈ Dom T and y⊗f Tξ T y⊗f ξ for any ξ ∈ Dom T .Since Dom T is dense in X and f / 0, there is ξ 0 ∈ Dom T such that f ξ 0 / 0. So y ⊗ f ξ 0 f ξ 0 y ∈ Dom T , f Tξ 0 y f ξ 0 T y , for any y ∈ X.
Lemma 3.6.Suppose that A is a subalgebra of B X and A is its norm closure.Then Lat A Lat A .
Proof.Clearly, Lat A ⊆ Lat A .Conversely, let M ∈ Lat A .For any B ∈ A, there exists a net Let A be a subalgebra of B X and δ be a bounded derivation from A A Dom δ into B X .For any B ∈ A, there exists a net of operators

3.17
Proof.Since A is transitive, A is also transitive.It is obvious that δ is a linear map.For any Then A

3.18
It follows that the linear map δ is a transitive derivation from

and Lat A δ
Lat A δ by Lemma 3.6.By 3.1 ,
Proof of Theorem 3.1.By 3.4 , Imp δ / ∅.Suppose that T ∈ Imp δ .Then T is a closed, densely defined operator on X and T μI : μ is a complex scalar ⊆ Imp δ .

3.20
Let δ 0 0 be the derivation with the same domain A of δ from A into B X , that is, δ 0 A 0 for any A ∈ A. If T 0 ∈ Imp δ 0 with domain Dom T 0 , then T 0 ∈ Imp δ 0 by 3.17 , and that is, T 0 Ax AT 0 x for any x ∈ Dom T 0 .Then T 0 μI for some scalar μ by Lemma 3.5.It follows that Imp δ 0 μI : μ is a complex scalar .

3.22
Let T 1 ∈ Imp δ be any closed, densely defined operators implementing δ.Then T 1 ∈ Imp δ by 3.17 .Then Dom T 1 is a nonzero Dom δ -invariant linear manifold of X by 1.3 ; that is, Dom T 1 is a nonzero A-invariant linear manifold of X. Therefore A Dom T 1 ⊆ Dom T 1 for any A ∈ A. If y ∈ X, y ⊗ f ∈ A by 3.14 and y ⊗ f ξ f ξ y ∈ Dom T 1 for any ξ ∈ Dom T 1 .Since Dom T 1 is dense in X and f / 0, there is ξ 0 ∈ Dom T 1 such that f ξ 0 / 0. So y ∈ Dom T 1 .Since y is arbitrary chosen, it follows that Dom T 1 X, the whole space.Then Dom T X also holds since T 1 is an arbitrary implementation of δ.It is obvious that Dom T 1 − T Dom T ∩ Dom T 1 X.Therefore the operator T 1 − T is everywhere defined.
Set T 2 T 1 − T .Then T 2 is closable.Indeed, suppose that ξ n ∈ X, ξ n → 0 and T 2 ξ n → η.Since T, T 1 ∈ Imp δ , δ A x T 1 A − AT 1 x and δ A x TA − AT x for any A ∈ Dom δ and any x ∈ X.So T 2 Aξ n AT 2 ξ n for any A ∈ A. As P x 1 ⊗f ∈ A by 3.10 and A is an algebra, T 2 PAξ n PAT 2 ξ n for any A ∈ A. Then
It follows that T 2 is a closable operator with domain Dom T 2 X, the whole space.So that T 2 is a closed operator.Clearly, T 2 T 1 − T ∈ Imp δ 0 .Then T 1 − T μI for some scalar μ by 3.22 , that is, T 1 T μI.It follows that Imp δ T μI : μ is a complex scalar .

3.25
Since T is closed and Dom T X, T is bounded by the closed graph theorem.It follows from 2.4 , 3.24 , and 3.25 that δ A TA − AT for each A ∈ A and the implementation T is unique only up to an additive constant.

Applications
Suppose that A is a subalgebra of B X and δ is a derivation from

4.2
It follows that δ * is a derivation. 2 Suppose that δ is transitive.Then A is transitive and Lat A {X, {0}}.
We have that Dom δ * is transitive, that is, δ * is transitive.Conversely, suppose that δ * is transitive.We obtain, as above, that δ * * is transitive.Since X is a reflexive Banach space, A * * A for any bounded operator A ∈ B X .It follows from 4.1 that δ δ * * is transitive by the definition of δ * .Set J 0 I I 0 , where I is the identity operator on X * and 0 is the zero operator on X * .Since the two operator algebras A δ * and A δ * are similar.We have that It is obvious that is a minimal left ideal of A, where x ⊗ f is the rank-one operator on H with x ⊗ f h h, f x for h ∈ H.By Theorem 3.1, there exists T ∈ B H such that δ A TA − AT for each A ∈ A, and the implementation T is unique only up to an additive constant.Remark 4.4.In a Banach space without the approximation property there exists such space as this, e.g., 17 , not all compact operators can be approximated by finite-rank operators in the norm operator topology.Therefore, Theorems 3.1 and 4.2 of this paper improve 5, Proposition 2.11 .

2 λProposition 3 . 7 .
. Therefore a linear map δ can be unambiguously defined by Dom δ Dom δ any B ∈ A and any net {A λ } in A such that A λ • − − → B, where δ B is the limit of δ A λ in the operator norm topology.It is obvious that δ δ if δ is bounded.Let δ be a bounded transitive derivation from A A Dom δ into B X .Then δ is a bounded transitive derivation from A into B X and Imp δ Imp δ .

4 . 6 If
T ∈ Imp δ , g −T * g ∈ G T ⊥ for any g ∈ Dom T * , since g, h ∈ X * such that g h ∈ G T ⊥ , then g Tx h x 0 for any x ∈ Dom T .4.8So that the functional x → g Tx −h x is a bounded functional on Dom T .Therefore g ∈ Dom T * and h −T * g holds on Dom T by 4.8 .As −T * g and h are bounded functionals on X and Dom T is dense in X, h −T * g on X.It follows thatG T ⊥ g −T * g : g ∈ Dom T * , δ * {0} ⊕ {0}, X * ⊕ {0}, X * ⊕ X * , G −T * : T ∈ Imp δ .4.10 Then B 22 x ∈ Dom T and B 11 Tx tB 11 x B 12 x T tI B 22 x for any x ∈ Dom T and any t ∈ C. It follows that B 11 x B 22 x ∈ Dom T and B 11 Tx B 12 x TB 22 x for all x ∈ Dom T .Since Dom T is dense in X and B 11 , B 22 ∈ B X , B 11 B 22 .So B 11 Dom T ⊆ Dom T and B 12 TB 11 − B 11 T on Dom T .Therefore, B 12 TB 11 − B 11 T is bounded on Dom T .It follows that B 11 B 22 ∈ Dom Δ T and B 12 Δ T B 11 .Thus B 2 ∈ G T tI for any x ∈ Dom T and any t ∈ C.∈ A Δ T .It follows that A Δ T is a reflexive algebra.Therefore Δ T is a reflexive transitive derivation.
Lemma 3.4.Let A be a transitive subalgebra of B X .If A contains a nonzero minimal left ideal I, then A ∩ F X is also a transitive subalgebra of B X .
A A Dom δ into B X .We can define the adjoint δ * of δ by Dom δ * A * {A * : A ∈ A}, δ * A * δ A * for any A ∈ A. 4.1 Lemma 4.1.Let δ be a derivation from A A Dom δ into B X .1 The adjoint δ * of δ is a derivation from A * into B X * .2Furthermore, suppose that X is a reflexive Banach space and A is a subalgebra of B X .Then δ * is a reflexive resp., transitive derivation from A * into B X * if and only if δ is a reflexive resp., transitive derivation from A into B X .And Imp δ * {−T * : T