Derivations, local and 2-local derivations on some algebras of operators on Hilbert C*-modules
Jun He, Jiankui Li, and Danjun Zhao

TL;DR
This paper investigates derivations and 2-local derivations on algebras of bounded and adjointable operators over Hilbert C*-modules, proving their linearity, continuity, and inner nature under certain conditions.
Contribution
It establishes that derivations on these operator algebras are inner and that 2-local derivations are actual derivations, extending understanding of their structure.
Findings
Derivations are $ ext{A}$-linear, continuous, and inner.
2-local derivations are actual derivations.
Conditions for local derivations to be derivations are identified.
Abstract
For a commutative C*-algebra with unit and a Hilbert~-module , denote by End the algebra of all bounded -linear mappings on , and by End the algebra of all adjointable mappings on . We prove that if is full, then each derivation on End is -linear, continuous, and inner, and each 2-local derivation on End or End is a derivation. If there exist in and in , such that , where denotes the set of all bounded -linear mappings from to , then each -linear local derivation on End is a derivation.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Holomorphic and Operator Theory
Derivations, local and 2-local derivations on some algebras of operators on Hilbert C*-modules
Jun He, Jiankui Li111Corresponding author. E-mail address: [email protected], and Danjun Zhao
Department of Mathematics, East China University of Science and Technology
Shanghai 200237, China
Abstract
For a commutative C*-algebra with unit and a Hilbert -module , denote by End the algebra of all bounded -linear mappings on , and by End the algebra of all adjointable mappings on . We prove that if is full, then each derivation on End is -linear, continuous, and inner, and each 2-local derivation on End or End is a derivation. If there exist in and in , such that , where denotes the set of all bounded -linear mappings from to , then each -linear local derivation on End is a derivation.
Keywords: Derivations, Hilbert C*-modules, inner derivations, local derivations, 2-local derivations
Mathematics Subject Classification(2010): 47B47; 47L10
1 Introduction and preliminaries
The structure of derivations on operator algebras is an important part of the theory of operator algebras.
Let be an algebra and be an -bimodule. Recall that a derivation is a linear mapping from into such that , for all in . For each in , one can define a derivation by , for all in . Such derivations are called inner derivations.
It is a classical problem to identify those algebras on which all derivations are inner derivations. Several authors investigate this topic. The following two results are classical. S. Sakai [17] proves that all derivations from a W*-algebra into itself are inner derivations. E. Christensen [3] proves that all derivations from a nest algebra into itself are inner derivations.
In 1990, R. Kadison [10] and D. Larson, A. Sourour [13] independently introduce the concept of local derivation in the following sense: a linear mapping from into such that for every , there exists a derivation , depending on , satisfying . In [10], R. Kadison proves that each continuous local derivation from a von Neumann algebra into its dual Banach module is a derivation. In [13], D. Larson and A. Sourour prove that each local derivation from into itself is a derivation, where is a Banach space. B. Jonson [8] proves that each local derivation from a C*-algebra into its Banach bimodule is a derivation. Z. Pan and the second author of this paper [14] prove that each local derivation from the algebra into is a derivation, where is a Hilbert space, is a von Neumann algebra acting on , and is a commutative subspace lattice in . For more information about this topic, we refer to [2, 4, 6].
In 1997, P. emrl [18] introduces the concept of 2-local derivations. Recall that a mapping : (not necessarily linear) is called a 2-local derivation if for each , there exists a derivation : such that and . Moreover, the author proves that every 2-local derivation on is a derivation for a separable Hilbert space . J. Zhang and H. Li [19] extend the above result for arbitrary symmetric digraph matrix algebras and construct an example of 2-local derivation which is not a derivation on the algebra of all upper triangular complex matrices. S. Ayupov and K. Kudaybergenov [1] prove that each 2-local derivation on a von Neumann algebra is a derivation. For more information about this topic, we refer to [2, 7, 11].
In this paper, we study derivations, local derivations and 2-local derivations on some algebras of operators on Hilbert C*-modules. There are few results in this topic. P. Li, D. Han and W. Tang [15] prove that each derivation on End is inner, where is a full Hilbert C*-module over a commutative unital C*-algebra . M. Moghadam, M. Miri and A. Janfada [16] prove that each -linear derivation on End is inner, where is a full Hilbert C*-module over a commutative unital C*-algebra with the property that there exist in and in such that .
Hilbert C*-modules provide a natural generalization of Hilbert spaces by replacing the complex field with an arbitrary C*-algebra. The theory of Hilbert C*-modules plays an important role in the theory of operator algebras, as it can be applied in many fields, such as index theory of elliptic operators, K- and K K-theory, noncommutative geometry, and so on.
In the following, we would firstly review some properties of Hilbert C*-modules [12].
Let be a C*-algebra and be a left -module.
is called a Pre-Hilbert -module if there exists a mapping with the following properties: for each ,
(1) , and implies that ,
(2) ,
(3) ,
(4) .
The mapping is called an -valued inner product. The inner product induces a norm on : . is called a Hilbert -module(or more exactly, a Hilbert C*-module over ), if it is complete with respect to this norm.
We denote by the closure of the linear span of all the elements of the form . is called a full Hilbert -module if .
For a full Hilbert -module , we have the following lemma.
Lemma 1.1**.**
Let be a C-algebra with unit and be a full Hilbert -module. There exists a sequence , such that .*
A linear mapping from into itself is said to be -linear if for each and . A bounded -linear mapping from into itself is called an operator on . Denote by End all operators on . End is a Banach algebra.
A mapping from into itself is said to be adjointable if there exists a mapping such that , for all . Notice that each adjointable mapping must be an operator. Denote by End all adjointable operators on . End is a C*-algebra.
Similarly, a linear mapping from into is said to be -linear if for each and . The set of all bounded -linear mappings from to is denoted by .
For each in , one can define a mapping from to by follows: , for all . Obviously, .
For each in and in , one can define a mapping from into itself by follows: , for all . Obviously, End.
In particular, for each in , we have , for all .
For the operators of the above forms, we have the following lemmas.
Lemma 1.2**.**
Let be a Hilbert C-module over a C*-algebra .
For all End, we have
(1) ,
(2) ,
(3) if in addition, is commutative, then , .*
Lemma 1.3**.**
Let be a Hilbert C-module over a C*-algebra .
For all End*, we have
(1) End*, and ,
(2) ,
(3) ,
(4) if in addition, is commutative, then , .*
For a commutative C*-algebra , for each in , one can define a mapping from into itself by follows: , for all . Obviously, End. It is worthwhile to notice that if is not commutative, then is not -linear. In this case, is not in End.
Lemma 1.4**.**
Let be a commutative C-algebra with unit and be a full Hilbert -module. Then End.*
Proof.
For each in End and in , since , we have . It is to say End.
On the other hand, assume End. By Lemma 1.1, there exists a sequence , such that . Thus we have
[TABLE]
and
[TABLE]
Let . Then we have . The proof is complete. ∎
For an algebra , if for each in , implies that , then it is said to be semi-prime.
Lemma 1.5**.**
Let be a C-algebra and be a Hilbert -module. Then End is a semi-prime Banach algebra.*
Proof.
Let be in End. Assume that for each in End. In particular, for each and , we have
[TABLE]
By taking and , we have . It follows that
[TABLE]
Since is a self-adjoint element, we have , and . Hence , and End is semi-prime. The proof is complete. ∎
2 Derivations on End
In this section, we study derivations on End. We begin with several lemmas.
Lemma 2.1**.**
Let be a commutative unital C-algebra and be a full Hilbert -module. Then each derivation on is -linear, i.e. for each and .*
Proof.
Suppose is a derivation on End.
By Lemma 1.4, we have End. For each in End, By
[TABLE]
and
[TABLE]
we obtain . Hence End, and EndEnd.
Since End is a commutative C*-algebra, and every derivation on a commutative C*-algebra is zero, we have .
It follows that
[TABLE]
which means that is -linear. The proof is complete. ∎
Lemma 2.2**.**
Let be a commutative unital C-algebra and be a full Hilbert -module. Then each derivation on End is continuous.*
Proof.
Suppose is a derivation on End. Assume that is a sequence converging to zero in End, and converges to .
According to the closed graph theorem, to show is continuous, it is sufficient to prove that .
By Lemma 2.1, we know is -linear. For , , we have
[TABLE]
and
[TABLE]
Since converges to zero and converges to , we have
[TABLE]
It follows that
Let , then we have . For each , we have
[TABLE]
By taking and in (2.1), we can obtain , i.e.
[TABLE]
By taking and in (2.2), we can obtain . i.e. . The proof is complete. ∎
Now we can prove our main theorem in this section.
Theorem 2.3**.**
Let be a commutative C-algebra with unit and be a full Hilbert -module. Then each derivation on End is an inner derivation.*
Proof.
Suppose is a derivation on End and is a sequence in such that .
Define a mapping from into itself by follows:
[TABLE]
for all
By Lemmas 2.1 and 2.2, is -linear and continuous, thus is also -linear and continuous. That is to say End.
Now it is sufficient to show that , for each End.
For each , we have
[TABLE]
It implies that . Hence is an inner derivation. The proof is complete. ∎
3 2-Local derivations on End and End
In this section, we characterize 2-local derivations on End and End. Firstly, we show the following lemma.
Lemma 3.1**.**
Let be a commutative unital C-algebra and be a Hilbert -module. For and , if , then .*
Proof.
Let and .
We have
[TABLE]
Since is a commutative unital C*-algebra, it is well known that is -isomorphic to for some compact Hausdorff space . Without loss of generality, we can assume .
Then for each , we have and .
Recall that for a matrix in , implies that , where denotes the trace of , i.e. the sum of all the diagonal elements.
Hence implies that . It follows that , that is to say . The proof is complete. ∎
Theorem 3.2**.**
Let be a commutative unital C-algebra and be a full Hilbert -module. Then each 2-local derivation on End is a derivation.*
Proof.
Denote by the linear span of the set . By Lemma 1.2, is a two-side ideal of End.
For each , define .
One can verify that is well defined by Lemma 3.1. And obviously, is -linear. Moreover, for each End, we have
[TABLE]
It follows that for each End and .
Suppose is a 2-local derivation on End. By the definition of 2-local derivation, there exists a derivation on End such that and . By Theorem 2.3, is an inner derivation, i.e. there exists an element End such that .
Thus we have
[TABLE]
Since is a two-side ideal of End, we know that .
Hence
[TABLE]
which follows that .
Now, for each End and , we have
[TABLE]
Let , we obtain .
By taking , we have
[TABLE]
It means that . That is to say is an additive mapping. In addition, by the definition of 2-local derivation, it is easy to show that is homogeneous and for each End. Hence is a Jordan derivation.
By Lemma 1.5, End is a semi-prime Banach algebra. According to the classical result that every Jordan derivation on a semi-prime Banach algebra is a derivation [5], we obtain that is a derivation. The proof is complete. ∎
Theorem 3.3**.**
Let be a commutative unital C-algebra and be a full Hilbert -module. Then each 2-local derivation on End* is a derivation.*
Proof.
Denote by the linear span of the set . By Lemma 1.3, is a two-side ideal of End.
For each , define .
By Lemma 3.1, is well defined. For each End, we have
[TABLE]
It follows that for each End and .
In [15], the authors prove that for a commutative unital C*-algebra and a full Hilbert -module , each derivation on End is an inner derivation.
The rest of the proof is similar to Theorem 3.2, so we omit it. ∎
4 Local derivations on End
In this section, we discuss local derivations on End. Through this section, we assume that is a commutative C*-algebra with unit , and is a Hilbert -module, and moreover, there exist in and in such that . Denote the unit of End by . Define , and
Lemma 4.1**.**
*(1) is an idempotent;
(2) each element in is an -linear combination of some idempotents in , and each element in is an -linear combination of some idempotents in ;
(3) is a left ideal of , and is a right ideal of ;
(4) is a left separating set of End, i.e. for each in End, implies that , and is a right separating set of End, i.e. for each in End, implies that *
Proof.
(1) .
(2) For each , there exists a non-zero complex number , such that is invertible in . Denote by , then we have
[TABLE]
By (1), we know that is an idempotent.
Thus we have
[TABLE]
That is to say is an -linear combination of idempotents in .
Similarly, for each , there exists a non-zero complex number , such that is invertible in . Denote by , then we have
[TABLE]
Again by (1), we know that is an idempotent.
Thus we have
[TABLE]
(3) For each End since , we know that is a left ideal of End. Similarly, is a right ideal of End since .
(4) Suppose End.
If , then
[TABLE]
i.e. .
If , then
[TABLE]
The proof is complete. ∎
Let be a left -module, and be a bilinear mapping from End End into .
We say that is -bilinear if for each End and .
We say that preserves zero product if implies that for each End.
Lemma 4.2**.**
Let be a left -module, and be an -bilinear mapping preserving zero product. Then for each , and , we have:
[TABLE]
and
[TABLE]
Proof.
Suppose is an idempotent in End. Let .
Since preserves zero product, we have
[TABLE]
By Lemma 4.1(2), each element in is an -linear combination of idempotents in . Considering is -bilinear, we obtain that .
By Lemma 4.1(3), is a left ideal, so . Hence .
Similarly, we can show the equation (4.2) is true. ∎
For an algebra with unit , a linear mapping on is said to be a generalized derivation if , for all in .
Theorem 4.3**.**
Suppose that is a commutative C-algebra with unit , and is a Hilbert -module, and moreover, there exist in and in such that . If is an -linear mapping from into itself such that: for each in End, implies that , then is a generalized derivation. In particular, if , where is the unit of End, then is a derivation.*
Proof.
Suppose are arbitrary elements in End, where , and are arbitrary elements in and , respectively.
Define a bilinear mapping : . Then is an -bilinear mapping preserving zero product.
By Lemma 4.2, we have
[TABLE]
i.e.
[TABLE]
Since is a right separating set of End, we have
[TABLE]
Now define a bilinear mapping : . Then is also an -bilinear mapping preserving zero product.
Again by Lemma 4.2, we have
[TABLE]
i.e.
[TABLE]
Since is a left separating set of End, we obtain that
[TABLE]
That is to say is a generalized derivation. The proof is complete. ∎
Applying the above Theorem, we can get the following corollary immediately.
Corollary 4.4**.**
Suppose is a commutative C-algebra with unit , is a Hilbert -module, and moreover, there exist in and in such that . Then each -linear local derivation on is a derivation.*
Proof.
For each in End, if , by the definition of local derivation, there exists a derivation such that . Thus we have
[TABLE]
Let be the unit of End, by the definition of local derivation, there exists a derivation such that .
By Theorem 4.3, is a derivation. The proof is complete. ∎
Acknowledgements. This paper was partially supported by National Natural Science Foundation of China(Grant No. 11371136).
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