Linear maps on C*-algebras which are derivations or triple derivations at a point
Ahlem Ben Ali Essaleh, Antonio M. Peralta

TL;DR
This paper investigates linear maps on C*-algebras that behave like derivations or triple derivations at specific points, establishing conditions under which they are generalized derivations or *-derivations, with automatic continuity results.
Contribution
It introduces the study of triple derivations at a point on C*-algebras and characterizes when such maps are generalized derivations or *-derivations, including automatic continuity results.
Findings
Triple derivation at the unit implies generalized derivation.
Additional conditions lead to *-derivation and triple derivation.
Automatic continuity holds for derivations at zero in von Neumann algebras.
Abstract
We prove new results on generalized derivations on C-algebras. By considering the triple product , we introduce the study of linear maps which are triple derivations or triple homomorphisms at a point. We prove that a continuous linear map on a unital C-algebra is a generalized derivation whenever it is a triple derivation at the unit element. If we additionally assume then is a -derivation and a triple derivation. Furthermore, a continuous linear map on a unital C-algebra which is a triple derivation at the unit element is a triple derivation. Similar conclusions are obtained for continuous linear maps which are derivations or triple derivations at zero. We also give an automatic continuity result, that is, we show that generalized derivations on a von Neumann algebra and linear maps on a von Neumann algebra…
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Linear maps on C∗-algebras which are derivations or triple derivations at a point
Ahlem Ben Ali Essaleh
Faculte des Sciences de Monastir, Dpartement de Mathmatiques, Avenue de L’environnement, 5019 Monastir, Tunisia
and
Antonio M. Peralta
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain.
Abstract.
We prove new results on generalized derivations on C∗-algebras. By considering the triple product , we introduce the study of linear maps which are triple derivations or triple homomorphisms at a point. We prove that a continuous linear map on a unital C∗-algebra is a generalized derivation whenever it is a triple derivation at the unit element. If we additionally assume then is a ∗-derivation and a triple derivation. Furthermore, a continuous linear map on a unital C∗-algebra which is a triple derivation at the unit element is a triple derivation. Similar conclusions are obtained for continuous linear maps which are derivations or triple derivations at zero.
We also give an automatic continuity result, that is, we show that generalized derivations on a von Neumann algebra and linear maps on a von Neumann algebra which are derivations or triple derivations at zero are all continuous even if not assumed a priori to be so.
Key words and phrases:
derivations; triple derivations; derivable mapping at a point; triple derivation at a point; generalized derivation; triple homomorphism at a point; orthogonality preserver
2010 Mathematics Subject Classification:
47B49, 46L05, 46L40, 46T20, 47L10, 47L35, 47L99, 47B47, 47B48
1. Introduction
Automorphisms and derivations on Banach algebras are among the most intensively studied classes of operators. Recent studies are concerned with the question of finding weaker conditions to characterize these maps. One of the most fruitful lines studies maps which are derivations or automorphisms at a certain point. More concretely, a linear map from a Banach algebra to a Banach -bimodule is said to be a derivation at a point if the identity
[TABLE]
holds for every with . In the literature a linear map which is a derivation at a point is also called derivable at . Clearly, a linear map from into is a derivation if and only if it is a derivation at every point of . We can similarly define linear maps which are Jordan derivations or generalized derivations at a point (see subsection 1.1 for detailed definitions).
Following the terminology set by J. Alaminos, M. Bresar, J. Extremera, and A. Villena in [1, §4] and J. Li and Z. Pan in [23], we shall say that a linear operator from a Banach algebra into a Banach -bimodule is a generalized derivation if there exists satisfying
[TABLE]
Every derivation is a generalized derivation, however there exist generalized derivations which are not derivation. This notion is very useful when characterizing (generalized) derivations in terms of annihilation of certain products of orthogonal elements (see, for example, Theorem 2.11 in [2, §2]).
The first results on linear maps that are derivable at zero appear in [4, Subsection 4.2] and [9, Theorem 2], where they were related to generalized derivations. In [19, Theorem 4] W. Jing, S.J. Lu, and P.T. Li prove that the implication
[TABLE]
holds for every continuous linear map on a von Neumann algebra. If, under the above hypothesis , then is a derivation. We shall prove in Corollary 2.16 that the hypothesis concerning continuity can be relaxed.
W. Jing proves in [18, Theorems 2.2 and 2.6] the following result: for an infinite dimensional Hilbert space , a linear map which is a generalized Jordan derivation at zero, or at 1, is a generalized derivation. We observe that, in the latter result, is not assumed to be continuous.
More related results read as follow. Let be a Banach -bimodule over a Banach algebra . In 2009, F. Lu establishes that a linear map is a derivation whenever it is continuous and a derivation at an element which is left (or right) invertible (see [24]). Is is further shown that is a derivation if it continuous and a derivation at an idempotent in such that for the condition implies and the condition gives . Here the linear map is assumed to be continuous.
Concerning our goals, J. Zhu, Ch. Xiong, and P. Li prove in [39] a significant result showing that, for a Hilbert space , a linear map is a derivation if and only if it is a derivation at a non-zero point in . It is further shown that a linear map which is a derivation at zero need not be a derivation (for example, the identity mapping on is a derivation at zero but it is not a derivation).
We refer to [17, 19, 32, 33, 34, 35, 36, 37, 38] and [40] for additional results on linear or additive maps on JSL algebras, finite CSL algebras, nest algebras or standard operator algebras.
In the present note we continue with the study of those linear maps which are derivable at zero. We shall introduce a new point of view by exploiting those properties of a C∗-algebra which are related to the ternary product defined by
[TABLE]
Every C∗-algebra is a JB∗-triple (in the sense of [21]) with respect to the triple product defined in (2). This is the natural triple product appearing in the study of J∗-algebras by L.A. Harris [13, 14] and the ternary rings of operators (TRO’s) in the sense of D.P. Blecher and M. Neal in [3] and M. Neal and B. Russo in [25].
A linear map between C∗-algebras preserving the previous triple product is called a triple homomorphism. A triple derivation on a C∗-algebra is a linear map satisfying the generalized Leibnitz’s rule
[TABLE]
for all .
We recall that a ∗-derivation on a C∗-algebra is a derivation satisfying for all . Examples of derivations on be given by fixing and defining as the linear map defined by . It is known that every ∗-derivation on a C∗-algebra is a triple derivation in the above sense. It is further known the existence of derivations on which are not triple derivations (compare [7, Comments after Lemma 3]).
On the other hand, for each in a C∗-algebra , the mapping is a triple derivation on , however, if and only if , and thus is not an associative derivation on for every .
In a recent paper M.J. Burgos, J.Cabello-Sánchez and the second author of this note explore those linear maps between C∗-algebras which are ∗-homomorphisms at certain points of the domain, for example, at the unit element and at zero (see the introduction of section 3 for more details).
In this paper we widen the scope by introducing linear maps which are triple derivations or triple homomorphism at a certain point. Our study will be conducted around the next two notions.
Definition 1.1**.**
Let be a linear map on a C∗-algebra, and let be an element in We shall say that is a triple derivation at if in implies that
The set of all linear maps on which are triple derivable at an element is a subspace of the space of all linear operators on .
Definition 1.2**.**
Let be a linear map between C∗-algebras, and let be an element in We shall say that is a triple homomorphism at if in implies that
Let be a continuous linear map on a unital C∗-algebra. In Theorem 2.3 we prove that being a triple derivation at the unit implies that is a generalized derivation. If we also assume that then is a ∗-derivation and a triple derivation (see Proposition 2.4). Among the consequences, we establish that a continuous linear map on a unital C∗-algebra which is a triple derivation at the unit element is a triple derivation (see Corollary 2.5).
When we study linear maps which are triple derivation at zero, our conclusions are stronger. We begin with an extension of [19, Theorem 4] to the setting of unital C∗-algebras. We show that a continuous linear map on a C∗-algebra is a generalized derivation whenever it is a derivation or a triple derivation at zero (see Theorem 2.9). Moreover, a bounded linear map on a C∗-algebra which is a triple derivation at zero with is a ∗-derivation, and hence a triple derivation (compare Corollary 2.10). We further show that a bounded linear map on a unital C∗-algebra which is a triple derivation at zero and satisfies is a triple derivation (see Corollary 2.11).
For linear maps whose domain is a von Neumann algebra the continuity assumptions can be dropped for certain maps. More concretely, generalized derivations on a von Neumann algebra, linear maps on a von Neumann algebra which are derivations (respectively, triple derivations) at zero are all continuous (see Corollary 2.13). Several characterizations of generalized derivations on von Neumann algebras are established in Corollary 2.15 without assuming continuity. In this particular setting, some hypothesis in [19, Theorem 4] and [24] can be relaxed.
In section 3 we study continuous linear maps on C∗-algebras which are triple homomorphisms at zero or at the unit element. Let be a continuous linear map between C∗-algebras, where is unital. We prove in Theorem 3.4 that if is a triple homomorphism at the unit of then is a triple homomorphism. Furthermore, is a partial isometry and is a Jordan ∗-homomorphism.
For triple homomorphisms at zero, we rediscover the orthogonality preserving operators. More concretely, let be a bounded linear map between two C∗-algebras. We shall revisit the main results in [6] to show that is orthogonality preserving if, and only if, preserves zero-triple-products (i.e. in implies in ) if, and only if, a triple homomorphism at zero.
1.1. Basic background and definitions
The class of C∗-algebras admits a Jordan analogous in the wider category of JB∗-algebras. More concretely, a real (resp., complex) Jordan algebra is an algebra over the real (resp., complex) field whose product is commutative (but, in general, non-associative) and satisfies the Jordan identity:
[TABLE]
A JB∗-algebra is a complex Jordan algebra which is also a Banach space and admits an isometric algebra involution ∗ satisfying and
[TABLE]
for all , where . Every C∗-algebra is a JB∗-algebra with respect to its natural norm and involution and the Jordan product given by . The self-adjoint part of a JB∗-algebra is a real Jordan Banach algebra which satisfies
[TABLE]
for every These axioms provide the precise definition of JB-algebras. A JBW∗-algebra (resp., a JBW-algebra) is a JB∗-algebra (resp., a JB-algebra) which is also a dual Banach space. The bidual of every JB∗-algebra is a JBW∗-algebra with a Jordan product and involution extending the original ones. The reader is referred to the monograph [12] for the basic background on JB∗- and JB-algebras.
Let be a JB∗-subalgebra of a JB∗-algebra . Accordingly to the notation in [1, 7, 8] a linear mapping will be called a generalized Jordan derivation if there exists satisfying
[TABLE]
for all in , where (). We shall write instead of . If is unital, every generalized Jordan derivation with is a Jordan derivation. Jordan derivations are generalized Jordan derivations.
2. Triple derivations at fixed point of a C∗-algebra
In this section we shall study linear maps between C∗-algebras which are triple derivations at a fixed point of the domain. There are two remarkable elements that every study should consider in a first stage, we refer to zero and the unit element of a C∗-algebra. We shall show later that linear maps between C∗-algebras which are triple derivations at zero or at the unit element are intrinsically related to generalized derivations.
Let be a bounded linear operator from a C∗-algebra into an essential Banach -bimodule. It is proved in [1, Theorem 4.5] and [8, Proposition 4.3] (see also [2, Theorem 2.11]) that is a generalized derivation if and only if one of the next statements holds:
is a generalized derivation; 2.
, whenever in ; 3.
, whenever in .
When in the above statement coincides with or with any C∗-algebra containing as a C∗-subalgebra with the same unit, the above equivalent statements admit another reformulation which is more interesting for our purposes. We shall isolate here an equivalence which was germinally contained in the proof of [10, Theorem 2.8]. More concretely, each statement in - is equivalent to any of the following:
, whenever in ; 2.
, whenever in . 3.
For each in we have in , where denotes the range projection of in .
For the proof we observe that and are clear. We shall prove . Suppose , whenever in . We shall focus on the commutative C∗-subalgebra generated by .
It is known from the Gelfand theory that , where denotes the spectrum of and the C∗-algebra of all continuous functions on vanishing at zero. For each natural , let denote the projection in corresponding to the characteristic function of the set Let us also pick a function such that and . Clearly, converges to , the range projection of in the strong∗-topology of (see [29, §1.8]). Let us take . Since , it follows from the hypothesis that .
On the other hand, it is known that is a closed projection in in the sense employed in [30, Definition III.6.19]. It is known that, under these circumstances, is weak∗-dense in (compare [26, Proposition 3.11.9]). By Kaplansky density theorem [29, Theorem 1.9.1], we can find a bounded net in converging to in the strong∗-topology of . We have seen above that for all . Since the product of is jointly strong∗-continuous (cf. [29, Proposition 1.8.12]), we deduce that for all natural . Since in the strong∗-topology and in norm, we have in .
We take with . We can easily see that and . Therefore, by assumptions, , which finishes the proof.
2.1. Triple derivations at the unit element of a C∗-algebra
Along the rest of this subsection, the symbol will denote a C∗-subalgebra of unital C∗-algebra , and we shall assume that contains the unit of .
Continuous linear maps which are derivations at 1 are derivations. This problem has been already studied in the literature, at least for continuous linear maps (see [24, Theorem 2.1 or Corollary 2.3]). Actually the next result follows from the just quoted reference and [20, Theorem 6.3].
Proposition 2.1**.**
([24, Theorem 2.1 or Corollary 2.3], [20, Theorem 6.3])* Let be a unital C∗-algebra, and be a unital Banach -bimodule. Suppose is a continuous linear map which is a derivation at the unit element. Then is a derivation.*
A common property of triple derivations and local triple derivations on C∗-algebras is that they map the unit of the domain C∗-algebra into a skew symmetric element (cf. [15, proof of Lemma 1], [16, Lemma 3.4] or [22, Lemma 2.1]). This good behavior is also true for linear maps which are derivations at the unit element.
Lemma 2.2**.**
Let be a triple derivation at the unit of Then the following statements hold:
** 2.
The identity holds for every projection in
Proof.
Since by assumptions, we have
[TABLE]
which proves the statement.
Let be a projection. The identity and the hypothesis prove that
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
This implies that ∎
There exist C∗-algebras containing no non-zero projections. For this reason, we need to deal with unitaries.
Theorem 2.3**.**
Let be a continuous linear map which is a triple derivation at the unit of Then is a generalized derivation.
Proof.
Let us take Since, for each is a unitary element in and we deduce that
[TABLE]
Taking the first derivative in we get
[TABLE]
[TABLE]
for every Taking a new derivative at in the last equality, we get
[TABLE]
[TABLE]
or equivalently,
[TABLE]
Lemma 2.2 assures that which implies that
[TABLE]
for every in
Finally, let us take with . If we write with and for all . Find in such that (). It is not hard to check (for example, by applying the orthogonality of the corresponding range projections) that for . Now applying (5) we get
[TABLE]
[TABLE]
We deduce from [2, Theorem 2.11] that is a generalized derivation. ∎
There exists generalized derivations which are not triple derivations at 1. For example, let be a non-zero symmetric element in and define (). Then for all , which assures that is a generalized derivation. However, together with Lemma 2.2 assure that is not a triple derivation at .
An appropriate change in the arguments given in the above theorem provide additional information when .
Proposition 2.4**.**
Let be a continuous linear map which is a triple derivation at 1 with Then is a ∗-derivation and a triple derivation.
Proof.
As in the above proof, let us take Since, for each is a unitary element in and we deduce that
[TABLE]
that is,
[TABLE]
Taking derivatives at we get
[TABLE]
which proves that for all , and thus is a symmetric map.
Finally, by Theorem 2.3, is a generalized derivation. Furthermore, since and is a symmetric operator, we deduce that is a ∗-derivation and a triple derivation as well. ∎
Corollary 2.5**.**
Let be a continuous linear map on a unital C∗-algebra which is a triple derivation at 1. Then is a triple derivation.
Proof.
Since, by Lemma 2.2, it is known that the mapping is a triple derivation (compare [15, Proof of Lemma 1]). Since the linear combination of linear maps which are triple derivations at is a triple derivation at , the mapping is a triple derivation at 1, and . Applying Proposition 2.4, we derive that is a ∗-derivation. Therefore is a triple derivation. ∎
Problem 2.6**.**
If is a triple derivation at the unit of a unital C∗-algebra , is continuous?
2.2. Triple derivations at zero
We begin this subsection exploring the basic properties of linear maps which are derivations at zero.
Lemma 2.7**.**
Suppose is a C∗-subalgebra of a C∗-algebra , and let be a linear map which is a derivation at zero. Then
[TABLE]
Proof.
Suppose satisfy the hypothesis of the lemma. Since is a derivation at zero we have ∎
Let us observe that under the hypothesis of the above lemma, we are not in a position to apply [1, Theorem 4.5] and [8, Proposition 4.3] (see also [2, Theorem 2.11]) and the reformulations we have reviewed in page 2 because is not assumed to be continuous. We shall see later that continuity can be relaxed when the domain is a von Neumann algebra.
Let we recall that and are orthogonal (written ) if and only if
Lemma 2.8**.**
Suppose is a C∗-subalgebra of a C∗-algebra , and let be a linear map which is a triple derivation at zero. Then
[TABLE]
Proof.
Let us take satisfying Since it follows from the hypothesis that
[TABLE]
which proves the statement because . ∎
We can now apply the reformulations of being a generalized derivation proved in page 2. Let us recall that as observed by J. Zhu, Ch. Xiong, and P. Li in [39] linear maps which are derivations at zero need not be derivations. We shall see next that continuous linear maps which are derivations at zero are always generalized derivations.
Theorem 2.9**.**
Let be a C∗-subalgebra of a unital C∗-algebra . Let be a bounded linear map. If is a derivation at zero or a triple derivation at zero, then is a generalized derivation.
Proof.
If is a triple derivation at zero, by Lemma 2.8, given with we have
[TABLE]
or equivalently,
[TABLE]
Lemma 2.7 assures that a similar conclusion holds when is a derivation at zero. It follows from the equivalence in page 2 that is a generalized derivation. ∎
We observe that Theorem 2.9 above extends [19, Theorem 4] to the setting of unital C∗-algebras.
Corollary 2.10**.**
Suppose is a unital C∗-algebra. Let be a bounded linear map which is a triple derivation at zero with Then is a ∗-derivation, and hence a triple derivation.
Proof.
Since , the previous Theorem 2.9 assures that is a derivation. We shall next show that is a symmetric mapping. It is well known that the bitransposed is a derivation too (see for example [29, Lemma 4.1.4]). To avoid confusion with the natural involution on , we shall denote by .
Fix with range projection . Applying the same arguments given in the proof of in page 2, we can find sequences and such that , is a closed projection in for every , , and for each natural , there exists a bounded net in converging to in the strong∗-topology (and hence in the weak∗-topology) of . By hypothesis
[TABLE]
Taking weak∗-limits in the above equality we get from the weak∗-continuity of that
[TABLE]
which implies, via norm continuity, that
[TABLE]
or equivalently
[TABLE]
Since the range projection of every power with coincides with the we can apply the above argument to deduce that
[TABLE]
and by linearity and norm continuity of the product we have
[TABLE]
A standard argument involving weak∗-continuity of gives
[TABLE]
Combining that is a derivation with (7) we get
[TABLE]
By [11, Proposition 3.7], we know that , and by the equivalence in page 2 we have
[TABLE]
and thus
[TABLE]
and
[TABLE]
Adding the last two identities we derive at
[TABLE]
We have proved that for every range projection of a hermitian element in
We return to . We observe that every projection of the form , with , is the range projection of an function in . Furthermore, every projection of the form with can be written as the difference of two projections of the previous type. We have shown in the previous paragraph that for every projection of the first type, and consequently for every projection of the second type. Since can be approximated in norm by finite linear combinations of mutually orthogonal projections of the second type, and is continuous, we conclude that , which finishes the proof. ∎
The conclusion after Corollary 2.10 is now clear.
Corollary 2.11**.**
Let be a bounded linear map on a unital C∗-algebra . Suppose is triple derivable at zero and . Then is a triple derivation.
Proof.
As in the proof of Corollary 2.5. Since the mapping is a triple derivation. Since the the linear combination of linear maps which are triple derivations at zero is a triple derivation at zero, the mapping is a triple derivation at zero, and . Applying Corollary 2.10, we derive that is a ∗-derivation, and hence a triple derivation. Therefore is a triple derivation. ∎
The above results, are somehow optimal, in the sense that there exists bounded linear maps which are triple derivations at zero but they are not triple derivations. For example, let be the center of a unital C∗-algebra , and let us pick with . We define a bounded linear mapping by . Clearly, implies that is not a triple derivation (cf. Lemma 2.2[15, proof of Lemma 1], [16, Lemma 3.4] or [22, Lemma 2.1]). Suppose in . Since is the center of , commutes with every element in , and then we have
[TABLE]
[TABLE]
witnessing that is a triple derivation at zero.
With the help of [10, Theorem 2.8 and Proposition 2.4] we can now throw some new light about the automatic continuity of generalized derivations and linear maps which are derivable at zero on a von Neumann algebra.
Theorem 2.12**.**
Let be a linear mapping on a von Neumann algebra. Suppose that for each in a commutative von Neumann subalgebra with we have . Then is continuous.
Proof.
We can mimic the ideas in the proof of [10, Theorem 2.8]. Assume that satisfies hypothesis (respectively ). Let be a commutative von Neumann subalgebra of containing the unit of . Fix with , and define a linear mapping , by . Let be elements in with , then, by hypothesis (because ). This proves that is a linear left-annihilator preserving. Proposition 2.4 in [10] assures that is continuous and a left-multiplier, that is, for every . This property assures that, for each the mapping , satisfies for every in . Consequently, is a linear right-annihilator preserving, and Proposition 2.4 in [10] proves that is a continuous right multiplier. We have shown that
[TABLE]
for every , or equivalently, is a derivation. Theorem 2 in [27] assures that is a bounded linear map, and this holds for every abelian von Neumann subalgebra of containing the unit of . The continuity of follows as a consequence of [28, Theorem 2.5]. ∎
Every generalized derivation on a von Neumann algebras satisfies the hypothesis of the above Theorem 2.12. Surprisingly, the linear maps on a von Neumann algebra which are triple derivable at zero also satisfy the same hypothesis.
Corollary 2.13**.**
Every generalized derivation on a von Neumann algebra is continuous. Every linear map on a von Neumann algebra which is a derivation (respectively, a triple derivation) at zero is continuous.
Proof.
The first statement is clear. The statement concerning (associative) derivations at zero is a consequence of Lemma 2.7. In order to prove the remaining statement, we assume that is a linear map on a von Neumann algebra which is triple derivable at zero. Let be a commutative von Neumann subalgebra of containing the unit, and let us take with . By the commutativity of we have and . By Lemma 2.8 we have and then .
Since is a commutative von Neumann algebra, the every element in is normal and its left and right range projections coincide and belong to . Let be the range projection of . Since , we deduce from what is proved in the first paragraph that . Finally, since and , we have . This shows that satisfies the hypothesis in Theorem 2.12, and thus is continuous. ∎
Combining Corollary 2.13 with Theorem 2.9 and Corollaries 2.10 and 2.11 we establish the following.
Corollary 2.14**.**
Let be a von Neumann algebra. Suppose is a linear map which is a triple derivation at zero. Then the following statements hold:
* is a continuous generalized derivation;* 2.
If then is a (continuous) ∗-derivation and a triple derivation; 3.
If then is a (continuous) triple derivation.
We can also apply the above results to relax some of the hypothesis in previous papers. We begin with a version of the results reviewed in page 2 for non-necessarily continuous linear maps.
Corollary 2.15**.**
Let be a linear map on a von Neumann algebra. The the following statements are equivalent:
* is a generalized derivation;* 2.
, whenever in ; 3.
, whenever in ; 4.
, whenever in ; 5.
, whenever in . 6.
For each in we have , where denotes the range projection of in .
Proof.
The implication follows from Lemma 2.7, while the implications are clear.
If we show that any linear mapping satisfying or is continuous then the remaining implications will follow from the arguments given in page 2. If satisfies then also satisfies because for every and .
Let be a commutative von Neumann subalgebra of , and let us take with . We can write , and with .
Suppose satisfies . Since is commutative and for all . Clearly and , for all , which assures that and for all . Therefore, by assumptions, we obtain
[TABLE]
[TABLE]
which finishes the proof. ∎
In [19, Theorem 4] W. Jing, S.J. Lu, and P.T. Li prove that a continuous linear map on a von Neumann algebra is a generalized derivation whenever it is a derivation at zero. If additionally , then is a derivation. Corollary 2.13 assures that the hypothesis concerning the continuity of can be relaxed.
Corollary 2.16**.**
Let be a linear map on a von Neumann algebra. Suppose is a derivation at zero. Then is a (continuous) generalized derivation. If additionally , then is a derivation.
3. Triple homomorphisms at a fixed point
Let and be C∗-algebras. According to the notation in [5], a linear map is said to be a ∗-homomorphism at if
[TABLE]
In [5, Theorem 2.5] it is shown that when is unital, a linear map which is a ∗-homomorphism at is continuous and a Jordan ∗-homomorphism. The same conclusion hold if there exists a non-zero projection such that is a ∗-homomorphism at and at [5, Corollary 2.12]. Furthermore, in the above setting, is a ∗-homomorphism if and only if is a ∗-homomorphism at [math] and at [5, Corollary 2.11]. If is assumed to be simple and infinite, then a linear map is a ∗-homomorphism if and only if is a ∗-homomorphism at the unit of (cf. [5, Theorem 2.8]). In the just quoted paper, it also studied when a continuous linear map which is a ∗-homomorphism at a unitary element is a Jordan ∗-homomorphism.
We recall some terminology needed in forthcoming results. For each partial isometry in a C∗-algebra (i.e., ), we can decompose as a direct sum of the form
[TABLE]
The above decomposition is called the Peirce decomposition of associated with . The subsets , and are called the Peirce subspaces associated with .
3.1. Triple homomorphisms at the unit element
We explore first the behavior on the projections of a linear map which is a triple homomorphism at the unit.
Lemma 3.1**.**
Let be a linear map between C∗-algebras, where is unital. Suppose is a triple homomorphism at the unit of Then the following statements hold:
* is a partial isometry;* 2.
The identity holds for every projection .
Proof.
The identity and the hypothesis imply
[TABLE]
Let be a projection. We know that . Thus
[TABLE]
[TABLE]
which combined with gives the desired statement. ∎
For the next result we shall assume continuity of our linear map.
Proposition 3.2**.**
Let be a continuous linear map between C∗-algebras, where is unital. Suppose is a triple homomorphism at the unit of Then the identity
[TABLE]
holds for all . Consequently, is a partial isometry and .
Proof.
Let us take and Since is a unitary in , we have and by the assumptions we get By taking derivative in , we obtain
[TABLE]
Since for we can write with it follows from the above that
[TABLE]
[TABLE]
It follows from Lemma 3.1 that is a partial isometry. For the final statement we observe that for each we have
[TABLE]
[TABLE]
which shows that and hence
[TABLE]
∎
For the next result we explore new arguments with higher derivatives.
Proposition 3.3**.**
Let be a continuous linear map between C∗-algebras, where is unital. Suppose is a triple homomorphism at the unit of Then is a partial isometry and .
Proof.
As in previous cases, we fix . Since for all by hypothesis we get By taking a first derivative in , we obtain
[TABLE]
for all . By taking subsequent derivatives at we get
[TABLE]
[TABLE]
and
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
By replacing with [math] we get
[TABLE]
[TABLE]
Now, by Proposition 3.2, we write , and , where for all . It is not hard to check that , and all lie in while with and . It follows from (9) that and hence . We have therefore shown that for all . The desired conclusion follows from the linearity of . ∎
We can now establish our main result for bounded linear maps which are triple homomorphisms at the unit element. We recall that given a partial isometry in a C∗-algebra , the Peirce subspace is a JB∗-algebra with Jordan product and involution .
Theorem 3.4**.**
Let be a continuous linear map between C∗-algebras, where is unital. Suppose is a triple homomorphism at the unit of Then is a triple homomorphism. Furthermore, is a partial isometry and is a Jordan ∗-homomorphism.
Proof.
By Lemma 3.1 the element is a partial isometry. Proposition 3.3 proves that , and consequently, Proposition 3.2 guarantees that for every . It is not hard to see from these properties that for every .
The proof will be completed if we show that is a Jordan ∗-homomorphism. We shall only prove that preserves the corresponding Jordan product. Following the arguments in the proof of Proposition 3.3, and replacing with [math] in (8) we arrive at
[TABLE]
[TABLE]
and then
[TABLE]
[TABLE]
for all . A standard polarization argument proves that is a Jordan ∗-homomorphism.∎
Problem 3.5**.**
Let be a linear map between C∗-algebras, where is unital. Suppose is a triple homomorphism at the unit of Is continuous?
Accordingly to the structure of this note, the reader is probably interested on bounded linear maps which are triple homomorphisms at zero. It is not a big surprise that these maps are directly connected with the so-called orthogonality preserving operators in the sense studied, for example, in [31, 6], and subsequent papers. We recall that a linear map between C∗-algebras is called orthogonality preserving if the equivalence
[TABLE]
It is known that elements in a C∗-algebra are orthogonal if, and only if, (see, for example, [6, Lemma 1 and comments in page 221]). The main result in [6] establishes a complete description of those continuous linear maps between C∗-algebras which preserver orthogonal elements. Let be a bounded linear map between two C∗-algebras, Corollary 18 in [6] proves that is orthogonality preserving if, and only if, preserves zero-triple-products (i.e. in implies in ), and the latter is precisely the notion of being a triple homomorphism at zero.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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