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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 415354, 9 pages
http://dx.doi.org/10.1155/2013/415354
Research Article

Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras

1Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
2Dipartimento di Matematica e Informatica, Università di Catania, 95125 Catania, Italy
3Departamento de Algebra y Análisis Matemático, Universidad de Almería, 04120 Almería, Spain

Received 1 October 2013; Accepted 27 November 2013

Academic Editor: Ngai-Ching Wong

Copyright © 2013 Jorge J. Garcés et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study holomorphic maps between C-algebras and , when is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball . If we assume that is orthogonality preserving and orthogonally additive on and contains an invertible element in , then there exist a sequence in and Jordan -homomorphisms such that uniformly in . When is abelian, the hypothesis of being unital and can be relaxed to get the same statement.

1. Introduction

The description of orthogonally additive -homogeneous polynomial on -spaces and on general C*-algebras, developed by Benyamini et al. [1], Pérez-García and Villanueva [2], and Palazuelos et al. [3], respectively (see also [4, 5], [6, Section 3] and [7]), made functional analysts study and explore orthogonally additive holomorphic functions on -spaces (see [8, 9]) and subsequently on general C*-algebras (cf. [10]).

We recall that a mapping from a C*-algebra into a Banach space is said to be orthogonally additive on a subset if for every in with , and we have , where elements ,   in are said to be orthogonal (denoted by ) whenever . We will say that is additive on elements having zero product if for every ,   in with , we have . Having this terminology in mind, the description of all -homogeneous polynomials on a general C*-algebra, , which are orthogonally additive on the self-adjoint part, , of reads as follows (see Section 2 for concrete definitions not explained here).

Theorem 1 (see [3]). Let be a C*-algebra and a Banach space, , and let be an -homogeneous polynomial. The following statements are equivalent.(a)There exists a bounded linear operator satisfying for every and .(b) is additive on elements having zero products.(c) is orthogonally additive on .

The task of replacing -homogeneous polynomials by polynomials or by holomorphic functions involves a higher difficulty. For example, as noticed by Carando et al. [8, Example 2.2], when denotes the closed unit disc in , there is no entire function such that the mapping , factorizes all degree-2 orthogonally additive scalar polynomials over . Furthermore, similar arguments show that defining , , we cannot find a triplet , where is a *-homomorphism and , satisfying that for every .

To avoid the difficulties commented above, Carando et al. introduce a factorization through an space. More concretely, for each compact Hausdorff space , a holomorphic mapping of bounded type is orthogonally additive if and only if there exist a Borel regular measure on , a sequence , and a holomorphic function of bounded type such that and for every (cf. [8, Theorem 3.3]).

When is replaced with a general C*-algebra , a holomorphic function of bounded type is orthogonally additive on if and only if there exist a positive functional in , a sequence in , and a power series holomorphic function in such that for every in , where denotes the unit element in and is a noncommutative -space (cf. [10]).

A very recent contribution due to Bu et al. [11] shows that, for holomorphic mappings between spaces, we can avoid the factorization through an -space by imposing additional hypothesis. Before stating the detailed result, we will set down some definitions.

Let and be C*-algebras. When is a map and the condition holds for every , we will say that   preserves orthogonality or it is orthogonality preserving (resp.,   preserves zero products) on . In the case we will simply say that is orthogonality preserving (resp.,   preserves zero products). Orthogonality preserving bounded linear maps between C*-algebras were completely described in [12, Theorem 17] (see [6] for completeness).

The following Banach-Stone type theorem for zero product preserving or orthogonality preserving holomorphic functions between spaces is established by Bu et al. in [11, Theorem 3.4].

Theorem 2 (see [11]). Let and be locally compact Hausdorff spaces and let be a bounded orthogonally additive holomorphic function. If is zero product preserving or orthogonality preserving, then there exist a sequence of open subsets of , a sequence of bounded functions from into , and a mapping such that for each natural the function is continuous and nonvanishing on and uniformly in .

The study developed by Bu et al. is restricted to commutative C*-algebras or to orthogonality preserving and orthogonally additive, -homogeneous polynomials between general C*-algebras. The aim of this paper is to extend their study to holomorphic maps between general C*-algebras. In Section 4, we determine the form of every orthogonality preserving and orthogonally additive holomorphic function from a general C*-algebra into a commutative C*-algebra (see Theorem 16).

In the wider setting of holomorphic mappings between general C*-algebras, we prove the following: let and be C*-algebras with unital and let be a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball . Suppose is orthogonality preserving and orthogonally additive on and contains an invertible element. Then there exist a sequence in and Jordan -homomorphisms such that uniformly in (see Theorem 18).

The main tool to establish our main results is a newfangled investigation on orthogonality preserving pairs of operators between C*-algebras developed in Section 3. Among the novelties presented in Section 3, we find an innovating alternative characterization of orthogonality preserving operators between C*-algebras which complements the original one established in [12] (see Proposition 14). Orthogonality preserving pairs of operators are also valid to determine orthogonality preserving operators and orthomorphisms or local operators on C*-algebras in the sense employed by Zaanen [13] and Johnson [14], respectively.

2. Orthogonally Additive, Orthogonality Preserving, and Holomorphic Mappings on C*-Algebras

Let and be Banach spaces. Given a natural , a (continuous) -homogeneous polynomial from to is a mapping for which there is a (continuous) -linear symmetric operator such that , for every  . All polynomials considered in this paper are assumed to be continuous. By a -homogeneous polynomial we mean a constant function. The symbol will denote the Banach space of all continuous -homogeneous polynomials from to , with norm given by .

Throughout the paper, the word operator will always stand for a bounded linear mapping.

We recall that, given a domain in a complex Banach space (i.e., an open, connected subset), a function from to another complex Banach space is said to be holomorphic if the Fréchet derivative of at exists for every point in . It is known that is holomorphic in if and only if for each there exists a sequence of polynomials from into , where each is -homogeneous, and a neighborhood of such that the series, converges uniformly to for every . Homogeneous polynomials on a C*-algebra constitute the most basic examples of holomorphic functions on . A holomorphic function is said to be of bounded type if it is bounded on all bounded subsets of ; in this case its Taylor series at zero, , has infinite radius of uniform convergence, that is, (compare [15, Section 6.2], see also [16]).

Suppose is a holomorphic function and let be its Taylor series at zero which is assumed to be uniformly convergent in . Given , it follows from Cauchy's integral formula that, for each , we have where is the circle forming the boundary of a disc in the complex plane , taken counterclockwise, such that . We refer to [15] for the basic facts and definitions used in this paper.

In this section we will study orthogonally additive, orthogonality preserving, and holomorphic mappings between C*-algebras. We begin with an observation which can be directly derived from Cauchy's integral formula. The statement in the next lemma was originally stated by Carando et al. in [8, Lemma 1.1] (see also [10, Lemma 3]).

Lemma 3. Let be a holomorphic mapping, where is a C*-algebra and is a complex Banach space, and let be its Taylor series at zero, which is uniformly converging in . Then the mapping is orthogonally additive on (resp., orthogonally additive on or additive on elements having zero product in ) if and only if all the 's satisfy the same property. In such a case, .

We recall that a functional in the dual of a C*-algebra is symmetric when , for every . Reciprocally, if for every symmetric functional , the element lies in . Having this in mind, our next lemma also is a direct consequence of Cauchy's integral formula and the power series expansion of . A mapping between C*-algebras is called symmetric whenever , or equivalently, , whenever .

Lemma 4. Let be a holomorphic mapping, where and are C*-algebras, and let be its Taylor series at zero, which is uniformly converging in . Then the mapping is symmetric on (i.e., ) if and only if is symmetric (i.e., ) for every .

Definition 5. Let be a couple of mappings between two C*-algebras. One will say that the pair is orthogonality preserving on a subset if whenever in . When in implies in , we will say that preserves zero products on .

We observe that a mapping is orthogonality preserving in the usual sense if and only if the pair is orthogonality preserving. We also notice that is orthogonality preserving (on ) if and only if is orthogonality preserving (on ).

Our next result assures that the -homogeneous polynomials appearing in the Taylor series of an orthogonality preserving holomorphic mapping between C*-algebras are pairwise orthogonality preserving.

Proposition 6. Let be a holomorphic mapping, where and are C*-algebras, and let be its Taylor series at zero, which is uniformly converging in . The following statements hold. (a)The mapping is orthogonally preserving on (resp., orthogonally preserving on ) if and only if and the pair is orthogonality preserving (resp., orthogonally preserving on ) for every .(b)The mapping preserves zero products on if and only if and for every , the pair preserves zero products.

Proof. (a) The “if” implication is clear. To prove the “only if” implication, let us fix with . Let us find two positive scalars ,   such that and for every . From the Cauchy estimates we have , for every . By hypothesis , for every , hence and by homogeneity Letting , we have . In particular, .
We will prove by induction on that the pair is orthogonality preserving on for every . Since , we also deduce that for every , which implies that for every , and hence Taking limit in , we get . Let us assume that is orthogonality preserving on for every . Following the argument above we deduce that for every . Taking limit in , we have Replacing with () we get for every , which implies that In a similar manner we prove that , for every . The equalities () follow similarly.
We have shown that for each , whenever with . Finally, taking ,   with , we can find a positive such that and , which implies that for every , witnessing that is orthogonality preserving for every .
The proof of (b) follows in a similar manner.

We can obtain now a corollary which is a first step toward the description of orthogonality preserving, orthogonally additive, and holomorphic mappings between C*-algebras.

Corollary 7. Let be a holomorphic mapping, where and are C*-algebras and let be its Taylor series at zero, which is uniformly converging in . Suppose is orthogonality preserving and orthogonally additive on (resp., orthogonally additive and zero products preserving) . Then there exists a sequence of operators from into satisfying that the pair is orthogonality preserving on (resp., zero products preserving on ) for every and uniformly in . In particular every is orthogonality preserving (resp., zero products preserving) on . Furthermore, is symmetric if and only if every is symmetric.

Proof. Combining Lemma 3 and Proposition 6, we deduce that , is orthogonally additive on , and is orthogonality preserving on for every in . By Theorem 1, for each natural there exists an operator such that and for every .
Consider now two positive elements with and fix . In this case there exist positive elements in with and and . Since the pair is orthogonality preserving on , we have . Now, noticing that given ,   in with , we can write and , where and are positive, , and ; for every , we deduce that . This shows that the pair is orthogonality preserving on .
When is orthogonally additive on and zero products preserving, then the pair is zero products preserving on for every . The final statement is clear from Lemma 4.

It should be remarked here that if a mapping is given by an expression of the form in (18) which uniformly converges in , where is a sequence of operators from into such that the pair is orthogonality preserving on (resp., zero products preserving on ) for every , then is orthogonally additive and orthogonality preserving on (resp., orthogonally additive on and zero products preserving).

3. Orthogonality Preserving Pairs of Operators

Let and be two C*-algebras. In this section we will study those pairs of operators satisfying that and the pair preserve orthogonality on . Our description generalizes some of the results obtained by Wolff in [17] because a (symmetric) mapping is orthogonality preserving on if and only if the pair enjoys the same property. In particular, for every -homomorphism , the pair preserves orthogonality. The same statement is true whenever is a -antihomomorphism, or a Jordan -homomorphism, or a triple homomorphism for the triple product .

We observe that being symmetric implies that is orthogonality preserving on if and only if is zero products preserving on . We shall present here a newfangled and simplified proof which is also valid for pairs of operators.

Let be an element in a von Neumann algebra . We recall that the left and right support projections of (denoted by and ) are defined as follows: (resp., ) is the smallest projection (resp., ) with the property that (resp., ). It is known that when is Hermitian is called the support or range projection of and is denoted by . It is also known that, for each , the sequence converges in the strong-topology of to (cf. [18, Sections 1.10 and 1.11]).

An element in a C*-algebra is said to be a partial isometry whenever (equivalently, or is a projection in ). For each partial isometry , the projections and are called the left and right support projections associated with , respectively. Every partial isometry in defines a Jordan product and an involution on given by and (). It is known that is a unital JB*-algebra with respect to its natural norm and is the unit element for the Jordan product .

Every element in a C*-algebra admits a polar decomposition in ; that is, decomposes uniquely as follows: , where and is a partial isometry in such that and (cf. [18, Theorem ]). Observe that . The unique partial isometry appearing in the polar decomposition of is called the range partial isometry of and is denoted by . Let us observe that taking , we have . It is also easy to check that for each with (resp., ) the condition (resp., ) implies . Furthermore, in if and only if in .

We begin with a basic argument in the study of orthogonality preserving operators between C*-algebras whose proof is inserted here for completeness reasons. Let us recall that for every C*-algebra , the multiplier algebra of , , is the set of all elements such that for each . We notice that is a C*-algebra and contains the unit element of .

Lemma 8. Let and be C*-algebras and let be a pair of operators. (a)The pair preserves orthogonality (on ) if and only if the pair preserves orthogonality (on ).(b)The pair preserves zero products (on ) if and only if the pair preserves zero products (on ).

Proof. (a) The “if” implication is clear. Let be two elements in with . We can find two elements and in satisfying , , and . Since , for every in , we have for every . By Goldstine's theorem we find two bounded nets and in , converging in the weak* topology of to and , respectively. Since , for every ,  , is weak*-continuous, the product of is separately weak*-continuous, and the involution of is also weak*-continuous, we get and hence , as desired.
The proof of (b) follows by a similar argument.

Proposition 9. Let be operators between C*-algebras such that is orthogonality preserving on . Let us denote and . Then the identities, hold for every .

Proof. By Lemma 8, we may assume, without loss of generality, that is unital. (a) for each , the continuous bilinear form , is orthogonal; that is, , whenever in . By Goldstein's theorem [19, Theorem 1.10], there exist functionals satisfying that for all . Taking and we have for every , respectively. Since was arbitrarily chosen, we get, by linearity, and , for every . The other identities follow in a similar way but replacing with .

Lemma 10. Let be Jordan -homomorphism between C*-algebras. The following statements are equivalent. (a)The pair is orthogonality preserving on .(b)The identity holds for every ,(c)The identity, holds for every .Furthermore, when is unital, , for every in .

Proof. The implications have been established in Proposition 9. To see , we observe that , for every . Therefore, given with , we have .

In [17, Proposition 2.5], Wolff establishes a uniqueness result for -homomorphisms between C*-algebras showing that for each pair of unital -homomorphisms from a unital C*-algebra into a unital C*-algebra , the condition orthogonality preserving on implies . This uniqueness result is a direct consequence of our previous lemma.

Orthogonality preserving pairs of operators can be also used to rediscover the notion of orthomorphism in the sense introduced by Zaanen in [13]. We recall that an operator on a C*-algebra is said to be an orthomorphism or a band preserving operator when the implication holds for every . We notice that when is regarded as an -bimodule, an operator is an orthomorphism if and only if it is a local operator in the sense used by Johnson in [14, Section 3]. Clearly, an operator is an orthomorphism if and only if is orthogonality preserving. The following noncommutative extension of [13, Theorem 5] follows from Proposition 9.

Corollary 11. Let be an operator on a C*-algebra . Then is an orthomorphism if and only if , for every in ; that is, is a multiple of the identity on by an element in its center.

We recall that two elements , and in a JB*-algebra are said to operator commute in if the multiplication operators and commute, where is defined by . That is, and operator commute if and only if for all in . A useful result in Jordan theory assures that self-adjoint elements and in generate a JB*-subalgebra that can be realized as a JC*-subalgebra of some (compare [20]) and, under this identification, and commute as elements in whenever they operator commute in , equivalently, (cf. Proposition 1 in [21]).

The next lemma contains a property which is probably known in C*-algebra, we include an sketch of the proof because we were unable to find an explicit reference.

Lemma 12. Let be a partial isometry in a C*-algebra and let , and be two elements in . Then , operator commute in the JB*-algebra if and only if and operator commute in the JB*-algebra , where , for every . Furthermore, when and are hermitian elements in , , and operator commute if and only if and commute in the usual sense (i.e., ).

Proof. We observe that the mapping , , is a Jordan -isomorphism between the above JB*-algebras. So, the first equivalence is clear. The second one has been commented before.

Our next corollary relies on the following description of orthogonality preserving operators between C*-algebras obtained in [12] (see also [6]).

Theorem 13 (see [12, Theorem 17], [6, Theorem 4.1 and Corollary 4.2]). If is an operator from a C*-algebra into another C*-algebra the following are equivalent.(a) is orthogonality preserving (on ).(b)There exists a unital Jordan -homomorphism such that and operator commute and where is the multiplier algebra of , is the range partial isometry of in , , and is the natural product making a JB*-algebra.
Furthermore, when is symmetric, is hermitian and hence decomposes as orthogonal sum of two projections in .

Our next result gives a new perspective for the study of orthogonality preserving (pairs of) operators between C*-algebras.

Proposition 14. Let and be C*-algebras. Let be operators and let and . Then the following statements hold.(a)The operator is orthogonality preserving if and only if there exist two Jordan -homomorphisms satisfying , , and , for every .(b), and are orthogonality preserving on if and only if the following statements hold.(b1)There exist Jordan -homomorphisms satisfying , , , ,, and , for every .(b2)The pairs and are orthogonality preserving on .

Proof. The “if” implications are clear in both statements. We will only prove the “only if” implication.(a)By Theorem 13, there exists a unital Jordan -homomorphism such that and operator commute in the JB*-algebra and Fix . Since and are hermitian elements in which operator commute, Lemma 12 assures that and commute in the usual sense of ; that is, or equivalently, Consequently, we have for every . The desired statement follows by considering and .(b)The statement in (b1) follows from (a). We will prove (b2). By hypothesis, given in with , we have Having in mind that and , we deduce that (compare the comments before Lemma 8), as we desired. In a similar fashion we prove , .

4. Holomorphic Mappings Valued in a Commutative C*-Algebra

The particular setting in which a holomorphic function is valued in a commutative C*-algebra provides enough advantages to establish a full description of the orthogonally additive, orthogonality preserving, and holomorphic mappings which are valued in .

Proposition 15. Let be operators between C*-algebras with commutative. Suppose that , and are orthogonality preserving, and let us denote and . Then there exists a Jordan -homomorphism satisfying , , and , for every .

Proof. Let be the Jordan -homomorphisms satisfying (b1) and (b2) in Proposition 14. By hypothesis, is commutative, and hence for every (compare the proof of Proposition 14). Since the pair is orthogonality preserving on , Lemma 10 assures that for every . In order to simplify notation, let us denote and .
We define an operator , given by Since , it can be easily checked that is a Jordan -homomorphism such that and , for every .

Theorem 16. Let be a holomorphic mapping, where and are C*-algebras with commutative and let be its Taylor series at zero, which is uniformly converging in . Suppose is orthogonality preserving and orthogonally additive on (equivalently, orthogonally additive on and zero products preserving). Then there exist a sequence in and a Jordan -homomorphism such that uniformly in .

Proof. By Corollary 7, there exists a sequence of operators from into satisfying that the pair is orthogonality preserving on (equivalently, zero products preserving on ) for every and uniformly in . Denote .
We will prove now the existence of the Jordan -homomorphism . We prove, by induction, that for each natural , there exists a Jordan -homomorphism such that and for every , . The statement for follows from Corollary 7 and Proposition 14. Let us assume that our statement is true for . Since for every in , , , and the pair are orthogonality preserving, we can easily check that , and are orthogonality preserving. By Proposition 15, there exists a Jordan -homomorphism satisfying , and for every . Since, for each ,
for every , as desired.
Let us consider a free ultrafilter on . By the Banach-Alaoglu theorem, any bounded set in is relatively weak*-compact, and thus the assignment defines a Jordan -homomorphism from into . If we fix a natural , we know that , for every and . Then it can be easily checked that , for every , which concludes the proof.

The Banach-Stone type theorem for orthogonally additive, orthogonality preserving, and holomorphic mappings between commutative C*-algebras, established in Theorem 2 (see [11, Theorem 3.4]), is a direct consequence of our previous result.

5. Banach-Stone Type Theorems for Holomorphic Mappings between General C*-Algebras

In this section we deal with holomorphic functions between general C*-algebras. In this more general setting we will require additional hypothesis to establish a result in the line of the above Theorem 16.

Given a unital C*-algebra , the symbol will denote the set of invertible elements in . The next lemma is a technical tool which is needed later. The proof is left to the reader and follows easily from the fact that is an open subset of .

Lemma 17. Let be a holomorphic mapping, where and are C*-algebras with unital and let be its Taylor series at zero, which is uniformly converging in . Let us assume that there exists with . Then there exists such that .

We can now state a description of those orthogonally additive, orthogonality preserving, and holomorphic mappings between C*-algebras whose image contains an invertible element.

Theorem 18. Let be a holomorphic mapping, where and are C*-algebras with unital and let be its Taylor series at zero, which is uniformly converging in . Suppose is orthogonality preserving and orthogonally additive on and . Then there exist a sequence in and Jordan -homomorphisms such that uniformly in .

Proof. By Corollary 7 there exists a sequence of operators from into satisfying that the pair is orthogonality preserving on for every and uniformly in .
Now, Proposition 14 (a), applied to (), implies the existence of sequences and of Jordan -homomorphisms from into satisfying , , where , and for every , . Moreover, from Proposition 14 (b), the pairs and are orthogonality preserving on , for every .
Since , it follows from Lemma 17 that there exists a natural and such that
We claim that in . Otherwise, we find a nonzero projection satisfying Since , this would imply that contradicting that is invertible in .
Consider now the mapping . It is clear that, for each natural , , , and the pair are orthogonality preserving. Applying Proposition 14 (b), we deduce the existence of Jordan -homomorphisms such that and are orthogonality preserving, , , , , for every , where . The condition , proved in the previous paragraph, shows that . Thus, since is orthogonality preserving, the last statement in Lemma 10 proves that for every , . The above identities guarantee that for every , .
A similar argument to the one given above, but replacing with , shows the existence of a Jordan -homomorphism such that for every , , which concludes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are partially supported by the Spanish Ministry of Economy and Competitiveness, D.G.I. Project no. MTM2011-23843, and Junta de Andalucía Grant FQM3737.

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