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Preserver Problems on Function Spaces, Operator Algebras, and Related Topics

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Volume 2013 |Article ID 415354 | https://doi.org/10.1155/2013/415354

Jorge J. Garcés, Antonio M. Peralta, Daniele Puglisi, María Isabel Ramírez, "Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras", Abstract and Applied Analysis, vol. 2013, Article ID 415354, 9 pages, 2013. https://doi.org/10.1155/2013/415354

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

Academic Editor: Ngai-Ching Wong
Received01 Oct 2013
Accepted27 Nov 2013
Published26 Dec 2013

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