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

Abstract

We study holomorphic maps between C∗-algebras 𝐴 and 𝐵, when 𝑓∶𝐵_𝐴(0,𝜚)→𝐵 is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball 𝑈=𝐵_𝐴(0,𝛿). If we assume that 𝑓 is orthogonality preserving and orthogonally additive on 𝐴_𝑠𝑎∩𝑈 and 𝑓(𝑈) contains an invertible element in 𝐵, then there exist a sequence (ℎ_𝑛) in 𝐵^∗∗ and Jordan ∗-homomorphisms Θ,Θ∶𝑀(𝐴)→𝐵^∗∗ such that 𝑓(𝑥)=∑^∞_𝑛=1ℎ_𝑛Θ(𝑎^𝑛)=∑^∞_𝑛=1Θ(𝑎^𝑛)ℎ_𝑛 uniformly in 𝑎∈𝑈. When 𝐵 is abelian, the hypothesis of 𝐵 being unital and 𝑓(𝑈)∩inv(𝐵)≠∅ can be relaxed to get the same statement.

1. Introduction

The description of orthogonally additive 𝑛-homogeneous polynomial on 𝐶(𝐾)-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 𝐶(𝐾)-spaces (see [8, 9]) and subsequently on general C*-algebras (cf. [10]).

We recall that a mapping 𝑓 from a C*-algebra 𝐴 into a Banach space 𝐵 is said to be orthogonally additive on a subset 𝑈⊆𝐴 if for every 𝑎,𝑏 in 𝑈 with 𝑎⟂𝑏, and 𝑎+𝑏∈𝑈 we have 𝑓(𝑎+𝑏)=𝑓(𝑎)+𝑓(𝑏), where elements 𝑎,  𝑏 in 𝐴 are said to be orthogonal (denoted by 𝑎⟂𝑏) whenever 𝑎𝑏^∗=𝑏^∗𝑎=0. We will say that 𝑓 is additive on elements having zero product if for every 𝑎,  𝑏 in 𝐴 with 𝑎𝑏=0, we have 𝑓(𝑎+𝑏)=𝑓(𝑎)+𝑓(𝑏). Having this terminology in mind, the description of all 𝑛-homogeneous polynomials on a general C*-algebra, 𝐴, which are orthogonally additive on the self-adjoint part, 𝐴_𝑠𝑎, of 𝐴 reads as follows (see Section 2 for concrete definitions not explained here).

Theorem 1 (see [3]). Let 𝐴 be a C*-algebra and 𝐵 a Banach space, 𝑛∈ℕ, and let 𝑃∶𝐴→𝐵 be an 𝑛-homogeneous polynomial. The following statements are equivalent.(a)There exists a bounded linear operator 𝑇∶𝐴→𝐵 satisfying 𝑃(𝑎)=𝑇𝑎^𝑛, (1) for every 𝑎∈𝐴 and ‖𝑃‖≤‖𝑇‖≤2‖𝑃‖.(b)𝑃 is additive on elements having zero products.(c)𝑃 is orthogonally additive on 𝐴_𝑠𝑎.

The task of replacing 𝑛-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 𝐾 denotes the closed unit disc in ℂ, there is no entire function Φ∶ℂ→ℂ such that the mapping ℎ∶𝐶(𝐾)→𝐶(𝐾), ℎ(𝑓)=Φ∘𝑓 factorizes all degree-2 orthogonally additive scalar polynomials over 𝐶(𝐾). Furthermore, similar arguments show that defining 𝑃∶𝐶([0,1])→ℂ, 𝑃(𝑓)=𝑓(0)+𝑓(1)², we cannot find a triplet (Φ,𝛼₁,𝛼₂), where Φ∶𝐶[0,1]→ℂ is a *-homomorphism and 𝛼₁,𝛼₂∈ℂ, satisfying that 𝑃(𝑓)=𝛼₁Φ(𝑓)+𝛼₂Φ(𝑓²) for every 𝑓∈𝐶([0,1]).

To avoid the difficulties commented above, Carando et al. introduce a factorization through an 𝐿₁(𝜇) space. More concretely, for each compact Hausdorff space 𝐾, a holomorphic mapping of bounded type 𝑓∶𝐶(𝐾)→ℂ is orthogonally additive if and only if there exist a Borel regular measure 𝜇 on 𝐾, a sequence (𝑔_𝑘)_𝑘⊆𝐿₁(𝜇), and a holomorphic function of bounded type ℎ∶𝐶(𝐾)→𝐿₁(𝜇) such that ℎ(𝑎)=∑^∞_𝑘=0𝑔_𝑘𝑎^𝑘 and 𝑓(𝑎)=_𝐾ℎ(𝑎)𝑑𝜇, (2) for every 𝑎∈𝐶(𝐾) (cf. [8, Theorem 3.3]).

When 𝐶(𝐾) is replaced with a general C*-algebra 𝐴, a holomorphic function of bounded type 𝑓∶𝐴→ℂ is orthogonally additive on 𝐴_𝑠𝑎 if and only if there exist a positive functional 𝜑 in 𝐴^∗, a sequence (𝜓_𝑛) in 𝐿₁(𝐴^∗∗,𝜑), and a power series holomorphic function ℎ in ℋ_𝑏(𝐴,𝐴^∗) such that ℎ(𝑎)=^∞_𝑘=1𝜓_𝑘⋅𝑎^𝑘,𝑓(𝑎)=1_𝐴^∗∗,ℎ(𝑎)=ℎ(𝑎)𝑑𝜑, (3) for every 𝑎 in 𝐴, where 1_𝐴^∗∗ denotes the unit element in 𝐴^∗∗ and 𝐿₁(𝐴^∗∗,𝜑) is a noncommutative 𝐿₁-space (cf. [10]).

A very recent contribution due to Bu et al. [11] shows that, for holomorphic mappings between 𝐶(𝐾) spaces, we can avoid the factorization through an 𝐿₁(𝜇)-space by imposing additional hypothesis. Before stating the detailed result, we will set down some definitions.

Let 𝐴 and 𝐵 be C*-algebras. When 𝑓∶𝑈⊆𝐴→𝐵 is a map and the condition 𝑎⟂𝑏⟹𝑓(𝑎)⟂𝑓(𝑏)resp.,𝑎𝑏=0⟹𝑓(𝑎)𝑓(𝑏)=0 (4) holds for every 𝑎,𝑏∈𝑈, we will say that 𝑓  preserves orthogonality or it is orthogonality preserving (resp., 𝑓  preserves zero products) on 𝑈. In the case 𝐴=𝑈 we will simply say that 𝑓 is orthogonality preserving (resp., 𝑓  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 𝐶₀(𝐿) spaces is established by Bu et al. in [11, Theorem 3.4].

Theorem 2 (see [11]). Let 𝐿₁ and 𝐿₂ be locally compact Hausdorff spaces and let 𝑓∶𝐵_𝐶0(𝐿1)(0,𝑟)→𝐶₀(𝐿₂) be a bounded orthogonally additive holomorphic function. If 𝑓 is zero product preserving or orthogonality preserving, then there exist a sequence (𝒪_𝑛) of open subsets of 𝐿₂, a sequence (ℎ_𝑛) of bounded functions from 𝐿₂∪{∞} into ℂ, and a mapping 𝜑∶𝐿₂→𝐿₁ such that for each natural 𝑛 the function ℎ_𝑛 is continuous and nonvanishing on 𝒪_𝑛 and 𝑓(𝑎)(𝑡)=^∞_𝑛=1ℎ_𝑛(𝑡)𝑎𝜑(𝑡)^𝑛,𝑡∈𝐿₂, (5) uniformly in 𝑎∈𝐵_𝐶0(𝐿1)(0,𝑟).

The study developed by Bu et al. is restricted to commutative C*-algebras or to orthogonality preserving and orthogonally additive, 𝑛-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 𝐴 and 𝐵 be C*-algebras with 𝐵 unital and let 𝑓∶𝐵_𝐴(0,𝜚)→𝐵 be a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball 𝑈=𝐵_𝐴(0,𝛿). Suppose 𝑓 is orthogonality preserving and orthogonally additive on 𝐴_𝑠𝑎∩𝑈 and 𝑓(𝑈) contains an invertible element. Then there exist a sequence (ℎ_𝑛) in 𝐵^∗∗ and Jordan ∗-homomorphisms Θ,Θ∶𝑀(𝐴)→𝐵^∗∗ such that 𝑓(𝑥)=^∞_𝑛=1ℎ_𝑛Θ𝑎^𝑛=^∞_𝑛=1Θ𝑎^𝑛ℎ_𝑛, (6) uniformly in 𝑎∈𝑈 (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 𝑋 and 𝑌 be Banach spaces. Given a natural 𝑛, a (continuous) 𝑛-homogeneous polynomial 𝑃 from 𝑋 to 𝑌 is a mapping 𝑃∶𝑋→𝑌 for which there is a (continuous) 𝑛-linear symmetric operator 𝐴∶𝑋×⋯×𝑋→𝑌 such that 𝑃(𝑥)=𝐴(𝑥,…,𝑥), for every  𝑥∈𝑋. All polynomials considered in this paper are assumed to be continuous. By a 0-homogeneous polynomial we mean a constant function. The symbol 𝒫(^𝑛𝑋,𝑌) will denote the Banach space of all continuous 𝑛-homogeneous polynomials from 𝑋 to 𝑌, with norm given by ‖𝑃‖=sup_{‖𝑥‖≤1}‖𝑃(𝑥)‖.

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

We recall that, given a domain 𝑈 in a complex Banach space 𝑋 (i.e., an open, connected subset), a function 𝑓 from 𝑈 to another complex Banach space 𝑌 is said to be holomorphic if the Fréchet derivative of 𝑓 at 𝑧₀ exists for every point 𝑧₀ in 𝑈. It is known that 𝑓 is holomorphic in 𝑈 if and only if for each 𝑧₀∈𝑋 there exists a sequence (𝑃_𝑘(𝑧₀))_𝑘 of polynomials from 𝑋 into 𝑌, where each 𝑃_𝑘(𝑧₀) is 𝑘-homogeneous, and a neighborhood 𝑉_𝑧0 of 𝑧₀ such that the series, ^∞_𝑘=0𝑃_𝑘𝑧₀𝑦−𝑧₀, (7) converges uniformly to 𝑓(𝑦) for every 𝑦∈𝑉_𝑧0. Homogeneous polynomials on a C*-algebra 𝐴 constitute the most basic examples of holomorphic functions on 𝐴. A holomorphic function 𝑓∶𝑋→𝑌 is said to be of bounded type if it is bounded on all bounded subsets of 𝑋; in this case its Taylor series at zero, 𝑓=∑^∞_𝑘=0𝑃_𝑘, has infinite radius of uniform convergence, that is, limsup_𝑘→∞‖𝑃_𝑘‖^1/𝑘=0 (compare [15, Section 6.2], see also [16]).

Suppose 𝑓∶𝐵_𝑋(0,𝛿)→𝑌 is a holomorphic function and let 𝑓=∑^∞_𝑘=0𝑃_𝑘 be its Taylor series at zero which is assumed to be uniformly convergent in 𝑈=𝐵_𝑋(0,𝛿). Given 𝜑∈𝑌^∗, it follows from Cauchy's integral formula that, for each 𝑎∈𝑈, we have 𝜑𝑃_𝑛(𝑎)=12𝜋𝑖_𝛾𝜑𝑓(𝜆𝑎)𝜆^𝑛+1𝑑𝜆, (8) where 𝛾 is the circle forming the boundary of a disc in the complex plane 𝐷_ℂ(0,𝑟₁), taken counterclockwise, such that 𝑎+𝐷_ℂ(0,𝑟₁)𝑎⊆𝑈. 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 𝑓∶𝐵_𝐴(0,𝜚)→𝐵 be a holomorphic mapping, where 𝐴 is a C*-algebra and 𝐵 is a complex Banach space, and let 𝑓=∑^∞_𝑘=0𝑃_𝑘 be its Taylor series at zero, which is uniformly converging in 𝑈=𝐵_𝐴(0,𝛿). Then the mapping 𝑓 is orthogonally additive on 𝑈 (resp., orthogonally additive on 𝐴_𝑠𝑎∩𝑈 or additive on elements having zero product in 𝑈) if and only if all the 𝑃_𝑘's satisfy the same property. In such a case, 𝑃₀=0.

We recall that a functional 𝜑 in the dual of a C*-algebra 𝐴 is symmetric when 𝜑(𝑎)∈ℝ, for every 𝑎∈𝐴_𝑠𝑎. Reciprocally, if 𝜑(𝑏)∈ℝ for every symmetric functional 𝜑∈𝐴^∗, the element 𝑏 lies in 𝐴_𝑠𝑎. Having this in mind, our next lemma also is a direct consequence of Cauchy's integral formula and the power series expansion of 𝑓. A mapping 𝑓∶𝐴→𝐵 between C*-algebras is called symmetric whenever 𝑓(𝐴_𝑠𝑎)⊆𝐵_𝑠𝑎, or equivalently, 𝑓(𝑎)=𝑓(𝑎)^∗, whenever 𝑎∈𝐴_𝑠𝑎.

Lemma 4. Let 𝑓∶𝐵_𝐴(0,𝜚)→𝐵 be a holomorphic mapping, where 𝐴 and 𝐵 are C*-algebras, and let 𝑓=∑^∞_𝑘=0𝑃_𝑘 be its Taylor series at zero, which is uniformly converging in 𝑈=𝐵_𝐴(0,𝛿). Then the mapping 𝑓 is symmetric on 𝑈 (i.e., 𝑓(𝐴_𝑠𝑎∩𝑈)⊆𝐵_𝑠𝑎) if and only if 𝑃_𝑘 is symmetric (i.e., 𝑃_𝑘(𝐴_𝑠𝑎)⊆𝐵_𝑠𝑎) for every 𝑘∈ℕ∪{0}.

Definition 5. Let 𝑆,𝑇∶𝐴→𝐵 be a couple of mappings between two C*-algebras. One will say that the pair (𝑆,𝑇) is orthogonality preserving on a subset 𝑈⊆𝐴 if 𝑆(𝑎)⟂𝑇(𝑏) whenever 𝑎⟂𝑏 in 𝑈. When 𝑎𝑏=0 in 𝑈 implies 𝑆(𝑎)𝑇(𝑏)=0 in 𝐵, we will say that (𝑆,𝑇) preserves zero products on 𝑈.

We observe that a mapping 𝑇∶𝐴→𝐵 is orthogonality preserving in the usual sense if and only if the pair (𝑇,𝑇) is orthogonality preserving. We also notice that (𝑆,𝑇) is orthogonality preserving (on 𝐴_𝑠𝑎) if and only if (𝑇,𝑆) is orthogonality preserving (on 𝐴_𝑠𝑎).

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

Proposition 6. Let 𝑓∶𝐵_𝐴(0,𝜚)→𝐵 be a holomorphic mapping, where 𝐴 and 𝐵 are C*-algebras, and let 𝑓=∑^∞_𝑘=0𝑃_𝑘 be its Taylor series at zero, which is uniformly converging in 𝑈=𝐵_𝐴(0,𝛿). The following statements hold. (a)The mapping 𝑓 is orthogonally preserving on 𝑈 (resp., orthogonally preserving on 𝐴_𝑠𝑎∩𝑈) if and only if 𝑃₀=0 and the pair (𝑃_𝑛,𝑃_𝑚) is orthogonality preserving (resp., orthogonally preserving on 𝐴_𝑠𝑎) for every 𝑛,𝑚∈ℕ.(b)The mapping 𝑓 preserves zero products on 𝑈 if and only if 𝑃₀=0 and for every 𝑛,𝑚∈ℕ, the pair (𝑃_𝑛,𝑃_𝑚) preserves zero products.

Proof. (a) The “if” implication is clear. To prove the “only if” implication, let us fix 𝑎,𝑏∈𝑈 with 𝑎⟂𝑏. Let us find two positive scalars 𝑟,  𝐶 such that 𝑎,𝑏∈𝐵(0,𝑟) and ‖𝑓(𝑥)‖≤𝐶 for every 𝑥∈𝐵(0,𝑟)⊂𝐵(0,𝑟)⊆𝑈. From the Cauchy estimates we have ‖𝑃_𝑚‖≤𝐶/𝑟^𝑚, for every 𝑚∈ℕ∪{0}. By hypothesis 𝑓(𝑡𝑎)⟂𝑓(𝑡𝑏), for every 𝑟>𝑡>0, hence 𝑃₀(𝑡𝑎)𝑃₀(𝑡𝑏)^∗+𝑃₀(𝑡𝑎)^∞_𝑘=1𝑃_𝑘(𝑡𝑏)^∗+^∞_𝑘=1𝑃_𝑘(𝑡𝑎)^∞_𝑘=0𝑃_𝑘(𝑡𝑏)^∗=0, (9) and by homogeneity 𝑃₀(𝑎)𝑃₀(𝑏)^∗=−𝑃₀(𝑎)^∞_𝑘=1𝑡^𝑘𝑃_𝑘(𝑏)^∗+^∞_𝑘=1𝑡^𝑘𝑃_𝑘(𝑎)^∞_𝑘=0𝑡^𝑘𝑃_𝑘(𝑏)^∗. (10) Letting 𝑡→0, we have 𝑃₀(𝑎)𝑃₀(𝑏)^∗=0. In particular, 𝑃₀=0.
We will prove by induction on 𝑛 that the pair (𝑃_𝑗,𝑃_𝑘) is orthogonality preserving on 𝑈 for every 1≤𝑗,𝑘≤𝑛. Since 𝑓(𝑡𝑎)𝑓(𝑡𝑏)^∗=0, we also deduce that 𝑃₁(𝑡𝑎)𝑃₁(𝑡𝑏)^∗+𝑃₁(𝑡𝑎)^∞_𝑘=2𝑃_𝑘(𝑡𝑏)^∗+^∞_𝑘=2𝑃_𝑘(𝑡𝑎)^∞_𝑘=1𝑃_𝑘(𝑡𝑏)^∗=0, (11) for every (min{‖𝑎‖,‖𝑏‖})/𝑟>𝑡>0, which implies that 𝑡²𝑃₁(𝑎)𝑃₁(𝑏)^∗=−𝑡𝑃₁(𝑎)^∞_𝑘=2𝑡^𝑘𝑃_𝑘(𝑏)^∗−^∞_𝑘=2𝑡^𝑘𝑃_𝑘(𝑎)^∞_𝑘=1𝑡^𝑘𝑃_𝑘(𝑏)^∗, (12) for every (min{‖𝑎‖,‖𝑏‖})/𝑟>𝑡>0, and hence 𝑃₁(𝑎)𝑃₁(𝑏)^∗≤𝑡𝐶𝑃₁(𝑎)^∞_𝑘=2‖𝑏‖^𝑘𝑟^𝑘𝑡^𝑘−2+𝑡𝐶^2∞_𝑘=2‖𝑎‖^𝑘𝑟^𝑘𝑡^𝑘−2∞_𝑘=1‖𝑏‖^𝑘𝑟^𝑘𝑡^𝑘−1. (13) Taking limit in 𝑡→0, we get 𝑃₁(𝑎)𝑃₁(𝑏)^∗=0. Let us assume that (𝑃_𝑗,𝑃_𝑘) is orthogonality preserving on 𝑈 for every 1≤𝑗,𝑘≤𝑛. Following the argument above we deduce that 𝑃₁(𝑎)𝑃_𝑛+1(𝑏)^∗+𝑃_𝑛+1(𝑎)𝑃₁(𝑏)^∗=−𝑡𝑃₁(𝑎)^∞_𝑗=𝑛+2𝑡^{𝑗−𝑛−2}𝑃_𝑗(𝑏)^∗−𝑡^𝑛_𝑘=2𝑡^𝑘−2𝑃_𝑘(𝑎)^∞_𝑗=𝑛+1𝑡^{𝑗−𝑛−1}𝑃_𝑗(𝑏)^∗−𝑡𝑃_𝑛+1(𝑎)^∞_𝑗=2𝑡^𝑗−2𝑃_𝑗(𝑏)^∗−𝑡^∞_𝑘=𝑛+2𝑡^{𝑘−𝑛−2}𝑃_𝑘(𝑎)^∞_𝑗=1𝑡^𝑗−1𝑃_𝑗(𝑏)^∗, (14) for every (min{‖𝑎‖,‖𝑏‖})/𝑟>|𝑡|>0. Taking limit in 𝑡→0, we have 𝑃₁(𝑎)𝑃_𝑛+1(𝑏)^∗+𝑃_𝑛+1(𝑎)𝑃₁(𝑏)^∗=0. (15) Replacing 𝑎 with 𝑠𝑎 (𝑠>0) we get 𝑠𝑃₁(𝑎)𝑃_𝑛+1(𝑏)^∗+𝑠^𝑛+1𝑃_𝑛+1(𝑎)𝑃₁(𝑏)^∗=0, (16) for every 𝑠>0, which implies that 𝑃₁(𝑎)𝑃_𝑛+1(𝑏)^∗=0. (17) In a similar manner we prove that 𝑃_𝑘(𝑎)𝑃_𝑛+1(𝑏)^∗=0, for every 1≤𝑘≤𝑛+1. The equalities 𝑃_𝑘(𝑏)^∗𝑃_𝑗(𝑎)=0 (1≤𝑗,𝑘≤𝑛+1) follow similarly.
We have shown that for each 𝑛,𝑚∈ℕ, 𝑃_𝑛(𝑎)⟂𝑃_𝑚(𝑏) whenever 𝑎,𝑏∈𝑈 with 𝑎⟂𝑏. Finally, taking 𝑎,  𝑏∈𝐴 with 𝑎⟂𝑏, we can find a positive 𝜌 such that 𝜌𝑎,𝜌𝑏∈𝑈 and 𝜌𝑎⟂𝜌𝑏, which implies that 𝑃_𝑛(𝜌𝑎)⟂𝑃_𝑚(𝜌𝑏) for every 𝑛,𝑚∈ℕ, witnessing that (𝑃_𝑛,𝑃_𝑚) is orthogonality preserving for every 𝑛,𝑚∈ℕ.
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 𝑓∶𝐵_𝐴(0,𝜚)→𝐵 be a holomorphic mapping, where 𝐴 and 𝐵 are C*-algebras and let 𝑓=∑^∞_𝑘=0𝑃_𝑘 be its Taylor series at zero, which is uniformly converging in 𝑈=𝐵_𝐴(0,𝛿). Suppose 𝑓 is orthogonality preserving and orthogonally additive on (resp., orthogonally additive and zero products preserving) 𝐴_𝑠𝑎∩𝑈. Then there exists a sequence (𝑇_𝑛) of operators from 𝐴 into 𝐵 satisfying that the pair (𝑇_𝑛,𝑇_𝑚) is orthogonality preserving on 𝐴_𝑠𝑎 (resp., zero products preserving on 𝐴_𝑠𝑎) for every 𝑛,𝑚∈ℕ and 𝑓(𝑥)=^∞_𝑛=1𝑇_𝑛𝑥^𝑛, (18) uniformly in 𝑥∈𝑈. In particular every 𝑇_𝑛 is orthogonality preserving (resp., zero products preserving) on 𝐴_𝑠𝑎. Furthermore, 𝑓 is symmetric if and only if every 𝑇_𝑛 is symmetric.

Proof. Combining Lemma 3 and Proposition 6, we deduce that 𝑃₀=0, 𝑃_𝑛 is orthogonally additive on 𝐴_𝑠𝑎, and (𝑃_𝑛,𝑃_𝑚) is orthogonality preserving on 𝐴_𝑠𝑎 for every 𝑛,𝑚 in ℕ. By Theorem 1, for each natural 𝑛 there exists an operator 𝑇_𝑛∶𝐴→𝐵 such that ‖𝑃_𝑛‖≤‖𝑇_𝑛‖≤2‖𝑃_𝑛‖ and 𝑃_𝑛(𝑎)=𝑇_𝑛𝑎^𝑛, (19) for every 𝑎∈𝐴.
Consider now two positive elements 𝑎,𝑏∈𝐴 with 𝑎⟂𝑏 and fix 𝑛,𝑚∈ℕ. In this case there exist positive elements 𝑐,𝑑 in 𝐴 with 𝑐^𝑛=𝑎 and 𝑑^𝑚=𝑏 and 𝑐⟂𝑑. Since the pair (𝑃_𝑛,𝑃_𝑚) is orthogonality preserving on 𝐴_𝑠𝑎, we have 𝑇_𝑛(𝑎)=𝑇_𝑛(𝑐^𝑛)=𝑃_𝑛(𝑐)⟂𝑃_𝑚(𝑑)=𝑇_𝑚(𝑑^𝑚)=𝑇_𝑚(𝑏). Now, noticing that given 𝑎,  𝑏 in 𝐴_𝑠𝑎 with 𝑎⟂𝑏, we can write 𝑎=𝑎⁺−𝑏⁻ and 𝑏=𝑏⁺−𝑏⁻, where 𝑎^𝜎 and 𝑏^𝜏 are positive, 𝑎⁺⟂𝑎⁻,𝑏⁺⟂𝑏⁻, and 𝑎^𝜎⟂𝑏^𝜏; for every 𝜎,𝜏∈{+,−}, we deduce that 𝑇_𝑛(𝑎)⟂𝑇_𝑚(𝑏). This shows that the pair (𝑇_𝑛,𝑇_𝑚) is orthogonality preserving on 𝐴_𝑠𝑎.
When 𝑓 is orthogonally additive on 𝐴_𝑠𝑎 and zero products preserving, then the pair (𝑇_𝑛,𝑇_𝑚) is zero products preserving on 𝐴_𝑠𝑎 for every 𝑛,𝑚∈ℕ. The final statement is clear from Lemma 4.

It should be remarked here that if a mapping 𝑓∶𝐵_𝐴(0,𝛿)→𝐵 is given by an expression of the form in (18) which uniformly converges in 𝑈=𝐵_𝐴(0,𝛿), where (𝑇_𝑛) is a sequence of operators from 𝐴 into 𝐵 such that the pair (𝑇_𝑛,𝑇_𝑚) is orthogonality preserving on 𝐴_𝑠𝑎 (resp., zero products preserving on 𝐴_𝑠𝑎) for every 𝑛,𝑚∈ℕ, then 𝑓 is orthogonally additive and orthogonality preserving on 𝐴_𝑠𝑎∩𝑈 (resp., orthogonally additive on 𝐴_𝑠𝑎∩𝑈 and zero products preserving).

3. Orthogonality Preserving Pairs of Operators

Let 𝐴 and 𝐵 be two C*-algebras. In this section we will study those pairs of operators 𝑆,𝑇∶𝐴→𝐵 satisfying that 𝑆,𝑇 and the pair (𝑆,𝑇) preserve orthogonality on 𝐴_𝑠𝑎. Our description generalizes some of the results obtained by Wolff in [17] because a (symmetric) mapping 𝑇∶𝐴→𝐵 is orthogonality preserving on 𝐴_𝑠𝑎 if and only if the pair (𝑇,𝑇) enjoys the same property. In particular, for every ∗-homomorphism Φ∶𝐴→𝐵, the pair (Φ,Φ) preserves orthogonality. The same statement is true whenever Φ is a ∗-antihomomorphism, or a Jordan ∗-homomorphism, or a triple homomorphism for the triple product {𝑎,𝑏,𝑐}=(1/2)(𝑎𝑏^∗𝑐+𝑐𝑏^∗𝑎).

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

Let 𝑎 be an element in a von Neumann algebra 𝑀. We recall that the left and right support projections of 𝑎 (denoted by 𝑙(𝑎) and 𝑑(𝑎)) are defined as follows: 𝑙(𝑎) (resp., 𝑑(𝑎)) is the smallest projection 𝑝∈𝑀 (resp., 𝑞∈𝑀) with the property that 𝑝𝑎=𝑎 (resp., 𝑎𝑞=𝑎). It is known that when 𝑎 is Hermitian 𝑑(𝑎)=𝑙(𝑎) is called the support or range projection of 𝑎 and is denoted by 𝑠(𝑎). It is also known that, for each 𝑎=𝑎^∗, the sequence (𝑎^1/3𝑛) converges in the strong∗-topology of 𝑀 to 𝑠(𝑎) (cf. [18, Sections 1.10 and 1.11]).

An element 𝑒 in a C*-algebra 𝐴 is said to be a partial isometry whenever 𝑒𝑒^∗𝑒=𝑒 (equivalently, 𝑒𝑒^∗ or 𝑒^∗𝑒 is a projection in 𝐴). For each partial isometry 𝑒, the projections 𝑒𝑒^∗ and 𝑒^∗𝑒 are called the left and right support projections associated with 𝑒, respectively. Every partial isometry 𝑒 in 𝐴 defines a Jordan product and an involution on 𝐴_𝑒(𝑒)∶=𝑒𝑒^∗𝐴𝑒^∗𝑒 given by 𝑎•_𝑒𝑏=(1/2)(𝑎𝑒^∗𝑏+𝑏𝑒^∗𝑎) and 𝑎^♯𝑒=𝑒𝑎^∗𝑒 (𝑎,𝑏∈𝐴₂(𝑒)). It is known that (𝐴₂(𝑒),•_𝑒,♯_𝑒) is a unital JB*-algebra with respect to its natural norm and 𝑒 is the unit element for the Jordan product •_𝑒.

Every element 𝑎 in a C*-algebra 𝐴 admits a polar decomposition in 𝐴^∗∗; that is, 𝑎 decomposes uniquely as follows: 𝑎=𝑢|𝑎|, where |𝑎|=(𝑎^∗𝑎)^1/2 and 𝑢 is a partial isometry in 𝐴^∗∗ such that 𝑢^∗𝑢=𝑠(|𝑎|) and 𝑢𝑢^∗=𝑠(|𝑎^∗|) (cf. [18, Theorem 1.12.1]). Observe that 𝑢𝑢^∗𝑎=𝑎𝑢^∗𝑢=𝑢. The unique partial isometry 𝑢 appearing in the polar decomposition of 𝑎 is called the range partial isometry of 𝑎 and is denoted by 𝑟(𝑎). Let us observe that taking 𝑐=𝑟(𝑎)|𝑎|^1/3, we have 𝑐𝑐^∗𝑐=𝑎. It is also easy to check that for each 𝑏∈𝐴 with 𝑏=𝑟(𝑎)𝑟(𝑎)^∗𝑏 (resp., 𝑏=𝑏𝑟(𝑎)^∗𝑟(𝑎)) the condition 𝑎^∗𝑏=0 (resp., 𝑏𝑎^∗=0) implies 𝑏=0. Furthermore, 𝑎⟂𝑏 in 𝐴 if and only if 𝑟(𝑎)⟂𝑟(𝑏) in 𝐴^∗∗.

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 𝐴, the multiplier algebra of 𝐴, 𝑀(𝐴), is the set of all elements 𝑥∈𝐴^∗∗ such that for each 𝐴𝑥,𝑥𝐴⊆𝐴. We notice that 𝑀(𝐴) is a C*-algebra and contains the unit element of 𝐴^∗∗.

Lemma 8. Let 𝐴 and 𝐵 be C*-algebras and let 𝑆,𝑇∶𝐴→𝐵 be a pair of operators. (a)The pair (𝑆,𝑇) preserves orthogonality (on 𝐴_𝑠𝑎) if and only if the pair (𝑆^∗∗|_𝑀(𝐴),𝑇^∗∗|_𝑀(𝐴)) preserves orthogonality (on 𝑀(𝐴)_𝑠𝑎).(b)The pair (𝑆,𝑇) preserves zero products (on 𝐴_𝑠𝑎) if and only if the pair (𝑆^∗∗|_𝑀(𝐴),𝑇^∗∗|_𝑀(𝐴)) preserves zero products (on 𝑀(𝐴)_𝑠𝑎).

Proof. (a) The “if” implication is clear. Let 𝑎,𝑏 be two elements in 𝑀(𝐴) with 𝑎⟂𝑏. We can find two elements 𝑐 and 𝑑 in 𝑀(𝐴) satisfying 𝑐𝑐^∗𝑐=𝑎, 𝑑𝑑^∗𝑑=𝑏, and 𝑐⟂𝑑. Since 𝑐𝑥𝑐⟂𝑑𝑦𝑑, for every 𝑥,𝑦 in 𝐴, we have 𝑇(𝑐𝑥𝑐)⟂𝑇(𝑑𝑦𝑑) for every 𝑥,𝑦∈𝐴. By Goldstine's theorem we find two bounded nets (𝑥_𝜆) and (𝑦_𝜇) in 𝐴, converging in the weak* topology of 𝐴^∗∗ to 𝑐^∗ and 𝑑^∗, respectively. Since 𝑇(𝑐𝑥_𝜆𝑐)𝑇(𝑑𝑦_𝜇𝑑)^∗=𝑇(𝑑𝑦_𝜇𝑑)^∗𝑇(𝑐𝑥_𝜆𝑐)=0, for every 𝜆,  𝜇, 𝑇^∗∗ is weak*-continuous, the product of 𝐴^∗∗ is separately weak*-continuous, and the involution of 𝐴^∗∗ is also weak*-continuous, we get 𝑇^∗∗(𝑐𝑐^∗𝑐)𝑇^∗∗(𝑑𝑑^∗𝑑)=𝑇^∗∗(𝑎)𝑇^∗∗(𝑏)^∗=0=𝑇^∗∗(𝑏)^∗𝑇^∗∗(𝑎) and hence 𝑇^∗∗(𝑎)⟂𝑇^∗∗(𝑏), as desired.
The proof of (b) follows by a similar argument.

Proposition 9. Let 𝑆,𝑇∶𝐴→𝐵 be operators between C*-algebras such that (𝑆,𝑇) is orthogonality preserving on 𝐴_𝑠𝑎. Let us denote ℎ∶=𝑆^∗∗(1) and 𝑘∶=𝑇^∗∗(1). Then the identities, 𝑆(𝑎)𝑇𝑎^∗∗=𝑆𝑎²𝑘^∗=ℎ𝑇𝑎^2∗∗,𝑇𝑎^∗∗𝑆(𝑎)=𝑘^∗𝑆𝑎²=𝑇𝑎^2∗∗𝑆(𝑎)𝑘ℎ,_∗=ℎ𝑇𝑎^∗∗,𝑘^∗𝑆(𝑎)=𝑇𝑎^∗∗ℎ, (20) hold for every 𝑎∈𝐴.

Proof. By Lemma 8, we may assume, without loss of generality, that 𝐴 is unital. (a) for each 𝜑∈𝐵^∗, the continuous bilinear form 𝑉_𝜑∶𝐴×𝐴→ℂ, 𝑉_𝜑(𝑎,𝑏)=𝜑(𝑆(𝑎)𝑇(𝑏^∗)^∗) is orthogonal; that is, 𝑉_𝜑(𝑎,𝑏)=0, whenever 𝑎𝑏=0 in 𝐴_𝑠𝑎. By Goldstein's theorem [19, Theorem 1.10], there exist functionals 𝜔₁,𝜔₂∈𝐴^∗ satisfying that 𝑉_𝜑(𝑎,𝑏)=𝜔₁(𝑎𝑏)+𝜔₂(𝑏𝑎), (21) for all 𝑎,𝑏∈𝐴. Taking 𝑏=1 and 𝑎=𝑏 we have 𝜑𝑆(𝑎)𝑘^∗=𝑉_𝜑(𝑎,1)=𝑉_𝜑(1,𝑎)=𝜑ℎ𝑇(𝑎)^∗,𝜑𝑆(𝑎)𝑇(𝑎)^∗=𝜑𝑆𝑎²𝑘^∗=𝜑ℎ𝑇𝑎^2∗, (22) for every 𝑎∈𝐴_𝑠𝑎, respectively. Since 𝜑 was arbitrarily chosen, we get, by linearity, 𝑆(𝑎)𝑘^∗=ℎ𝑇(𝑎^∗)^∗ and 𝑆(𝑎)𝑇(𝑎^∗)^∗=𝑆(𝑎²)𝑘^∗=ℎ𝑇((𝑎²)^∗)^∗, for every 𝑎∈𝐴. The other identities follow in a similar way but replacing 𝑉_𝜑(𝑎,𝑏)=𝜑(𝑆(𝑎)𝑇(𝑏^∗)^∗) with 𝑉_𝜑(𝑎,𝑏)=𝜑(𝑇(𝑏^∗)^∗𝑆(𝑎)).

Lemma 10. Let 𝐽₁,𝐽₂∶𝐴→𝐵 be Jordan ∗-homomorphism between C*-algebras. The following statements are equivalent. (a)The pair (𝐽₁,𝐽₂) is orthogonality preserving on 𝐴_𝑠𝑎.(b)The identity 𝐽₁(𝑎)𝐽₂(𝑎)=𝐽₁𝑎²𝐽₂^∗∗(1)=𝐽₁^∗∗(1)𝐽₂𝑎², (23) holds for every 𝑎∈𝐴_𝑠𝑎,(c)The identity, 𝐽₁^∗∗(1)𝐽₂(𝑎)=𝐽₁(𝑎)𝐽₂^∗∗(1), (24) holds for every 𝑎∈𝐴_𝑠𝑎.Furthermore, when 𝐽₁^∗∗ is unital, 𝐽₂(𝑎)=𝐽₁(𝑎)𝐽₂^∗∗(1)=𝐽₂^∗∗(1)𝐽₁(𝑎), for every 𝑎 in 𝐴.

Proof. The implications (a)⇒(b)⇒(c) have been established in Proposition 9. To see (c)⇒(a), we observe that 𝐽_𝑖(𝑥)=𝐽_𝑖^∗∗(1)𝐽_𝑖(𝑥)𝐽_𝑖^∗∗(1)=𝐽_𝑖(𝑥)𝐽_𝑖^∗∗(1)=𝐽_𝑖^∗∗(1)𝐽_𝑖(𝑥), for every 𝑥∈𝐴. Therefore, given 𝑎,𝑏∈𝐴_𝑠𝑎 with 𝑎⟂𝑏, we have 𝐽₁(𝑎)𝐽₂(𝑏)=𝐽₁(𝑎)𝐽₁^∗∗(1)𝐽₂(𝑏)=𝐽₁(𝑎)𝐽₁(𝑏)𝐽₂^∗∗(1)=0.

In [17, Proposition 2.5], Wolff establishes a uniqueness result for ∗-homomorphisms between C*-algebras showing that for each pair (𝑈,𝑉) of unital ∗-homomorphisms from a unital C*-algebra 𝐴 into a unital C*-algebra 𝐵, the condition (𝑈,𝑉) orthogonality preserving on 𝐴_𝑠𝑎 implies 𝑈=𝑉. 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 𝑇 on a C*-algebra 𝐴 is said to be an orthomorphism or a band preserving operator when the implication 𝑎⟂𝑏⇒𝑇(𝑎)⟂𝑏 holds for every 𝑎,𝑏∈𝐴. We notice that when 𝐴 is regarded as an 𝐴-bimodule, an operator 𝑇∶𝐴→𝐴 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 𝑇∶𝐴→𝐴 is an orthomorphism if and only if (𝑇,𝐼𝑑_𝐴) is orthogonality preserving. The following noncommutative extension of [13, Theorem 5] follows from Proposition 9.

Corollary 11. Let 𝑇 be an operator on a C*-algebra 𝐴. Then 𝑇 is an orthomorphism if and only if 𝑇(𝑎)=𝑇^∗∗(1)𝑎=𝑎𝑇^∗∗(1), for every 𝑎 in 𝐴; that is, 𝑇 is a multiple of the identity on 𝐴 by an element in its center.

We recall that two elements 𝑎, and 𝑏 in a JB*-algebra 𝐴 are said to operator commute in 𝐴 if the multiplication operators 𝑀_𝑎 and 𝑀_𝑏 commute, where 𝑀_𝑎 is defined by 𝑀_𝑎(𝑥)∶=𝑎∘𝑥. That is, 𝑎 and 𝑏 operator commute if and only if (𝑎∘𝑥)∘𝑏=𝑎∘(𝑥∘𝑏) for all 𝑥 in 𝐴. A useful result in Jordan theory assures that self-adjoint elements 𝑎 and 𝑏 in 𝐴 generate a JB*-subalgebra that can be realized as a JC*-subalgebra of some 𝐵(𝐻) (compare [20]) and, under this identification, 𝑎 and 𝑏 commute as elements in 𝐿(𝐻) whenever they operator commute in 𝐴, equivalently, 𝑎²∘𝑏=2(𝑎∘𝑏)∘𝑎−𝑎²∘𝑏 (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 𝑒 be a partial isometry in a C*-algebra 𝐴 and let 𝑎, and 𝑏 be two elements in 𝐴₂(𝑒)=𝑒𝑒^∗𝐴𝑒^∗𝑒. Then 𝑎, 𝑏 operator commute in the JB*-algebra (𝐴₂(𝑒),•_𝑒,♯_𝑒) if and only if 𝑎𝑒^∗ and 𝑏𝑒^∗ operator commute in the JB*-algebra (𝐴₂(𝑒𝑒^∗),•_𝑒𝑒∗,♯_𝑒𝑒∗), where 𝑥•_𝑒𝑒∗𝑦=𝑥∘𝑦=(1/2)(𝑥𝑦+𝑦𝑥), for every 𝑥,𝑦∈𝐴₂(𝑒𝑒^∗). Furthermore, when 𝑎 and 𝑏 are hermitian elements in (𝐴₂(𝑒),•_𝑒,♯_𝑒), 𝑎, and 𝑏 operator commute if and only if 𝑎𝑒^∗ and 𝑏𝑒^∗ commute in the usual sense (i.e., 𝑎𝑒^∗𝑏𝑒^∗=𝑏𝑒^∗𝑎𝑒^∗).

Proof. We observe that the mapping 𝑅_𝑒∗∶(𝐴₂(𝑒),•_𝑒)→(𝐴₂(𝑒𝑒^∗),•_𝑒𝑒∗), 𝑥↦𝑥𝑒^∗, 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 𝑇 is an operator from a C*-algebra 𝐴 into another C*-algebra 𝐵 the following are equivalent.(a)𝑇 is orthogonality preserving (on 𝐴_𝑠𝑎).(b)There exists a unital Jordan ∗-homomorphism 𝐽∶𝑀(𝐴)→𝐵₂^∗∗(𝑟(ℎ)) such that 𝐽(𝑥) and ℎ=𝑇^∗∗(1) operator commute and 𝑇(𝑥)=ℎ•_𝑟(ℎ)𝐽(𝑥),𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑥∈𝐴, (25) where 𝑀(𝐴) is the multiplier algebra of 𝐴, 𝑟(ℎ) is the range partial isometry of ℎ in 𝐵^∗∗, 𝐵₂^∗∗(𝑟(ℎ))=𝑟(ℎ)𝑟(ℎ)^∗𝐵^∗∗𝑟(ℎ)^∗𝑟(ℎ), and •_𝑟(ℎ) is the natural product making 𝐵₂^∗∗(𝑟(ℎ)) a JB*-algebra.
Furthermore, when 𝑇 is symmetric, ℎ is hermitian and hence 𝑟(ℎ) decomposes as orthogonal sum of two projections in 𝐵^∗∗.

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

Proposition 14. Let 𝐴 and 𝐵 be C*-algebras. Let 𝑆,𝑇∶𝐴→𝐵 be operators and let ℎ=𝑆^∗∗(1) and 𝑘=𝑇^∗∗(1). Then the following statements hold.(a)The operator 𝑆 is orthogonality preserving if and only if there exist two Jordan ∗-homomorphisms Φ,Φ∶𝑀(𝐴)→𝐵^∗∗ satisfying Φ(1)=𝑟(ℎ)𝑟(ℎ)^∗, Φ(1)=𝑟(ℎ)^∗𝑟(ℎ), and 𝑆(𝑎)=Φ(𝑎)ℎ=ℎΦ(𝑎), for every 𝑎∈𝐴.(b)𝑆, 𝑇 and (𝑆,𝑇) are orthogonality preserving on 𝐴_𝑠𝑎 if and only if the following statements hold.(b1)There exist Jordan ∗-homomorphisms Φ₁,Φ₁,Φ₂,Φ₂∶𝑀(𝐴)→𝐵^∗∗ satisfying Φ₁(1)=𝑟(ℎ)𝑟(ℎ)^∗, Φ₁(1)=𝑟(ℎ)^∗𝑟(ℎ), Φ₂(1)=𝑟(𝑘)𝑟(𝑘)^∗, Φ₂(1)=𝑟(𝑘)^∗𝑟(𝑘),𝑆(𝑎)=Φ₁(𝑎)ℎ=ℎΦ₁(𝑎), and 𝑇(𝑎)=Φ₂(𝑎)𝑘=𝑘Φ₂(𝑎), for every 𝑎∈𝐴.(b2)The pairs (Φ₁,Φ₂) and (Φ₁,Φ₂) are orthogonality preserving on 𝐴_𝑠𝑎.

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 𝐽₁∶𝑀(𝐴)→𝐵₂^∗∗(𝑟(ℎ)) such that 𝐽₁(𝑥) and ℎ operator commute in the JB*-algebra (𝐵₂^∗∗(𝑟(ℎ)),•_𝑟(ℎ)) and 𝑆(𝑥)=ℎ•_𝑟(𝑎)𝐽₁(𝑎)forevery𝑎∈𝐴. (26) Fix 𝑎∈𝐴_𝑠𝑎. Since ℎ and 𝐽₁(𝑎) are hermitian elements in (𝐵₂^∗∗(𝑟(ℎ)),•_𝑟(ℎ)) which operator commute, Lemma 12 assures that ℎ𝑟(ℎ)^∗ and 𝐽₁(𝑎)𝑟(ℎ)^∗ commute in the usual sense of 𝐵^∗∗; that is, ℎ𝑟(ℎ)^∗𝐽₁(𝑎)𝑟(ℎ)^∗=𝐽₁(𝑎)𝑟(ℎ)^∗ℎ𝑟(ℎ)^∗, (27) or equivalently, ℎ𝑟(ℎ)^∗𝐽₁(𝑎)=𝐽₁(𝑎)𝑟(ℎ)^∗ℎ. (28) Consequently, we have 𝑆(𝑎)=ℎ•_𝑟(ℎ)𝐽₁(𝑎)=ℎ𝑟(ℎ)^∗𝐽₁(𝑎)=𝐽₁(𝑎)𝑟(ℎ)^∗ℎ, (29) for every 𝑎∈𝐴. The desired statement follows by considering Φ₁(𝑎)=𝐽₁(𝑎)𝑟(ℎ)^∗ and Φ₁(𝑎)=𝑟(ℎ)^∗𝐽₁(𝑎).(b)The statement in (b1) follows from (a). We will prove (b2). By hypothesis, given 𝑎,𝑏 in 𝐴_𝑠𝑎 with 𝑎⟂𝑏, we have 0=𝑆(𝑎)𝑇(𝑏)^∗=ℎΦ₁(𝑎)𝑘Φ₂(𝑏)^∗=ℎΦ₁(𝑎)Φ₂(𝑏)^∗𝑘^∗. (30) Having in mind that Φ₁(𝐴)⊆𝑟(ℎ)^∗𝑟(ℎ)𝐵^∗∗ and Φ₂(𝐴)⊆𝐵^∗∗𝑟(𝑘)^∗𝑟(𝑘), we deduce that Φ₁(𝑎)Φ₂(𝑏)^∗=0 (compare the comments before Lemma 8), as we desired. In a similar fashion we prove Φ₂(𝑏)^∗Φ₁(𝑎)=0, Φ₂(𝑏)^∗Φ₁(𝑎)=0=Φ₁(𝑎)Φ₂(𝑏)^∗.

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

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

Proposition 15. Let 𝑆,𝑇∶𝐴→𝐵 be operators between C*-algebras with 𝐵 commutative. Suppose that 𝑆, 𝑇 and (𝑆,𝑇) are orthogonality preserving, and let us denote ℎ=𝑆^∗∗(1) and 𝑘=𝑇^∗∗(1). Then there exists a Jordan ∗-homomorphism Φ∶𝑀(𝐴)→𝐵^∗∗ satisfying Φ(1)=𝑟(|ℎ|+|𝑘|), 𝑆(𝑎)=Φ(𝑎)ℎ, and 𝑇(𝑎)=Φ(𝑎)𝑘, for every 𝑎∈𝐴.

Proof. Let Φ₁,Φ₁,Φ₂,Φ₂∶𝑀(𝐴)→𝐵^∗∗ be the Jordan ∗-homomorphisms satisfying (b1) and (b2) in Proposition 14. By hypothesis, 𝐵 is commutative, and hence Φ_𝑖=Φ_𝑖 for every 𝑖=1,2 (compare the proof of Proposition 14). Since the pair (Φ₁,Φ₂) is orthogonality preserving on 𝐴_𝑠𝑎, Lemma 10 assures that Φ₁^∗∗(1)Φ₂(𝑎)=Φ₁(𝑎)Φ₂^∗∗(1), (31) for every 𝑎∈𝐴_𝑠𝑎. In order to simplify notation, let us denote 𝑝=Φ₁^∗∗(1) and 𝑞=Φ₂^∗∗(1).
We define an operator Φ∶𝑀(𝐴)→𝐵^∗∗, given by Φ(𝑎)=𝑝𝑞Φ₁(𝑎)+𝑝1−𝑞Φ₁(𝑎)+𝑞1−𝑝Φ₂(𝑎). (32) Since 𝑝Φ₂(𝑎)=Φ₁(𝑎)𝑞, it can be easily checked that Φ is a Jordan ∗-homomorphism such that 𝑆(𝑎)=Φ(𝑎)ℎ and 𝑇(𝑎)=Φ(𝑎)𝑘, for every 𝑎∈𝐴.

Theorem 16. Let 𝑓∶𝐵_𝐴(0,𝜚)→𝐵 be a holomorphic mapping, where 𝐴 and 𝐵 are C*-algebras with 𝐵 commutative and let 𝑓=∑^∞_𝑘=0𝑃_𝑘 be its Taylor series at zero, which is uniformly converging in 𝑈=𝐵_𝐴(0,𝛿). Suppose 𝑓 is orthogonality preserving and orthogonally additive on 𝐴_𝑠𝑎∩𝑈 (equivalently, orthogonally additive on 𝐴_𝑠𝑎∩𝑈 and zero products preserving). Then there exist a sequence (ℎ_𝑛) in 𝐵^∗∗ and a Jordan ∗-homomorphism Φ∶𝑀(𝐴)→𝐵^∗∗ such that 𝑓(𝑥)=^∞_𝑛=1ℎ_𝑛Φ𝑎^𝑛=^∞_𝑛=1ℎ_𝑛Φ𝑎^𝑛, (33) uniformly in 𝑎∈𝑈.

Proof. By Corollary 7, there exists a sequence (𝑇_𝑛) of operators from 𝐴 into 𝐵 satisfying that the pair (𝑇_𝑛,𝑇_𝑚) is orthogonality preserving on 𝐴_𝑠𝑎 (equivalently, zero products preserving on 𝐴_𝑠𝑎) for every 𝑛,𝑚∈ℕ and 𝑓(𝑥)=^∞_𝑛=1𝑇_𝑛𝑥^𝑛, (34) uniformly in 𝑥∈𝑈. Denote ℎ_𝑛=𝑇_𝑛^∗∗(1).
We will prove now the existence of the Jordan ∗-homomorphism Φ. We prove, by induction, that for each natural 𝑛, there exists a Jordan ∗-homomorphism Ψ_𝑛∶𝑀(𝐴)→𝐵^∗∗ such that 𝑟(Ψ_𝑛(1))=𝑟(|ℎ₁|+⋯+|ℎ_𝑛|) and 𝑇_𝑘(𝑎)=ℎ_𝑘Ψ_𝑛(𝑎) for every 𝑘≤𝑛, 𝑎∈𝐴. The statement for 𝑛=1 follows from Corollary 7 and Proposition 14. Let us assume that our statement is true for 𝑛. Since for every 𝑘,𝑚 in ℕ, 𝑇_𝑘, 𝑇_𝑚, and the pair (𝑇_𝑘,𝑇_𝑚) are orthogonality preserving, we can easily check that 𝑇_𝑛+1, 𝑇₁+⋯+𝑇_𝑛 and (𝑇_𝑛+1,𝑇₁+⋯+𝑇_𝑛)=(𝑇_𝑛+1,(ℎ₁+⋯+ℎ_𝑛)Ψ_𝑛) are orthogonality preserving. By Proposition 15, there exists a Jordan ∗-homomorphism Ψ_𝑛+1∶𝑀(𝐴)→𝐵^∗∗ satisfying 𝑟(Ψ_𝑛+1(1))=𝑟(|ℎ₁|+⋯+|ℎ_𝑛|+|ℎ_𝑛+1|), 𝑇_𝑛+1(𝑎)=ℎ_𝑛+1Ψ_𝑛+1(𝑎^𝑛+1) and (𝑇₁+⋯+𝑇_𝑛)(𝑎)=(ℎ₁+⋯+ℎ_𝑛)Ψ_𝑛+1(𝑎) for every 𝑎∈𝐴. Since, for each 1≤𝑘≤𝑛, ℎ_𝑘Ψ_𝑛+1(𝑎)=ℎ_𝑘𝑟ℎ₁+⋯+ℎ_𝑛+ℎ_𝑛+1Ψ_𝑛+1(𝑎)=ℎ_𝑘𝑟ℎ₁+⋯+ℎ_𝑛Ψ_𝑛+1(𝑎)=ℎ_𝑘𝑟ℎ₁+⋯+ℎ_𝑛Ψ_𝑛(𝑎)=ℎ_𝑘Ψ_𝑛(𝑎)=𝑇_𝑘(𝑎), (35)
for every 𝑎∈𝐴, as desired.
Let us consider a free ultrafilter 𝒰 on ℕ. By the Banach-Alaoglu theorem, any bounded set in 𝐵^∗∗ is relatively weak*-compact, and thus the assignment 𝑎↦Φ(𝑎)∶=𝑤^∗−lim_𝒰Ψ_𝑛(𝑎) defines a Jordan ∗-homomorphism from 𝑀(𝐴) into 𝐵^∗∗. If we fix a natural 𝑘, we know that 𝑇_𝑘(𝑎)=ℎ_𝑘Ψ_𝑛(𝑎), for every 𝑛≥𝑘 and 𝑎∈𝐴. Then it can be easily checked that 𝑇_𝑘(𝑎)=ℎ_𝑘Φ(𝑎), for every 𝑎∈𝐴, 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 𝐴, the symbol inv(𝐴) will denote the set of invertible elements in 𝐴. 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 inv(𝐴) is an open subset of 𝐴.

Lemma 17. Let 𝑓∶𝐵_𝐴(0,𝜚)→𝐵 be a holomorphic mapping, where 𝐴 and 𝐵 are C*-algebras with 𝐵 unital and let 𝑓=∑^∞_𝑘=0𝑃_𝑘 be its Taylor series at zero, which is uniformly converging in 𝑈=𝐵_𝐴(0,𝛿). Let us assume that there exists 𝑎₀∈𝑈 with 𝑓(𝑎₀)∈inv(𝐵). Then there exists 𝑚₀∈ℕ such that ∑^𝑚0_𝑘=0𝑃_𝑘(𝑎₀)∈inv(𝐵).

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 𝑓∶𝐵_𝐴(0,𝜚)→𝐵 be a holomorphic mapping, where 𝐴 and 𝐵 are C*-algebras with 𝐵 unital and let 𝑓=∑^∞_𝑘=0𝑃_𝑘 be its Taylor series at zero, which is uniformly converging in 𝑈=𝐵_𝐴(0,𝛿). Suppose 𝑓 is orthogonality preserving and orthogonally additive on 𝐴_𝑠𝑎∩𝑈 and 𝑓(𝑈)∩inv(𝐵)≠∅. Then there exist a sequence (ℎ_𝑛) in 𝐵^∗∗ and Jordan ∗-homomorphisms Θ,Θ∶𝑀(𝐴)→𝐵^∗∗ such that 𝑓(𝑎)=^∞_𝑛=1ℎ_𝑛Θ𝑎^𝑛=^∞_𝑛=1Θ𝑎^𝑛ℎ_𝑛, (36) uniformly in 𝑎∈𝑈.

Proof. By Corollary 7 there exists a sequence (𝑇_𝑛) of operators from 𝐴 into 𝐵 satisfying that the pair (𝑇_𝑛,𝑇_𝑚) is orthogonality preserving on 𝐴_𝑠𝑎 for every 𝑛,𝑚∈ℕ and 𝑓(𝑥)=^∞_𝑛=1𝑇_𝑛𝑥^𝑛, (37) uniformly in 𝑥∈𝑈.
Now, Proposition 14 (a), applied to 𝑇_𝑛 (𝑛∈ℕ), implies the existence of sequences (Φ_𝑛) and (Φ_𝑛) of Jordan ∗-homomorphisms from 𝑀(𝐴) into 𝐵^∗∗ satisfying Φ_𝑛(1)=𝑟(ℎ_𝑛)𝑟(ℎ_𝑛)^∗, Φ_𝑛(1)=𝑟(ℎ_𝑛)^∗𝑟(ℎ_𝑛), where ℎ_𝑛=𝑇_𝑛^∗∗(1), and 𝑇_𝑛(𝑎)=Φ_𝑛(𝑎)ℎ_𝑛=ℎ_𝑛Φ_𝑛(𝑎), (38) for every 𝑎∈𝐴, 𝑛∈ℕ. Moreover, from Proposition 14 (b), the pairs (Φ_𝑛,Φ_𝑚) and (Φ_𝑛,Φ_𝑚) are orthogonality preserving on 𝐴_𝑠𝑎, for every 𝑛,𝑚∈ℕ.
Since 𝑓(𝑈)∩inv(𝐵)≠∅, it follows from Lemma 17 that there exists a natural 𝑚₀ and 𝑎₀∈𝐴 such that ^𝑚0_𝑘=1𝑃_𝑘𝑎₀=^𝑚0_𝑘=1Φ_𝑘𝑎^𝑘₀ℎ_𝑘=^𝑚0_𝑘=1ℎ_𝑘Φ_𝑘𝑎^𝑘₀∈inv(𝐵). (39)
We claim that 𝑟(𝑟(ℎ₁)^∗𝑟(ℎ₁)+⋯+𝑟(ℎ_𝑚0)^∗𝑟(ℎ_𝑚0))=1 in 𝐵^∗∗. Otherwise, we find a nonzero projection 𝑞∈𝐵^∗∗ satisfying 𝑟𝑟ℎ₁^∗𝑟ℎ₁+⋯+𝑟ℎ_𝑚0^∗𝑟ℎ_𝑚0𝑞=0. (40) Since 𝑟(ℎ_𝑖)^∗𝑟(ℎ_𝑖)≤𝑟(𝑟(ℎ₁)^∗𝑟(ℎ₁)+⋯+𝑟(ℎ_𝑚0)^∗𝑟(ℎ_𝑚0)), this would imply that ^𝑚0_𝑘=1𝑃_𝑘𝑎₀𝑞=^𝑚0_𝑘=1Φ_𝑘𝑎^𝑘₀ℎ_𝑘𝑞=0, (41) contradicting that ∑^𝑚0_𝑘=1𝑃_𝑘(𝑎₀)=∑^𝑚0_𝑘=1Φ_𝑘(𝑎^𝑘₀)ℎ_𝑘 is invertible in 𝐵.
Consider now the mapping Ψ=∑^𝑚0_𝑘=1Φ_𝑘. It is clear that, for each natural 𝑛, Ψ, Φ_𝑛, and the pair (Ψ,Φ_𝑛) are orthogonality preserving. Applying Proposition 14 (b), we deduce the existence of Jordan ∗-homomorphisms Θ,Θ,Θ_𝑛,Θ_𝑛∶𝑀(𝐴)→𝐵^∗∗ such that (Θ,Θ_𝑛) and (Θ,Θ_𝑛) are orthogonality preserving, Θ(1)=𝑟(𝑘)𝑟(𝑘)^∗, Θ(1)=𝑟(𝑘)^∗𝑟(𝑘), Θ_𝑛(1)=𝑟(ℎ_𝑛)𝑟(ℎ_𝑛)^∗, Θ_𝑛(1)=𝑟(ℎ_𝑛)^∗𝑟(ℎ_𝑛), Ψ(𝑎)(𝑎)Θ(𝑎),Φ=Θ𝑘=𝑘_𝑛(𝑎)=Θ_𝑛(𝑎)𝑟ℎ_𝑛^∗𝑟ℎ_𝑛=𝑟ℎ_𝑛^∗𝑟ℎ_𝑛Θ_𝑛(𝑎), (42) for every 𝑎∈𝐴, where 𝑘=Ψ(1)=𝑟(ℎ₁)^∗𝑟(ℎ₁)+⋯+𝑟(ℎ_𝑚0)^∗𝑟(ℎ_𝑚0). The condition 𝑟(𝑘)=1, proved in the previous paragraph, shows that Θ(1)=1. Thus, since (Θ,Θ_𝑛) is orthogonality preserving, the last statement in Lemma 10 proves that Θ_𝑛(𝑎)=Θ_𝑛(1)Θ(𝑎)=Θ(𝑎)Θ_𝑛(1), (43) for every 𝑎∈𝐴, 𝑛∈ℕ. The above identities guarantee that Φ_𝑛(𝑎)=Θ(𝑎)𝑟ℎ_𝑛^∗𝑟ℎ_𝑛=𝑟ℎ_𝑛^∗𝑟ℎ_𝑛Θ(𝑎), (44) for every 𝑎∈𝐴, 𝑛∈ℕ.
A similar argument to the one given above, but replacing Φ_𝑘 with Φ_𝑘, shows the existence of a Jordan ∗-homomorphism Θ∶𝑀(𝐴)→𝐵^∗∗ such that Φ_𝑛(𝑎)=Θ(𝑎)𝑟ℎ_𝑛𝑟ℎ_𝑛^∗=𝑟ℎ_𝑛𝑟ℎ_𝑛^∗Θ(𝑎), (45) for every 𝑎∈𝐴, 𝑛∈ℕ, 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.