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

Authors: Jorge J. Garcés; Antonio M. Peralta; Daniele Puglisi; María Isabel Ramírez
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study holomorphic maps between C*-algebras A and B, when f:BA(0,ϱ)→B is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball U=BA(0,δ). If we assume that f is orthogonality preserving and orthogonally additive on Asa∩U and f(U) contains an invertible element in B, then there exist a sequence (hn) in B** and Jordan *-homomorphisms Θ,Θ~:M(A)→B** such that f(x)=∑n=1∞hnΘ~(an)=∑n=1∞Θ(an)hn 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.


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## Body

## 1. Introduction

The description of orthogonally additiven-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 mappingf 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, Asa, of A reads as follows (see Section 2 for concrete definitions not explained here).Theorem 1 (see [3]).
LetA 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 operatorT:A→B satisfying
(1)P(a)=T(an),
for every a∈A and ∥P∥≤∥T∥≤2∥P∥.(b)
P is additive on elements having zero products.(c)
P is orthogonally additive on Asa.The task of replacingn-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 anL1(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 (gk)k⊆L1(μ), and a holomorphic function of bounded type h:C(K)→L1(μ) such that h(a)=∑k=0∞‍gkak and
(2)f(a)=∫K‍h(a)dμ,
for every a∈C(K) (cf. [8, Theorem 3.3]).WhenC(K) is replaced with a general C*-algebra A, a holomorphic function of bounded type f:A→ℂ is orthogonally additive on Asa if and only if there exist a positive functional φ in A*, a sequence (ψn) in L1(A**,φ), and a power series holomorphic function h in ℋb(A,A*) such that
(3)h(a)=∑k=1∞‍ψk·ak,f(a)=〈1A**,h(a)〉=∫‍h(a)dφ,
for every a in A, where 1A** 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.LetA and B be C*-algebras. When f:U⊆A→B is a map and the condition
(4)a⊥b⟹f(a)⊥f(b)(resp.,ab=0⟹f(a)f(b)=0)
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 betweenC0(L) spaces is established by Bu et al. in [11, Theorem 3.4].Theorem 2 (see [11]).
LetL1 and L2 be locally compact Hausdorff spaces and let f:BC0(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 (𝒪n) of open subsets of L2, a sequence (hn) of bounded functions from L2∪{∞} into ℂ, and a mapping φ:L2→L1 such that for each natural n the function hn is continuous and nonvanishing on 𝒪n and
(5)f(a)(t)=∑n=1∞‍hn(t)(a(φ(t)))n,(t∈L2),
uniformly in a∈BC0(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: letA and B be C*-algebras with B unital and let f:BA(0,ϱ)→B be a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball U=BA(0,δ). Suppose f is orthogonality preserving and orthogonally additive on Asa∩U and f(U) contains an invertible element. Then there exist a sequence (hn) in B** and Jordan *-homomorphisms Θ,Θ~:M(A)→B** such that
(6)f(x)=∑n=1∞‍hnΘ~(an)=∑n=1∞‍Θ(an)hn,
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 Section3. 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

LetX 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 domainU 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 (Pk(z0))k of polynomials from X into Y, where each Pk(z0) is k-homogeneous, and a neighborhood Vz0 of z0 such that the series,
(7)∑k=0∞‍Pk(z0)(y-z0),
converges uniformly to f(y) for every y∈Vz0. 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=0∞‍Pk, has infinite radius of uniform convergence, that is, limsupk→∞∥Pk∥1/k=0 (compare [15, Section 6.2], see also [16]).Supposef:BX(0,δ)→Y is a holomorphic function and let f=∑k=0∞‍Pk be its Taylor series at zero which is assumed to be uniformly convergent in U=BX(0,δ). Given φ∈Y*, it follows from Cauchy's integral formula that, for each a∈U, we have
(8)φPn(a)=12πi∫γ‍φf(λa)λn+1dλ,
where γ is the circle forming the boundary of a disc in the complex plane Dℂ(0,r1), taken counterclockwise, such that a+Dℂ(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.
Letf:BA(0,ϱ)→B be a holomorphic mapping, where A is a C*-algebra and B is a complex Banach space, and let f=∑k=0∞‍Pk be its Taylor series at zero, which is uniformly converging in U=BA(0,δ). Then the mapping f is orthogonally additive on U (resp., orthogonally additive on Asa∩U or additive on elements having zero product in U) if and only if all the Pk's satisfy the same property. In such a case, P0=0.We recall that a functionalφ in the dual of a C*-algebra A is symmetric when φ(a)∈ℝ, for every a∈Asa. Reciprocally, if φ(b)∈ℝ for every symmetric functional φ∈A*, the element b lies in Asa. 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(Asa)⊆Bsa, or equivalently, f(a)=f(a)*, whenever a∈Asa.Lemma 4.
Letf:BA(0,ϱ)→B be a holomorphic mapping, where A and B are C*-algebras, and let f=∑k=0∞‍Pk be its Taylor series at zero, which is uniformly converging in U=BA(0,δ). Then the mapping f is symmetric on U (i.e., f(Asa∩U)⊆Bsa) if and only if Pk is symmetric (i.e., Pk(Asa)⊆Bsa) for every k∈ℕ∪{0}.Definition 5.
LetS,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 mappingT: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 Asa) if and only if (T,S) is orthogonality preserving (on Asa).Our next result assures that then-homogeneous polynomials appearing in the Taylor series of an orthogonality preserving holomorphic mapping between C*-algebras are pairwise orthogonality preserving.Proposition 6.
Letf:BA(0,ϱ)→B be a holomorphic mapping, where A and B are C*-algebras, and let f=∑k=0∞‍Pk be its Taylor series at zero, which is uniformly converging in U=BA(0,δ). The following statements hold. (a)
The mappingf is orthogonally preserving on U (resp., orthogonally preserving on Asa∩U) if and only if P0=0 and the pair (Pn,Pm) is orthogonality preserving (resp., orthogonally preserving on Asa) for every n,m∈ℕ.(b)
The mappingf preserves zero products on U if and only if P0=0 and for every n,m∈ℕ, the pair (Pn,Pm) preserves zero products.Proof.
(a) The “if” implication is clear. To prove the “only if” implication, let us fixa,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 ∥Pm∥≤C/rm, for every m∈ℕ∪{0}. By hypothesis f(ta)⊥f(tb), for every r>t>0, hence
(9)P0(ta)P0(tb)*+P0(ta)(∑k=1∞‍Pk(tb))*+(∑k=1∞‍Pk(ta))(∑k=0∞‍Pk(tb))*=0,
and by homogeneity
(10)P0(a)P0(b)*=-P0(a)(∑k=1∞‍tkPk(b))*+(∑k=1∞‍tkPk(a))(∑k=0∞‍tkPk(b))*.
Letting t→0, we have P0(a)P0(b)*=0. In particular, P0=0.
We will prove by induction onn that the pair (Pj,Pk) is orthogonality preserving on U for every 1≤j,k≤n. Since f(ta)f(tb)*=0, we also deduce that
(11)P1(ta)P1(tb)*+P1(ta)(∑k=2∞‍Pk(tb))*+(∑k=2∞‍Pk(ta))(∑k=1∞‍Pk(tb))*=0,
for every (min{∥a∥,∥b∥})/r>t>0, which implies that
(12)t2P1(a)P1(b)*=-tP1(a)(∑k=2∞‍tkPk(b))*-(∑k=2∞‍tkPk(a))(∑k=1∞‍tkPk(b))*,
for every (min{∥a∥,∥b∥})/r>t>0, and hence
(13)∥P1(a)P1(b)*∥≤tC∥P1(a)∥∑k=2∞‍∥b∥krktk-2+tC2(∑k=2∞‍∥a∥krktk-2)(∑k=1∞‍∥b∥krktk-1).
Taking limit in t→0, we get P1(a)P1(b)*=0. Let us assume that (Pj,Pk) is orthogonality preserving on U for every 1≤j,k≤n. Following the argument above we deduce that
(14)P1(a)Pn+1(b)*+Pn+1(a)P1(b)*=-tP1(a)(∑j=n+2∞‍tj-n-2Pj(b))*-t∑k=2n‍tk-2Pk(a)(∑j=n+1∞‍tj-n-1Pj(b))*-tPn+1(a)(∑j=2∞‍tj-2Pj(b))*-t(∑k=n+2∞‍tk-n-2Pk(a))(∑j=1∞‍tj-1Pj(b))*,
for every (min{∥a∥,∥b∥})/r>|t|>0. Taking limit in t→0, we have
(15)P1(a)Pn+1(b)*+Pn+1(a)P1(b)*=0.
Replacing a with sa (s>0) we get
(16)sP1(a)Pn+1(b)*+sn+1Pn+1(a)P1(b)*=0,
for every s>0, which implies that
(17)P1(a)Pn+1(b)*=0.
In a similar manner we prove that Pk(a)Pn+1(b)*=0, for every 1≤k≤n+1. The equalities Pk(b)*Pj(a)=0 (1≤j,k≤n+1) follow similarly.
We have shown that for eachn,m∈ℕ, Pn(a)⊥Pm(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 Pn(ρa)⊥Pm(ρb) for every n,m∈ℕ, witnessing that (Pn,Pm) 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.
Letf:BA(0,ϱ)→B be a holomorphic mapping, where A and B are C*-algebras and let f=∑k=0∞‍Pk be its Taylor series at zero, which is uniformly converging in U=BA(0,δ). Suppose f is orthogonality preserving and orthogonally additive on (resp., orthogonally additive and zero products preserving) Asa∩U. Then there exists a sequence (Tn) of operators from A into B satisfying that the pair (Tn,Tm) is orthogonality preserving on Asa (resp., zero products preserving on Asa) for every n,m∈ℕ and
(18)f(x)=∑n=1∞‍Tn(xn),
uniformly in x∈U. In particular every Tn is orthogonality preserving (resp., zero products preserving) on Asa. Furthermore, f is symmetric if and only if every Tn is symmetric.Proof.
Combining Lemma3 and Proposition 6, we deduce that P0=0, Pn is orthogonally additive on Asa, and (Pn,Pm) is orthogonality preserving on Asa for every n,m in ℕ. By Theorem 1, for each natural n there exists an operator Tn:A→B such that ∥Pn∥≤∥Tn∥≤2∥Pn∥ and
(19)Pn(a)=Tn(an),
for every a∈A.
Consider now two positive elementsa,b∈A with a⊥b and fix n,m∈ℕ. In this case there exist positive elements c,d in A with cn=a and dm=b and c⊥d. Since the pair (Pn,Pm) is orthogonality preserving on Asa, we have Tn(a)=Tn(cn)=Pn(c)⊥Pm(d)=Tm(dm)=Tm(b). Now, noticing that given a,  b in Asa with a⊥b, we can write a=a+-b- and b=b+-b-, where aσ and bτ are positive, a+⊥a-,b+⊥b-, and aσ⊥bτ; for every σ,τ∈{+,-}, we deduce that Tn(a)⊥Tm(b). This shows that the pair (Tn,Tm) is orthogonality preserving on Asa.
Whenf is orthogonally additive on Asa and zero products preserving, then the pair (Tn,Tm) is zero products preserving on Asa for every n,m∈ℕ. The final statement is clear from Lemma 4.It should be remarked here that if a mappingf:BA(0,δ)→B is given by an expression of the form in (18) which uniformly converges in U=BA(0,δ), where (Tn) is a sequence of operators from A into B such that the pair (Tn,Tm) is orthogonality preserving on Asa (resp., zero products preserving on Asa) for every n,m∈ℕ, then f is orthogonally additive and orthogonality preserving on Asa∩U (resp., orthogonally additive on Asa∩U and zero products preserving).


## 3. Orthogonality Preserving Pairs of Operators

LetA 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 Asa. Our description generalizes some of the results obtained by Wolff in [17] because a (symmetric) mapping T:A→B is orthogonality preserving on Asa 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 thatS,T being symmetric implies that (S,T) is orthogonality preserving on Asa if and only if (S,T) is zero products preserving on Asa. We shall present here a newfangled and simplified proof which is also valid for pairs of operators.Leta 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 (a1/3n) converges in the strong*-topology of M to s(a) (cf. [18, Sections 1.10 and 1.11]).An elemente 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 Ae(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 elementa 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*-algebraA, 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.
LetA and B be C*-algebras and let S,T:A→B be a pair of operators. (a)
The pair(S,T) preserves orthogonality (on Asa) 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 Asa) 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. Leta,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 (xλ) and (yμ) in A, converging in the weak* topology of A** to c* and d*, respectively. Since T(cxλc)T(dyμd)*=T(dyμd)*T(cxλc)=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.
LetS,T:A→B be operators between C*-algebras such that (S,T) is orthogonality preserving on Asa. Let us denote h:=S**(1) and k:=T**(1). Then the identities,
(20)S(a)T(a*)*=S(a2)k*=hT((a2)*)*,T(a*)*S(a)=k*S(a2)=T((a2)*)*h,S(a)k*=hT(a*)*,k*S(a)=T(a*)*h,
hold for every a∈A.Proof.
By Lemma8, we may assume, without loss of generality, that A is unital. (a) for each φ∈B*, the continuous bilinear form Vφ:A×A→ℂ, Vφ(a,b)=φ(S(a)T(b*)*) is orthogonal; that is, Vφ(a,b)=0, whenever ab=0 in Asa. By Goldstein's theorem [19, Theorem 1.10], there exist functionals ω1,ω2∈A* satisfying that
(21)Vφ(a,b)=ω1(ab)+ω2(ba),
for all a,b∈A. Taking b=1 and a=b we have
(22)φ(S(a)k*)=Vφ(a,1)=Vφ(1,a)=φ(hT(a)*),φ(S(a)T(a)*)=φ(S(a2)k*)=φ(hT(a2)*),
for every a∈Asa, 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 Vφ(a,b)=φ(S(a)T(b*)*) with Vφ(a,b)=φ(T(b*)*S(a)).Lemma 10.
LetJ1,J2:A→B be Jordan *-homomorphism between C*-algebras. The following statements are equivalent. (a)
The pair(J1,J2) is orthogonality preserving on Asa.(b)
The identity(23)J1(a)J2(a)=J1(a2)J2**(1)=J1**(1)J2(a2),
holds for every a∈Asa,(c)
The identity,(24)J1**(1)J2(a)=J1(a)J2**(1),
holds for every a∈Asa.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 Ji(x)=Ji**(1)Ji(x)Ji**(1)=Ji(x)Ji**(1)=Ji**(1)Ji(x), for every x∈A. Therefore, given a,b∈Asa 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 Asa 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,IdA) is orthogonality preserving. The following noncommutative extension of [13, Theorem 5] follows from Proposition 9.Corollary 11.
LetT 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 elementsa, and b in a JB*-algebra A are said to operator commute in A if the multiplication operators Ma and Mb commute, where Ma is defined by Ma(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.
Lete 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 mappingRe*:(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]).
IfT is an operator from a C*-algebra A into another C*-algebra B the following are equivalent.(a)
T is orthogonality preserving (on Asa).(b)
There exists a unital Jordan*-homomorphism J:M(A)→B2**(r(h)) such that J(x) and h=T**(1) operator commute and
(25)T(x)=h•r(h)J(x),foreveryx∈A,
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, whenT 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.
LetA 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 operatorS 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 Asa if and only if the following statements hold.
(b1)
There exist Jordan*-homomorphisms Φ1,Φ~1,Φ2,Φ~2:M(A)→B** satisfying Φ1(1)=r(h)r(h)*, Φ~1(1)=r(h)*r(h), Φ2(1)=r(k)r(k)*, Φ~2(1)=r(k)*r(k),S(a)=Φ1(a)h=hΦ~1(a), and T(a)=Φ2(a)k=kΦ~2(a), for every a∈A.(b2)
The pairs(Φ1,Φ2) and (Φ~1,Φ~2) are orthogonality preserving on Asa.Proof.
The “if” implications are clear in both statements. We will only prove the “only if” implication.(a)
By Theorem13, 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
(26)S(x)=h•r(a)J1(a)foreverya∈A.
Fix a∈Asa. 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,
(27)hr(h)*J1(a)r(h)*=J1(a)r(h)*hr(h)*,
or equivalently,
(28)hr(h)*J1(a)=J1(a)r(h)*h.
Consequently, we have
(29)S(a)=h•r(h)J1(a)=hr(h)*J1(a)=J1(a)r(h)*h,
for every a∈A. The desired statement follows by considering Φ1(a)=J1(a)r(h)* and Φ~1(a)=r(h)*J1(a).(b)
The statement in (b1)  follows from (a). We will prove (b2). By hypothesis, givena,b in Asa with a⊥b, we have
(30)0=S(a)T(b)*=(hΦ~1(a))(kΦ~2(b))*=hΦ~1(a)Φ~2(b)*k*.
Having in mind that Φ~1(A)⊆r(h)*r(h)B** and Φ~2(A)⊆B**r(k)*r(k), we deduce that Φ~1(a)Φ~2(b)*=0 (compare the comments before Lemma 8), as we desired. In a similar fashion we prove Φ~2(b)*Φ~1(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*-algebraB provides enough advantages to establish a full description of the orthogonally additive, orthogonality preserving, and holomorphic mappings which are valued in B.Proposition 15.
LetS,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.
LetΦ1,Φ~1,Φ2,Φ~2:M(A)→B** be the Jordan *-homomorphisms satisfying (b1) and (b2) in Proposition 14. By hypothesis, B is commutative, and hence Φi=Φ~i for every i=1,2 (compare the proof of Proposition 14). Since the pair (Φ1,Φ2) is orthogonality preserving on Asa, Lemma 10 assures that
(31)Φ1**(1)Φ2(a)=Φ1(a)Φ2**(1),
for every a∈Asa. In order to simplify notation, let us denote p=Φ1**(1) and q=Φ2**(1).
We define an operatorΦ:M(A)→B**, given by
(32)Φ(a)=pqΦ1(a)+p(1-q)Φ1(a)+q(1-p)Φ2(a).
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.
Letf:BA(0,ϱ)→B be a holomorphic mapping, where A and B are C*-algebras with B commutative and let f=∑k=0∞‍Pk be its Taylor series at zero, which is uniformly converging in U=BA(0,δ). Suppose f is orthogonality preserving and orthogonally additive on Asa∩U (equivalently, orthogonally additive on Asa∩U and zero products preserving). Then there exist a sequence (hn) in B** and a Jordan *-homomorphism Φ:M(A)→B** such that
(33)f(x)=∑n=1∞‍hnΦ(an)=∑n=1∞‍hnΦ(an),
uniformly in a∈U.Proof.
By Corollary7, there exists a sequence (Tn) of operators from A into B satisfying that the pair (Tn,Tm) is orthogonality preserving on Asa (equivalently, zero products preserving on Asa) for every n,m∈ℕ and
(34)f(x)=∑n=1∞‍Tn(xn),
uniformly in x∈U. Denote hn=Tn**(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 Ψn:M(A)→B** such that r(Ψn(1))=r(|h1|+⋯+|hn|) and Tk(a)=hkΨn(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, Tk, Tm, and the pair (Tk,Tm) are orthogonality preserving, we can easily check that Tn+1, T1+⋯+Tn and (Tn+1,T1+⋯+Tn)=(Tn+1,(h1+⋯+hn)Ψn) are orthogonality preserving. By Proposition 15, there exists a Jordan *-homomorphism Ψn+1:M(A)→B** satisfying r(Ψn+1(1))=r(|h1|+⋯+|hn|+|hn+1|), Tn+1(a)=hn+1Ψn+1(an+1) and (T1+⋯+Tn)(a)=(h1+⋯+hn)Ψn+1(a) for every a∈A. Since, for each 1≤k≤n,
(35)hkΨn+1(a)=hkr(|h1|+⋯+|hn|+|hn+1|)Ψn+1(a)=hkr(|h1|+⋯+|hn|)Ψn+1(a)=hkr(|h1|+⋯+|hn|)Ψn(a)=hkΨn(a)=Tk(a),
for everya∈A, as desired.
Let us consider a free ultrafilter𝒰 on ℕ. By the Banach-Alaoglu theorem, any bounded set in B** is relatively weak*-compact, and thus the assignment a↦Φ(a):=w*-lim𝒰Ψn(a) defines a Jordan *-homomorphism from M(A) into B**. If we fix a natural k, we know that Tk(a)=hkΨn(a), for every n≥k and a∈A. Then it can be easily checked that Tk(a)=hkΦ(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 Theorem2 (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 Theorem16.Given a unital C*-algebraA, 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.
Letf:BA(0,ϱ)→B be a holomorphic mapping, where A and B are C*-algebras with B unital and let f=∑k=0∞‍Pk be its Taylor series at zero, which is uniformly converging in U=BA(0,δ). Let us assume that there exists a0∈U with f(a0)∈inv(B). Then there exists m0∈ℕ such that ∑k=0m0‍Pk(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.
Letf:BA(0,ϱ)→B be a holomorphic mapping, where A and B are C*-algebras with B unital and let f=∑k=0∞‍Pk be its Taylor series at zero, which is uniformly converging in U=BA(0,δ). Suppose f is orthogonality preserving and orthogonally additive on Asa∩U and f(U)∩inv(B)≠∅. Then there exist a sequence (hn) in B** and Jordan *-homomorphisms Θ,Θ~:M(A)→B** such that
(36)f(a)=∑n=1∞‍hnΘ~(an)=∑n=1∞‍Θ(an)hn,
uniformly in a∈U.Proof.
By Corollary7 there exists a sequence (Tn) of operators from A into B satisfying that the pair (Tn,Tm) is orthogonality preserving on Asa for every n,m∈ℕ and
(37)f(x)=∑n=1∞‍Tn(xn),
uniformly in x∈U.
Now, Proposition14 (a), applied to Tn (n∈ℕ), implies the existence of sequences (Φn) and (Φ~n) of Jordan *-homomorphisms from M(A) into B** satisfying Φn(1)=r(hn)r(hn)*, Φ~n(1)=r(hn)*r(hn), where hn=Tn**(1), and
(38)Tn(a)=Φn(a)hn=hnΦ~n(a),
for every a∈A, n∈ℕ. Moreover, from Proposition 14 (b), the pairs (Φn,Φm) and (Φ~n,Φ~m) are orthogonality preserving on Asa, for every n,m∈ℕ.
Sincef(U)∩inv(B)≠∅, it follows from Lemma 17 that there exists a natural m0 and a0∈A such that
(39)∑k=1m0‍Pk(a0)=∑k=1m0‍Φk(a0k)hk=∑k=1m0‍hkΦ~k(a0k)∈inv(B).
We claim thatr(r(h1)*r(h1)+⋯+r(hm0)*r(hm0))=1 in B**. Otherwise, we find a nonzero projection q∈B** satisfying
(40)r(r(h1)*r(h1)+⋯+r(hm0)*r(hm0))q=0.
Since r(hi)*r(hi)≤r(r(h1)*r(h1)+⋯+r(hm0)*r(hm0)), this would imply that
(41)(∑k=1m0‍Pk(a0))q=(∑k=1m0‍Φk(a0k)hk)q=0,
contradicting that ∑k=1m0‍Pk(a0)=∑k=1m0‍Φk(a0k)hk is invertible in B.
Consider now the mappingΨ=∑k=1m0‍Φ~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~,Θn,Θ~n:M(A)→B** such that (Θ,Θn) and (Θ~,Θ~n) are orthogonality preserving, Θ(1)=r(k)r(k)*, Θ~(1)=r(k)*r(k), Θn(1)=r(hn)r(hn)*, Θ~n(1)=r(hn)*r(hn),
(42)Ψ(a)=Θ(a)k=kΘ~(a),Φ~n(a)=Θn(a)r(hn)*r(hn)=r(hn)*r(hn)Θ~n(a),
for every a∈A, where k=Ψ(1)=r(h1)*r(h1)+⋯+r(hm0)*r(hm0). 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
(43)Θ~n(a)=Θ~n(1)Θ~(a)=Θ~(a)Θ~n(1),
for every a∈A, n∈ℕ. The above identities guarantee that
(44)Φ~n(a)=Θ(a)r(hn)*r(hn)=r(hn)*r(hn)Θ~(a),
for every a∈A, n∈ℕ.
A similar argument to the one given above, but replacingΦ~k with Φk, shows the existence of a Jordan *-homomorphism Θ:M(A)→B** such that
(45)Φn(a)=Θ(a)r(hn)r(hn)*=r(hn)r(hn)*Θ(a),
for every a∈A, n∈ℕ, which concludes the proof.



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*Source: 415354-2013-12-26.xml*