On symmetry of Birkhoff orthogonality in the positive cones of C⁎-algebras with applications

J. Math. Anal. Appl. 474 (2019) 1488–1497

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

On symmetry of Birkhoff orthogonality in the positive cones of C ∗ -algebras with applications Naoto Komuro a , Kichi-Suke Saito b , Ryotaro Tanaka c,∗ a

Department of Mathematics, Hokkaido University of Education, Asahikawa Campus, Asahikawa 070-8621, Japan b Department of Mathematical Sciences, Institute of Science and Technology, Niigata University, Niigata 950-2181, Japan c Faculty of Industrial Science and Technology, Tokyo University of Science, Oshamanbe, Hokkaido 049-3514, Japan

a r t i c l e

i n f o

Article history: Received 25 October 2018 Available online 12 February 2019 Submitted by R.M. Aron Keywords: C ∗ -algebra Birkhoff orthogonality Symmetric point Positive cone Preserver

a b s t r a c t In this paper, local symmetry of Birkhoff orthogonality is considered in the positive cones of C ∗ -algebras. For the positive cone A+ of a C ∗ -algebra A, the notion of A+ -local left (or right) symmetric points for Birkhoff orthogonality is introduced. Some simple characterizations of them are given. As an application, we determine the form of Birkhoff orthogonality preservers between the positive cones of C ∗ -algebras by using Jordan ∗-isomorphisms. © 2019 Elsevier Inc. All rights reserved.

1. Introduction This paper is concerned with the following (generalized) orthogonality relation in Banach spaces, the so-called Birkhoff(-James) orthogonality. Definition 1.1 (Birkhoff(-James) orthogonality). Let X be a Banach space with the scalar field K, and let x, y ∈ X. Then x is Birkhoff(-James) orthogonal to y, denoted by x ⊥B y, if x + λy ≥ x for each λ ∈ K. In Hilbert spaces, the usual orthogonality relation defined by using inner products plays a crucial role. It provides much information about geometric positioning of vectors, subspaces and operators. Such viewpoints are, in fact, often important and useful in Banach space theory; but, in the general setting, the usual orthogonality cannot be considered in the original form. Thus, instead, various generalized orthogonality * Corresponding author. E-mail addresses: [email protected] (N. Komuro), [email protected] (K.-S. Saito), [email protected] (R. Tanaka). https://doi.org/10.1016/j.jmaa.2019.02.033 0022-247X/© 2019 Elsevier Inc. All rights reserved.

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relations have been introduced in terms of norm (in)equalities characterizing the usual orthogonality in Hilbert spaces. The notion of Birkhoff orthogonality is introduced by Birkhoff [5] from the viewpoint of best approximation vectors; and many of its useful properties were given by James [11,12]. If an element x of a Banach space X is Birkhoff orthogonal to another y, then x is the nearest point to the origin in the line L = {x + λy : λ ∈ K}, where K is the scalar field of X. If x is additionally a unit vector, then L is tangent to the unit ball of X; from this, Birkhoff orthogonality is closely related to support hyperplanes and support functionals of the unit balls. For further information about Birkhoff orthogonality, Alonso–Martini–Wu [1] gives a good exposition. Birkhoff orthogonality ⊥B is similar to the usual orthogonality ⊥ in the sense of homogeneity, that is, x ⊥B y implies that αx ⊥B βy for each scalars α, β. On the other hand, ⊥B is completely different from ⊥ in their symmetry. Indeed, as James [11, Theorem 1] was proved, if x ⊥B y implies that y ⊥B x for each x, y in a Banach space X with dim X ≥ 3, then X is a Hilbert space. Consequently, in the most Banach spaces, Birkhoff orthogonality is not symmetric. However, Birkhoff orthogonality has the possibility of being symmetric in local senses. For example, Turnšek [21] showed the following: In the algebra B(H) of all bounded linear operators on a complex Hilbert space H, an element V ∈ B(H) is a scalar multiple of an isometry or a coisometry if and only if A ∈ B(H) and A ⊥B V imply that V ⊥B A. Later, this last property was named “right symmetric”, while its reflection was called “left symmetric”. See Sain [19] and Sain–Ghosh–Paul [20]. Let X be a Banach space. • An element x of X is called a left symmetric point for Birkhoff orthogonality if y ∈ X and x ⊥B y imply that y ⊥B x. • An element x of X is called a right symmetric point for Birkhoff orthogonality if y ∈ X and y ⊥B x imply that x ⊥B y. These notions (with some variations) are, recently, extensively studied by several authors [2,3,8,10,17,18,22]. The first aim of this paper is to consider further localizations of such symmetry. For this, the following notions are introduced. Definition 1.2. Let X be a Banach space, and let M be a subset of X. Then x ∈ M is an M -local left symmetric point for Birkhoff orthogonality if y ∈ M and x ⊥B y imply that y ⊥B x. Definition 1.3. Let X be a Banach space, and let M be a subset of X. Then x ∈ M is an M -local right symmetric point for Birkhoff orthogonality if y ∈ M and y ⊥B x imply that x ⊥B y. In this paper, particularly, X and M are assumed to be a C ∗ -algebra A and its positive cone A+ , respectively. Then, it is shown for a C ∗ -algebra A that an A+ -local left symmetric point is a “very small” projection, while an A+ -local right symmetric point has properties near the strict positivity. In particular, if A is unital then A+ -local right symmetric points are characterized as (positive) invertible elements. The second aim of this paper is concerned with the following theorem. Theorem 1.4 (Blanco and Turnšek [6]). Let X, Y be Banach spaces, and let T : X → Y be a linear mapping such that T x ⊥B T y whenever x ⊥B y. Then T is a scalar multiple of an isometry. Our aim is to localize this theorem in the setting of C ∗ -algebras. For this, in particular, local right symmetric points play an important role. It is shown that every additive bijection between the positive cones of unital C ∗ -algebras that preserves Birkhoff orthogonality in both directions extends to a scalar multiple of a Jordan ∗-isomorphism.

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2. Preliminaries Let X be a Banach space. Then X  denotes the dual space of X. If H is a Hilbert space and x, y ∈ H, then x, y means the inner product of x and y. For a locally compact Hausdorff space K, the symbol C0 (K) stands for the algebra of all continuous functions on K vanishing at infinity. In the case that K is compact, C0 (K) coincides with C(K), the space of all continuous functions on K. Suppose that A is a C ∗ -algebra. An element A ∈ A is said to be self-adjoint if A∗ = A, and positive if A = B ∗ B for some B ∈ A. The subset Ar of A consisting of all self-adjoint elements is called the real part of A and it forms a real Banach space. The positive cone A+ is the closed cone of all positive elements of A. An element ρ ∈ A is said to be positive if ρ(A) ≥ 0 for each A ∈ A+ . If A is unital, that is, A contains the multiplicative unit I, then ρ ∈ A is positive if and only if ρ(I) = ρ. In particular, functionals satisfying ρ(I) = ρ = 1 are called states of A. If a C ∗ -algebra A is not unital, we can consider a unitization AI of A. Here, AI is a unital C ∗ -algebra containing A as a two-sided ideal of codimension one. Moreover, it is well-known that every C ∗ -algebra can be viewed as a C ∗ -subalgebra of B(H) for some Hilbert space H by using the GNS-construction. For comprehensive knowledge of operator algebras, see, for example, [14,15]. The rest of this section is devoted to presenting some useful characterizations of Birkhoff orthogonality. The first one gives a relationship between Birkhoff orthogonality and support functionals. Lemma 2.1 (James [12]). Let X be a Banach space, and let x, y be elements of X. Then x ⊥B y if and only if there exists an f ∈ X  satisfying f  = 1, f (x) = x and f (y) = 0. In operator algebras, Birkhoff orthogonality has another frequently used characterization. Lemma 2.2 (Bhatia and Šemrl [4]). Let H be a complex Hilbert space, and let A, B ∈ B(H). Then A ⊥B B if and only if there exists a sequence (xn ) of unit vectors in H such that Axn  → A and Axn , Bxn → 0 as n → ∞. For positive elements of a (unital) C ∗ -algebra, Lemma 2.1 has the following slight improvement. Lemma 2.3. Let A be a unital C ∗ -algebra, and let A, B ∈ A+ . If A ⊥B B, then there exists a state of A such that ρ(A) = A and ρ(B) = 0. Proof. By Lemma 2.1, there exists an ω ∈ A such that ω = 1, ω(A) = A and ω(B) = 0. Let ω ∗ (C) = ω(C ∗ ) for each C ∈ A. Then ω ∗ is also an element of A and ω ∗  = 1. Let ρ = 2−1 (ω + ω ∗ ). Then ρ(C ∗ ) = ρ(C) for each C, that is, ρ is a hermitian functional on A. By [14, Theorem 4.3.6], there exists a pair ρ+ , ρ− of positive linear functionals such that ρ = ρ+ − ρ− and ρ = ρ+  + ρ− . Since A is positive and ρ(A) = A, we have ρ = 1 and ρ+ A ≥ ρ+ (A) = ρ(A) + ρ− (A) ≥ A, which implies that ρ+  = 1 (and ρ−  = 0). Hence ρ is a state of A. Moreover, one obtains ρ(B) = 0 since B ∗ = B. 2 In this paper, we mainly deal with C ∗ -algebras that are complex Banach spaces. On the other hand, we often restrict our arguments to the positive cones or real parts, which are (subsets) of real Banach spaces. Here, the following problem occurs: Which scalars should be taken for Birkhoff orthogonality? The following result prevents such ambiguity.

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Lemma 2.4. Let A be a C ∗ -algebra, and let A, B ∈ Ar . Then A + λB ≥ A for each λ ∈ C if and only if A + αB ≥ A for each α ∈ R. Proof. The forward implication is clear. For the converse, suppose that A + αB ≥ A for each α ∈ R. Considering a faithful representation, we may assume that A is a C ∗ -subalgebra of B(H) for some Hilbert space H. For given α, β ∈ R, let λ = α + iβ. Since A + αB is self-adjoint, one has A + αB = sup{|(A + αB)x, x | : x = 1}; see, for example [9, Proposition II.2.13]. Let (xn ) be a sequence of unit vectors such that |(A + αB)xn , xn | → A + αB as n → ∞. It follows from A + λB ≥ |(A + αB)xn , xn + iβBxn , xn | = (|(A + αB)xn , xn |2 + |βBxn , xn |2 )1/2 ≥ |(A + αB)xn , xn | for each n that A + λB ≥ A + αB ≥ A. This completes the proof. 2 3. Local left symmetric points We start this section with the following fundamental result for the commutative case. The proof is a slight modification of arguments in [17, Theorem 3.6]. Lemma 3.1. Let K be a locally compact Hausdorff space, and let f ∈ C0 (K)+ with f ∞ = 1. If f is a C0 (K)+ -local left symmetric point for Birkhoff orthogonality, then f = χ{t0 } for some isolated point t0 ∈ K. Proof. Suppose that f is a C0 (K)+ -local left symmetric point for Birkhoff orthogonality. We assume that K is non-compact. In the case that K is compact, the proof is appropriately simplified. Let K∞ = K ∪ {t∞ } be the one-point compactification of K. We first note that C0 (K) can be naturally identified with the subalgebra A = {g ∈ C(K∞ ) : g(t∞ ) = 0} of C(K∞ ) by the ∗-isomorphism A g → g|K ∈ C0 (K). In particular, putting f (t∞ ) = 0 yields an element of C(K∞ ). Since f ∞ = 1, there exists a t0 ∈ K such that f (t0 ) = 1. To see f = χ{t0 } , suppose to the contrary that f (t1 ) > 0 for some t1 (= t0 ). Since K is Hausdorff and t1 ∈ K, there exist disjoint subsets U, V that are open in K (and hence, in K∞ ) such that t0 ∈ U and t1 ∈ V . Let W = {t ∈ K∞ : f (t) > f (t1 )/2} ∩V . We note that t0 , t∞ ∈ / W and t1 ∈ W . By Urysohn’s lemma, we have an element g ∈ C(K∞ )+ such that g∞ = g(t1 ) = 1 and g(W c ) = {0}. In particular, g is essentially an element of C0 (K)+ since g(t∞ ) = 0. Since g(t0 ) = 0, one obtains that f ⊥B g. However, we have |(g − 2−1 f )(t)| ≤ max{1 − f (t1 )/4, 1/2} < 1 for each t ∈ W , while |(g − 2−1 f )(t)| = |f (t)|/2 ≤ 1/2 for each t ∈ W c . This shows that g ⊥B f , a contradiction. Hence it follows that f = χ{t0 } . Since f is continuous, t0 must be isolated. 2 From this, we obtain a characterization of A+ -local left symmetric points for Birkhoff orthogonality. Theorem 3.2. Let A be a C ∗ -algebra, and let A ∈ A+ with A = 1. Then A is an A+ -local left symmetric point for Birkhoff orthogonality if and only if A is a projection satisfying AAA = CA. Proof. Suppose that A is an A+ -local left symmetric point for Birkhoff orthogonality. Let A1 be an abelian C ∗ -subalgebra generated by A. Then A1 is ∗-isomorphic to C0 (K1 ) for some locally compact Hausdorff space K1 . Since A is identified with a C0 (K1 )+ -local left symmetric point for Birkhoff orthogonality, at least, A is a projection.

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Now take an arbitrary B ∈ A+ . We consider the abelian C ∗ -subalgebra A2 generated by A and ABA. Let K2 be a locally compact Hausdorff space such that A2 is ∗-isomorphic to C0 (K2 ), and let f and g be elements of C0 (K2 ) that are identified with A and ABA, respectively. Then f is C0 (K2 )+ -local left symmetric for Birkhoff orthogonality; and hence f = χ{t0 } for some t0 ∈ K2 . On the other hand, since f g = g, it follows that g = g(t0 )χ{t0 } = g(t0 )f . Therefore ABA = g(t0 )A ∈ CA. Finally, since each element of A can be represented as a linear combination of positive elements, we have AAA = CA. Conversely, let A be a projection in A satisfying AAA = CA. Suppose that B ∈ A+ and A ⊥B B. By Lemma 2.2, there exists a sequence (xn ) of unit vectors such that Axn  → A = 1 and |Axn , Bxn | → 0 as n → ∞. Since A is a projection, it follows that xn − Axn 2 = 1 − Axn 2 → 0 as n → ∞, which in turn implies that |Axn , BAxn | → 0 as n → ∞. On the other hand, we have ABA = αA for some α. From this, one obtains |Axn , BAxn | = |α|Axn 2 for each n. It follows that |α| = lim |α|Axn 2 = lim |Axn , BAxn | = 0. n

n

Hence ABA = 0. Now, we have BA = B 1/2 (B 1/2 A) = 0 by B 1/2 A2 = ABA = 0, that is, B = B(I − A). Therefore the inequality B + λA ≥ B(I − A) = B holds for each λ, which shows that B ⊥B A.

2

4. Local right symmetric points In this section, we first present the following general characterization of A+ -local right symmetric points. Theorem 4.1. Let A be a C ∗ -algebra, and let A ∈ A+ with A = 1. Then the following are equivalent: (i) A is an A+ -local right symmetric point for Birkhoff orthogonality. (ii) B ∈ A+ and B ⊥B A implies that B = 0. (iii) B − BA < B for each nonzero B ∈ A+ . Proof. Throughout this proof, we use the symbol AI as the unitization A ⊕ C of A if A is non-unital. If A contains the unit, then we put AI = A. Let A be an A+ -local right symmetric point for Birkhoff orthogonality. Suppose that there exists a nonzero positive B ∈ A such that B ⊥B A. We may assume that B = 1. By Lemma 2.3, we have ρ(B) = 1 and ρ(A) = 0 for some state ρ ∈ AI . We first note that ρ vanishes on the whole of the C ∗ -subalgebra A0 generated by A. For this, it is enough to see that 0 ≤ An ≤ A for each n ∈ N by A ≤ 1. On the other hand, since A0 is abelian, there is a compact Hausdorff space K such that A0 is identified with C(K) or its closed ideal K = {f ∈ C(K) : f (t∞ ) = 0}. Let fA be the continuous function identified with A, and let K1 = {t ∈ K : fA (t) ≥ 1/3}, K2 = {t ∈ K : fA (t) ≥ 1/2} and K3 = {t ∈ K : fA (t) ≤ 1/3}. By Urysohn’s lemma, there are positive functions g0 , h0 ∈ C(K) such that g0  = h0  = 1, g(K1 ) = {1}, h(K2 ) = {1} and h(K3 ) = 0. If A0 ∼ = C(K), then we put g = g0 and h = h0 . For the other case, one has that t∞ ∈ / K1 since fA ∈ K. Hence, again by Urysohn’s lemma, we obtain a positive function k ∈ C(K) such that k(K1 ) = {1} and k(t∞ ) = 0 (that is, k ∈ K). Putting g = g0 k and h = h0 k yields elements of K. We note that, in either case, (1 − g)h = 0 and 0 ≤ fA (1 − 2−1 h) ≤ 2−1 1. Let C, D be elements of A0 identified with g and h, respectively. Then A, C, D are all commute, D = CD(≤ C), A − 2−1 AD ≥ 0 and A − 2−1 AD ≤ 1/2.

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Now let B0 = (I − C)B(I − C) + AD. Then B0 is a positive element of A since A is an ideal of AI . By considering a faithful representation, we may suppose that AI is a C ∗ -subalgebra of B(H) for some Hilbert space H. If x is a unit vector of H, then B0 x2 = (I − C)B(I − C)x + ADx2 = (I − C)B(I − C)x2 + ADx2 ≤ (I − C)B2 (I − C)x2 + A2 Dx2 ≤ (I − C)x2 + Dx2 = (I − C + D)x2 ≤1 by 0 ≤ I − C + D ≤ I. This shows that B0  ≤ 1. Moreover, since ρ vanishes on the whole of A0 , it follows that B0  ≥ ρ(B0 ) = ρ(B) − ρ(BC) − ρ(CB) + ρ(CBC) + ρ(AD) = ρ(B) = 1, that is, ρ(B0 ) = B0  = 1. Hence B0 ⊥B A holds. On the other hand, since −2−1 I ≤ −2−1 (I − C)B(I − C) ≤ A − 2−1 B0 ≤ A − 2−1 AD ≤ 2−1 I, it follows that A − 2−1 B0  ≤ 1/2 < 1. Therefore A ⊥B B0 , which contradicts the A+ -local right symmetry of A. Thus (i) ⇒ (ii) holds. Since the converse obviously holds, we can conclude that (i) ⇔ (ii). Suppose that (ii) holds. If there exists a nonzero B ∈ A such that B −BA = B, then B(I −A)2 B = (I − A)B 2 (I − A) = B 2 . It may be assumed that B = 1. Since B(I − A)2 B is positive, we have ρ(B(I − A)2 B) = B(I − A)2 B = 1 for some state ρ of AI . On the other hand, since 0 ≤ A ≤ I, the inequality 1 ≥ ρ(B(I − A)B) ≥ ρ(B(I − A)2 B) = 1 holds. It follows that ρ(B 2 ) = 1 and ρ(BAB) = 0. From this, one obtains |ρ(AB)| = |ρ(A1/2 (A1/2 B))| ≤ ρ(A)1/2 ρ(BAB)1/2 = 0 and |ρ(A(I − B))| = 0 by ρ(B) = 1. Hence ρ(A) = 0; however, which implies that B ⊥B A by Lemma 2.1. This contradicts (ii). Therefore (ii) ⇒ (iii) holds. Finally, we assume that (iii) holds. If B ⊥B A for some B ∈ A+ , by Lemma 2.3, we have a state ρ of AI such that ρ(B) = B and ρ(A) = 0. In this case, one obtains ρ(BA) = 0. It follows from B ≥ B(I − A) ≥ ρ(B(I − A)) = B that B(I − A) = B. Thus B must be 0, which shows that (iii) ⇒ (ii). The proof is complete.

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The preceding result is valid for general C ∗ -algebras, and states characteristic properties of A+ -local right symmetric points for Birkhoff orthogonality. However, we note that Theorem 4.1 contains no information

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about the existence of such elements. For effective use of local right symmetry of Birkhoff orthogonality in C ∗ -algebras, for the moment, we need some assumptions on (approximate) units. At least, in the unital case, we have a simpler characterization. Corollary 4.2. Let A be a unital C ∗ -algebra, and let A ∈ A+ be such that A = 0. Then A is an A+ -local right symmetric point for Birkhoff orthogonality if and only if A is invertible. Proof. We may assume that A = 1. Then the corollary follows from Theorem 4.1 and the fact that A is invertible if and only if I − A < 1. 2 Recall that a C ∗ -algebra A is σ-unital if there exists a sequence (An ) in A+ such that limn A −An A = 0 for each A ∈ A. An element A of a C ∗ -algebra is said to be strictly positive if A = AAA, where the closure is considered in the norm topology. It is known that a C ∗ -algebra is σ-unital if and only if it has a strictly positive element; see, for example, [7, Proposition II.4.2.4]. Moreover, as was commented in [7, II.4.2.3], a strictly positive element A in a C ∗ -algebra is characterized as a positive element satisfying ρ(A) > 0 for each nonzero positive functional ρ on A. In the σ-unital case, we have the following implication. Proposition 4.3. Let A be a σ-unital C ∗ -algebra, and let A be a strictly positive element of A. Then A is an A+ -local right symmetric point for Birkhoff orthogonality. Proof. Suppose that B ⊥B A for some B ∈ A+ . Then, by Lemma 2.3, there exists a state ρ of a unitization AI of A such that ρ(B) = B and ρ(A) = 0. Since A is strictly positive, we have B ∈ AAA, which implies that B = ρ(B) = 0. Hence A is A+ -local right symmetric for Birkhoff orthogonality. 2 It follows from the preceding proposition that σ-unital C ∗ -algebras always contain A+ -local right symmetric points for Birkhoff orthogonality. Finally, we consider the abelian case. Theorem 4.4. Let A be an abelian C ∗ -algebra, and let A ∈ A+ be such that A = 0. Then A is an A+ -local right symmetric point for Birkhoff orthogonality if and only if A is strictly positive. Proof. We may assume that A = 1. Suppose that A is A+ -local right symmetric. Since A is abelian, we can identify A with C0 (K) for some locally compact Hausdorff space K. If A is not strictly positive, then A(t) = 0 for some t ∈ K. However, then, one obtains that B − BA = B for some nonzero B ∈ A+ such that B(t) = B. This is a contradiction by Theorem 4.1. Hence A is strictly positive. The converse follows from Proposition 4.3 and the fact that a C ∗ -algebra with a strictly positive element is σ-unital. 2 It turns out that an abelian C ∗ -algebra contains an A+ -local right symmetric point for Birkhoff orthogonality if and only if it is σ-unital. Example 4.5. Let I be an uncountable set, and let K be the closed ideal of ∞ (I) consisting of all elements that have countable supports. Then, as an easy consequence of Theorem 4.4, K has no K+ -local right symmetric point for Birkhoff orthogonality. 5. Applications We now proceed to our second problem on Birkhoff orthogonality preservers between the positive cones of unital C ∗ -algebras. The following proposition provides a key ingredient.

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Proposition 5.1. Let A be a unital C ∗ -algebra, and let A ∈ A+ . Then A ⊥B B for each non-invertible B ∈ A+ if and only if A = AI. Proof. Without loss of generality, we may assume that A = 1. Suppose first that A ⊥B B for each non-invertible B ∈ A+ . Let A0 be the C ∗ -subalgebra generated by A and I. We identify A0 with C(K), where K is a compact Hausdorff space. If f is the element of C(K) identified with A, we have f ⊥B g for each non-invertible g ∈ C(K)+ . Take an arbitrary t ∈ K. Let U be an open neighborhood of t. Then, by Urysohn’s lemma, there exists an g ∈ C(K)+ such that g∞ = 1, g(t) = 1 and g(U c ) = {0}. Since f (1 − g) is non-invertible, we have f ⊥B f (1 − g). In particular, it follows from f − f (1 − g)∞ = f g∞ = 1 that f (tU ) = g(tU ) = 1 for some tU ; and tU ∈ U since g(U c ) = {0}. Now (tU )U is a net in K that converges to t, and f (tU ) = 1 for each U . Thus one obtains f (t) = 1. Since t is arbitrary, we have f = 1, that is, A = I. Conversely, if B is a non-invertible element in A, then 0 ∈ σ(B). Since 1 ∈ σ(I +λB), we have I+λB ≥ 1 for each λ ∈ C. This proves that I ⊥B B. 2 For unital C ∗ -algebras A and B, a linear mapping J : A → B is called a Jordan ∗-homomorphism if J(A2 ) = J(A)2 and J(A∗ ) = J(A)∗ for each A ∈ A. Bijective Jordan ∗-homomorphism is called a Jordan ∗-isomorphism. If J : A → B is a Jordan ∗-isomorphism, then (i) J is a linear order isomorphism; (ii) J is an isometry; and (iii) J(AB) = J(A)J(B) whenever AB = BA. See, for example, [15, Exercise 10.5.22 and 10.5.31] and their solutions in [16]. The following is the main purpose in this section. Theorem 5.2. Let A and B be unital C ∗ -algebras. Suppose that f : A+ → B+ is a bijective additive mapping that preserves Birkhoff orthogonality in both directions. Then there exists a Jordan ∗-isomorphism J : A → B such that f (A) = f (I)J(A) for each A ∈ A+ . Proof. We first note that f preserves the order. Indeed, if A1 , A2 ∈ A+ with A1 ≤ A2 , then f (A2 −A1 ) ∈ B+ and f (A2 ) = f (A2 − A1 + A1 ) = f (A2 − A1 ) + f (A1 ) ≥ f (A1 ). Next, let A ∈ A+ . Then it is routine to verify that f

 n A = f (A) m m

n

for each m, n ∈ N. Now let r be an arbitrary positive number. Then there exist two sequences (pn ) and (qn ) of positive rational numbers such that both converge to r and pn ≤ r ≤ qn for each n. Since f preserves the order, it follows that rf (A) = lim pn f (A) = lim f (pn A) ≤ f (rA) n

n

≤ lim f (qn A) = lim qn f (A) = rf (A), n

n

which implies that f (rA) = rf (A). On the other hand, we obtain f (0) = 0 from the additivity of f . Therefore f is positive homogeneous.

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Let Ar and Br be the real parts of A and B, respectively. Recall that each element of Ar is represented as the difference of two positive elements. Moreover, if H1 , H2 , K1 , K2 ∈ A+ and H1 − K1 = H2 − K2 , then f (H1 ) + f (K2 ) = f (H1 + K2 ) = f (H2 + K1 ) = f (H2 ) + f (K1 ), which implies that f (H1 ) − f (K1 ) = f (H2 ) − f (K2 ). Thus we can define a mapping T : Ar → Br by T (H − K) = f (H) − f (K) for each H, K ∈ A+ . It is easy to check that T is real-linear, bijective and T |A+ = f . Now let A ∈ A+ . Since f preserves Birkhoff orthogonality in the both directions, A is an A+ -local right symmetric point for Birkhoff orthogonality if and only if f (A) has the same property. From this and Corollary 4.2, it follows that f preserves invertible elements; and then, Proposition 5.1 assures that f (I) = f (I)I. Put T  = f (I)−1 T . Then T  is real-linear, bijective, T  (A+ ) = B+ and T  (I) = I. These conditions guarantee that T  is an isometry. Indeed, if A ∈ Ar then −AI ≤ A ≤ AI. By T  (I) = I, we obtain −AI ≤ T  (A) ≤ AI, that is, T  (A) ≤ A. Conversely, −T  (A)I ≤ T  (A) ≤ T  (A)I implies that A ≤ T  (A). Therefore T  (A) = A. Thus, by the result of Kadison [13, Theorem 2], T  extends to a Jordan ∗-isomorphism J : A → B; and, in which case, we have f (A) = T (A) = f (I)T  (A) = f (I)J(A) for each A ∈ A+ .

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Remark 5.3. If we lack the additivity of f in Theorem 5.2, then f can be strange. Let A be a unital C ∗ -algebra with a nontrivial projection P , and let ρ1 and ρ2 be states of A such that ρ1 (P ) = 1 and ρ2 (I − P ) = 1. Put ρ = 2−1 (ρ1 − ρ2 ). Then ρ(P ) = 1/2 and ρ(I − P ) = −1/2. From this, the sets D+ = {A ∈ A+ : ρ(A) ≥ 0} D− = {A ∈ A+ : ρ(A) < 0} are both nonempty. Now suppose that α and β are bijections on the set R+ of all nonnegative real numbers satisfying α(0) = β(0) = 0. Let f be a mapping on A+ defined by ⎧ ⎪ (A = 0) ⎨0 f (A) = α(A)A−1 A (A ∈ D+ \ {0}) . ⎪ ⎩ β(A)A−1 A (A ∈ D ) − Then, at least, f is a bijection and preserves Birkhoff orthogonality in the both directions. (I) If α(t) = t and β(t) = 2t for each t ≥ 0, then f is further positive homogeneous. However, it cannot be additive by f (P + I − P ) = f (I) = I = 2I − P = P + 2(I − P ) = f (P ) + f (I − P ). (II) If α(t) = β(t) = t2 for each t ≥ 0, then f (A) = AA for each A ∈ A+ , which is continuous but not positive homogeneous.

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We conclude this paper with the following natural problem. Problem 5.4. Is Theorem 5.2 true for non-unital C ∗ -algebras? How about the case of σ-unital? Acknowledgments This work was supported in part by Grants-in-Aid for Scientific Research Grant Numbers 17K05287, 15K04920, Japan Society for the Promotion of Science. References [1] J. Alonso, H. Martini, S. Wu, On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces, Aequationes Math. 83 (2012) 153–189. [2] L. Arambašić, R. Rajić, Operators preserving the strong Birkhoff–James orthogonality on B(H), Linear Algebra Appl. 471 (2015) 394–404. [3] L. Arambašić, R. Rajić, On symmetry of the (strong) Birkhoff–James orthogonality in Hilbert C ∗ -modules, Ann. Funct. Anal. 7 (2016) 17–23. [4] R. Bhatia, P. Šemrl, Orthogonality of matrices and some distance problem, Linear Algebra Appl. 287 (1999) 77–85. [5] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935) 169–172. [6] A. Blanco, A. Turnšek, On maps that preserves orthogonality in normed spaces, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 709–716. [7] B. Blackadar, Operator Algebras. Theory of C ∗ -Algebras and von Neumann Algebras, Springer-Verlag, Berlin, 2006. [8] J. Chmieliński, P. Wójcik, Approximate symmetry of Birkhoff orthogonality, J. Math. Anal. Appl. 461 (2018) 625–640. [9] J.B. Conway, A Course in Functional Analysis, second edition, Springer-Verlag, New York, 1990. [10] P. Ghosh, D. Sain, K. Paul, On symmetry of Birkhoff–James orthogonality of linear operators, Adv. Oper. Theory 2 (2017) 428–434. [11] R.C. James, Inner product in normed linear spaces, Bull. Amer. Math. Soc. 53 (1947) 559–566. [12] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947) 265–292. [13] R.V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2) 56 (1952) 494–503. [14] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. I: Elementary Theory, Grad. Stud. Math., vol. 15, American Mathematical Society, Providence, RI, 1997. Reprint of the 1983 original. [15] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. II: Advanced Theory, Pure Appl. Math., vol. 100, Academic Press, Inc., Orlando, FL, 1986. [16] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras. Vol. IV: Special Topics. Advanced Theory—An Exercise Approach, Birkhäuser Boston, Inc., Boston, MA, 1992. [17] N. Komuro, K.-S. Saito, R. Tanaka, Symmetric points for (strong) Birkhoff orthogonality in von Neumann algebras with applications to preserver problems, J. Math. Anal. Appl. 463 (2018) 1109–1131. [18] N. Komuro, K.-S. Saito, R. Tanaka, Left symmetric points for Birkhoff orthogonality in the preduals of von Neumann algebras, Bull. Aust. Math. Soc. 98 (3) (2018) 494–501. [19] D. Sain, Birkhoff–James orthogonality of linear operators on finite dimensional Banach spaces, J. Math. Anal. Appl. 447 (2017) 860–866. [20] D. Sain, P. Ghosh, K. Paul, On symmetry of Birkhoff–James orthogonality of linear operators on finite-dimensional real Banach spaces, Oper. Matrices 11 (2017) 1087–1095. [21] A. Turnšek, On operators preserving James’ orthogonality, Linear Algebra Appl. 407 (2005) 189–195. [22] A. Turnšek, A remark on orthogonality and symmetry of operators in B(H), Linear Algebra Appl. 535 (2017) 141–150.