Orthogonal forms and orthogonality preservers on real function algebras

Garcés, Jorge J.; Peralta, Antonio M.

doi:10.1080/03081087.2013.772998

Cited by 7 publications

(16 citation statements)

References 43 publications

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“…A bilinear form V : A × A → C is said to be orthogonal if V (a, b) = 0 whenever a ⊥ b. In [1] we establish a generalization of a celebrated result due to S. Goldstein (see [2,Theorem 1.10]) by proving the following result: Theorem 1.1. [1, Theorem 2.4] Let V : A × A → R be a continuous orthogonal form on a commutative real C * -algebra, then there exist ϕ 1 and ϕ 2 in A * satisfying V (x, y) = ϕ 1 (xy) + ϕ 2 (xy * ), for every x, y ∈ A.…”

Section: Introductionmentioning

confidence: 99%

“…A bilinear form V : A × A → C is said to be orthogonal if V (a, b) = 0 whenever a ⊥ b. In [1] we establish a generalization of a celebrated result due to S. Goldstein (see [2,Theorem 1.10]) by proving the following result: We recently realized the presence of a "gap" affecting some of the technical results given in [1]. The concrete difficulties appear in the following arguments: By the Gelfand theory for commutative real C * -algebras, every commutative unital real C * -algebra A is C * -isomorphic (and hence isometric) to a real function algebra of the form C(K) τ = {f ∈ C(K) : τ (f ) = f }, where K is a compact Hausdorff space, τ is a conjugation (i.e.…”

Section: Introductionmentioning

confidence: 99%

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Orthogonal forms and orthogonality preservers on real function algebras revisited

Peralta

2016

Linear and Multilinear Algebra

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In [1] we determine the precise form of a continuous orthogonal form on a commutative real C * -algebra. We also describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between commutative unital real C * -algebras. Among the consequences, we show that every orthogonality preserving linear bijection between commutative unital real C * -algebras is continuous. In this note we revisit these results and their proofs with the idea of filling a gap in the arguments, and to extend the original conclusions.

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“…A bilinear form V : A × A → C is said to be orthogonal if V (a, b) = 0 whenever a ⊥ b. In [1] we establish a generalization of a celebrated result due to S. Goldstein (see [2,Theorem 1.10]) by proving the following result: Theorem 1.1. [1, Theorem 2.4] Let V : A × A → R be a continuous orthogonal form on a commutative real C * -algebra, then there exist ϕ 1 and ϕ 2 in A * satisfying V (x, y) = ϕ 1 (xy) + ϕ 2 (xy * ), for every x, y ∈ A.…”

Section: Introductionmentioning

confidence: 99%

“…A bilinear form V : A × A → C is said to be orthogonal if V (a, b) = 0 whenever a ⊥ b. In [1] we establish a generalization of a celebrated result due to S. Goldstein (see [2,Theorem 1.10]) by proving the following result: We recently realized the presence of a "gap" affecting some of the technical results given in [1]. The concrete difficulties appear in the following arguments: By the Gelfand theory for commutative real C * -algebras, every commutative unital real C * -algebra A is C * -isomorphic (and hence isometric) to a real function algebra of the form C(K) τ = {f ∈ C(K) : τ (f ) = f }, where K is a compact Hausdorff space, τ is a conjugation (i.e.…”

Section: Introductionmentioning

confidence: 99%

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Orthogonal forms and orthogonality preservers on real function algebras revisited

Peralta

2016

Linear and Multilinear Algebra

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“…a ⊥ b ⇔ T (a) ⊥ T (b)) is automatically continuous (see [6,Theorem 19]). Additional results on automatic continuity of linear biorthogonality preservers on some other structures can be seen in [21,22,10,24] and [18]. For additional details on orthogonality preservers the reader is referred to the recent survey [19].…”

Section: Introductionmentioning

confidence: 99%

Contractive linear preservers of absolutely compatible pairs between C*-algebras

Jana¹,

Karn²,

Peralta³

2018

Preprint

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Let a and b be elements in the closed ball of a unital C *algebra A (if A is not unital we consider its natural unitization). We shall say that a and b are domain (respectively, range) absolutely compatibleWe shall say that a and b are absolutely compatible (a△b in short) if they are both range and domain absolutely compatible. In general, a△ d b (respectively, a△rb and a△b) is strictly weaker than ab * = 0 (respectively, a * b = 0 and a ⊥ b). Let T : A → B be a contractive linear mapping between C * -algebras. We prove that if T preserves domain absolutely compatible elements (i.e., a△ d b ⇒ T (a)△ d T (b)) then T is a triple homomorphism. A similar statement is proved when T preserves range absolutely compatible elements. It is finally shown that T is a triple homomorphism if, and only if, T preserves absolutely compatible elements.

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“…The applications of Goldstein's theorem appear in many different contexts ( [5,6,18,21]). Quite recently, an extension of Goldstein's theorem for commutative real C * -algebras has been published in [17]. 13).…”

Section: Introductionmentioning

confidence: 99%

Jordan weak amenability and orthogonal forms on JB$^*$-algebras

Jamjoom¹,

Peralta²,

Siddiqui³

2015

Banach J. Math. Anal.

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We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB *algebra J and the Banach space of all purely Jordan generalized derivations from J into J * . We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all antisymmetric orthogonal forms on J , and of all Lie Jordan derivations from J into J * .Making use of the weak amenability of every C * -algebra, U. Haagerup and N.J. Laustsen gave a simplified proof of Goldstein's theorem in [21]. In the third section of the just quoted paper, and more concretely, in the proof of [21, Proposition 3.5], the above mentioned authors pointed out that for every anti-symmetric form V on a C * -algebra A which is orthogonal on A sa , the mappingReciprocally, the weak amenability of A also implies that every derivation δ from A into A * is inner and hence of the form δ(a) = adj φ (a) = φa − aφ for a functional φ ∈ A * . In particular, the form V δ (a, b) = δ(a)(b) is antisymmetric and orthogonal.The above results are the starting point and motivation of the present note. In the setting of C * -algebras we shall complete the above picture showing that symmetric orthogonal forms on a C * -algebra A are in bijective correspondence with the purely Jordan generalized derivations from A into A * (see Section 2 for definitions). However, the main goal of this note is to explore the orthogonal forms on a JB * -algebra and the similarities and differences between the associative setting of C * -algebras and the wider class of JB * -algebras.In Section 2 we revisit the basic theory and results on Jordan modules and derivations from the associative derivations on C * -algebras to Jordan derivations on C * -algebras and JB * -algebras. The novelties presented in this section include a new study about generalized Jordan derivations from a JB * -algebra J into a Jordan Banach J -module in the line explored in [31], [1, §4], and [8, §3]. We recall that, given a Jordan Banach J -module X over a JB * -algebra, a generalized Jordan derivation from J into X is a linear mapping G : J → X for which there exists ξ ∈ X * * satisfyingfor every a, b in J . We show how the results on automatic continuity of Jordan derivations from a JB * -algebra J into itself or into its dual, established by S. Hejazian, A. Niknam [23] and B. Russo and the second author of this paper in [33], can be applied to prove that every generalized Jordan derivation from J into J or into J * is continuous (see Proposition 2.1).Section 3 contains the main results of the paper. In Proposition 3.8 we prove that for every generalized Jordan derivation G : J → J * , where J is a JB * -algebra, the form V G : J × J → C, V G (a, b) = G(a)(b) is orthogonal on the whole J . We introduce the two new subclasses of purely Jordan generalized derivations and Lie Jordan derivations. A generalized derivation G : J → J * is said to be a purely Jordan generalized derivation if G(a)(b) = G(b)(a), for every a, b ∈ J ; while a Li...

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Orthogonal forms and orthogonality preservers on real function algebras

Cited by 7 publications

References 43 publications

Orthogonal forms and orthogonality preservers on real function algebras revisited

Orthogonal forms and orthogonality preservers on real function algebras revisited

Contractive linear preservers of absolutely compatible pairs between C*-algebras

Jordan weak amenability and orthogonal forms on JB$^*$-algebras

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