Towards a goldie theory for jordan pairs

López, Antonio Fernández; Rus, Eulalia García; Jaa, Omar

doi:10.1007/bf02678016

Cited by 7 publications

(7 citation statements)

References 14 publications

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“…and [FGM2]) all them closely follow to those of the analogous results for nondegenerate Jordan algebras, which can be found in [FGM1].…”

Section: 1supporting

confidence: 85%

“…In this section we recall the definition of the singular set for Jordan pairs, which is analogous to that given in [FGM1] for Jordan algebras. The contents of this section correspond to an unpublished work by the first author together with Fernández López and García Rus [FGM2]. The main result we provide here is that the set of elements of a Jordan pair having essential annihilator form an ideal when the Jordan pair is nondegenerate.…”

Section: The Singular Ideal Of a Jordan Pairmentioning

confidence: 75%

“…Then, in Section 2, we include a series of results on the set of elements of a Jordan pair having essential annihilator. This set was proved to be an ideal of any nondegenerate Jordan pair by Fernández López, García Rus and Montaner [FGM2].…”

Section: Introductionmentioning

confidence: 96%

“…A similar local approach was followed by Fernández López, García Rus, and Jaa in [FGJ], where the authors introduced a notion of order in nondegenerate linear Jordan pairs with descending chain condition on principal inner ideals. Later Fernández López, García Rus, and Montaner provided a first approach to a Goldie theory for quadratic Jordan pairs in their unpublished manuscript [FGM2].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The ideal of the Lesieur-Croisot elements of a Jordan algebra. II

Montaner¹,

Tocón²

2009

Algebras, Representations and Applications

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“…and [FGM2]) all them closely follow to those of the analogous results for nondegenerate Jordan algebras, which can be found in [FGM1].…”

Section: 1supporting

confidence: 85%

Section: The Singular Ideal Of a Jordan Pairmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The ideal of the Lesieur-Croisot elements of a Jordan algebra. II

Montaner¹,

Tocón²

2009

Algebras, Representations and Applications

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“…Thus, D'Amour and McCrimmon extended a substantial part of Zelmanov's results [31] to arbitrary quadratic Jordan systems by making use of local algebras [9,10]; Anquela and Cortés characterized the primitivity of a Jordan system in terms of their local algebras and, as a consequence, gave a full classification of primitive systems [1]; the relationship between generalized polynomial identities and the existence of socle in primitive Jordan systems can actually be seen, after the works of D'Amour and McCrimmon [9], and Montaner [26], as a consequence of the existence of local algebras which are PI; and local algebras of a Jordan system seem to be a crucial notion in order to develop a local Goldie theory for Jordan systems [13]. Local algebras (or their related notion of subquotient) have also proved their usefulness in some questions involving Jordan Banach systems.…”

Section: Introductionmentioning

confidence: 97%

The Jordan algebras of a Lie algebra

2007

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We attach a Jordan algebra L x to any ad-nilpotent element x of index of nilpotence at most 3 in a Lie algebra L. This Jordan algebra has a behavior similar to that of the local algebra of a Jordan system at an element. Thus, L x inherits nice properties from L and keeps relevant information about the element x. McCrimmon [9]) have played a prominent role in the recent structure theory of Jordan systems, mainly due to the fact that niceness properties flow between the system and their local algebras. Local algebras of a Jordan system (introduced by Meyberg [25], used by Zelmanov as a minor part of his brilliant classification of Jordan systems [31], and revisited by D'Amour and

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Spectrum preserving linear maps on JBW*-triples

Neal

2002

Archiv der Mathematik

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In this paper, we will consider a linear mapping between two JBW*-triples with the property that it preserves spectrum with respect to two arbitrary homotopes. We will show that such a map is a jordan homomorphism of those homotopes modulo the radical, which we characterize explicitly. 1. Introduction. The general problem of characterizing spectrum preserving linear maps between Banach algebras and Jordan-Banach algebras goes back to G. Frobenius and has seen much progress recently (see [4]-[8], [9], [10], and [25]). In [20], Frobenius showed that linear spectrum preserving maps φ from M n (C) onto M n (C) are of the forms φ(x) = axa −1 or φ(x) = ax t a −1 for some invertible matrix a. Hence, φ is a Jordan isomorphism. Aupetit, in [8], Theorem 1.3, has recently generalized this result to infinite dimensions, showing that all linear surjective spectrum preserving maps between W*-algebras and JBW*-algebras are Jordan isomorphisms. In the proof, the key point is that the spectrum function is lipschitzian around an idempotent in any semi-primitive Jordan-Banach algebra. It follows that spectrum preserving maps preserve idempotents. Consequently, the result is true whenever a linearly dense collection of idempotents are present.In this paper, we will generalize Aupetit's result to the larger class of JBW*-triples, which includes W*-algebras and JBW*-algebras. JB*-triples are a natural generalization of C*-algebras because, as shown by Friedman and Russo in [17], the range of a contractive projection on a C*-algebra is a JB*-triple. This fact also follows from the work of Kaup, who showed [26] that JB*-triples are are exactly those Banach spaces whose unit ball is a bounded symmetric domain. These objects have been much studied in the last twenty years from the point of view of both complex analysis [2], [26], [30] and C*-algebra theory [3], [11]-[13], [16]-[19].JB*-triples possess a ternary multiplication and thus the the spectrum must be defined with respect to a homotope as in [23]. We will prove that surjective linear maps which preserve such spectrum are Jordan isomorphisms of the homotope structure modulo the radical. More specifically, if φ is a linear surjective map between two JBW*-triples A and B which preserves spectrum with respect to the homotopes A a and B b , for elements a in A and b in B, then the cannonical map φ from A a onto B b / Rad(B b ) is a Jordan homomorphism. Furthermore, the kernel of φ coincides with Rad( A a ) and φ restricts to an isomorphism from A a / Rad( A a ) onto B b / Rad(B b ).

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Towards a goldie theory for jordan pairs

Cited by 7 publications

References 14 publications

The ideal of the Lesieur-Croisot elements of a Jordan algebra. II

The ideal of the Lesieur-Croisot elements of a Jordan algebra. II

The Jordan algebras of a Lie algebra

Spectrum preserving linear maps on JBW*-triples

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