Structure des algèbres de Jordan-Banach non commutatives complexes régulières ou semi-simples à spectre fini

Benslimane, M.; Kaidi, El Amin

doi:10.1016/0021-8693(88)90189-5

Cited by 9 publications

(7 citation statements)

References 9 publications

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“…Ž. algebra by the Cayley᎐Dickson process . Then the set H C of all 3 = 3 3 t matrices in C which are hermitian under the involution X * s X is a simple 27-dimensional exceptional Jordan algebra. Let J be a Jordan algebra.…”

Section: Algebraic Preliminaries On Jordan Algebrasmentioning

confidence: 99%

Noetherian Jordan Banach Algebras Are Finite-Dimensional

Benslimane

Boudi

1999

Journal of Algebra

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Section: Algebraic Preliminaries On Jordan Algebrasmentioning

confidence: 99%

Noetherian Jordan Banach Algebras Are Finite-Dimensional

Benslimane

Boudi

1999

Journal of Algebra

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“…Since F is an algebraically closed field, the triple product (abc) = h a cba is isomorphic to cba. Consider the mapping a -> h fi a with p 2 = or 1 . Finally, the mapping (a, b)…”

Section: Moreover B Is Strongly Simple If and Only If B Is As In (I)mentioning

confidence: 99%

“…By Zelmanov's theorem for simple JTS, M is one of the following: (1) a hermitian JTS y(B, *), where B is a ""-simple ATS (say of the second kind); (2) a quadratic-form triple J(Q, rf) or the polarized JTS J(Q) © J(Q); or (3) a simple exceptional JTS which is finite-dimensional over its centroid. In case (1) we have that either J = N + or J = Sf(N, *), where TV is a simple ATS of the second kind with minimal inner ideals (5.2i). Now we can prove, either directly by using (4.1) or reducing the problem to Loos's embedding C of N [20] (where C is an associative algebra coinciding with its socle [4, (1.9)] and hence is regular by [5,Theorem 1]) that N, and therefore J too, is regular.…”

Section: Von Neumann Regularity In Jordan Banach Triple Systemsmentioning

confidence: 99%

“…Kaplansky proved in [12] that von Neumann regular associative Banach algebras are finite-dimensional. Recently, Benslimane and Kaidi showed in [1] that every von Neumann regular non-commutative Jordan Banach algebra is a direct sum of a finite number of closed simple ideals each of which is either finite-dimensional or an infinite-dimensional flexible quadratic complex algebra. Von Neumann regularity in Jordan triple systems was first considered by Meyberg [20,21], who proved that nondegenerate Jordan triple systems satisfying descending chain conditions on principal inner ideals are von Neumann regular.…”

Section: Introductionmentioning

confidence: 99%

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Von Neumann Regular Jordan Banach Triple Systems

López

Rus

Campos

1990

Journal of the London Mathematical Society

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“…Von Neumann regularity as such is not a finiteness condition but becomes so in the Banach case, as shown by the following result, due to Kaplansky [15] for Banach algebras, to Benslimane and Kaidi [4] for not necessarily commutative Banach Jordan algebras and to Fernandez et al [8] for Banach Jordan triple systems.…”

Section: Regular Jordan Pairsmentioning

confidence: 94%

Recent results on finiteness conditions in Jordan pairs

Loos¹

Jordan Algebras

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This article collects a number of recent results related to various types of finiteness conditions in Jordan theory. With a few exceptions, the conditions are at least as strong as the descending chain condition on principal inner ideals; in particular, they imply regularity in the nondegenerate case. Thus finiteness conditions of Noetherian type will not be considered, nor conditions like finite generation, algebraicity, or polynomial identities.1991 Mathematics Subject Classification: 17C10, 17C27, 17C30, 17C55. Finite dimensionThis is the condition that the underlying module of a Jordan system J (algebra, triple system or pair) over a commutative ring of scalars k be a finite-dimensional vector space; in particular, that k be a field. From a purely algebraic point of view the condition is not very satisfactory because it is not tied to the Jordan structure. However, finite-dimensionality allows methods from algebraic geometry and algebraic group theory to be used; see, e.g., the books by Braun-Koecher [5] and Springer [41] and papers too numerous to mention, starting with the original paper by Jordan, von Neumann and Wigner.A slight weakening of finite-dimensionality but in the same spirit is the requirement that, for k an arbitrary commutative ring, J be finitely generated and projective as a Axmodule. Jordan systems of this type allow again the use of algebraicgeometric methods [20,19], but classical algebraic geometry has to be supplanted by scheme theory. There are the beginnings of a theory of separable Jordan systems over commutative rings [18], in analogy to the theory of Azumaya algebras, but a full theory remains to be worked out.The logical continuation of this train of thought is to globalize the theory, i.e. to replace the commutative base ring k, or rather its prime spectrum, by an arbitrary base scheme X and let J be a locally free Οχ-module of finite rank. H.P. Petersson [40] has recently started to develop a theory of composition algebras in this setup.

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Structure des algèbres de Jordan-Banach non commutatives complexes régulières ou semi-simples à spectre fini

Cited by 9 publications

References 9 publications

Noetherian Jordan Banach Algebras Are Finite-Dimensional

Noetherian Jordan Banach Algebras Are Finite-Dimensional

Von Neumann Regular Jordan Banach Triple Systems

Recent results on finiteness conditions in Jordan pairs

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