Commutativity of<i>C</i>*-algebras and associativity of<i>JB</i>*-algebras

Iochum, Bruno; Loupias, G.; Rodríguez-Palacios, A.

doi:10.1017/s0305004100078117

Cited by 8 publications

(7 citation statements)

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“…If Condition (1) is fulfilled, then X , endowed with the product x • y := {xuy} and the involution x * := {uxu}, becomes a unital JB * -algebra whose unit is precisely u (see [15]), and hence (2) holds by [56,Theorem 26] (see also [36,Theorem 4]). On the other hand, the implication (2) In relation to the above proof, it is worth mentioning that, contrarily to what happens in the case of C * -algebras, the group of all surjective linear isometries on a JB * -triple X need not act transitively on the set of all unitary elements of X [15,Example 5.7].…”

Section: Unitaries In Jb * -Triplesmentioning

confidence: 99%

Banach space characterizations of unitaries: A survey

Palacios

2010

Journal of Mathematical Analysis and Applications

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Section: Unitaries In Jb * -Triplesmentioning

confidence: 99%

Banach space characterizations of unitaries: A survey

Palacios

2010

Journal of Mathematical Analysis and Applications

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“…Rodríguez-Palacios proved in [69,Theorem 26] that if A is a (unital) non-commutative Jordan V -algebra, we have n(A) = 1 if A is associative and commutative, and n(A) = 1 2 otherwise. The same author in collaboration with Iochum and Loupias showed that the same conclusion holds when A is a (non-necessarily commutative) JB * -algebra [51]. The reader can take a look at [34], and [19,Proposition 3.5.44 and comments in §2.1.47 and page 422] for a more recent approach.…”

Section: Introductionmentioning

confidence: 89%

“…The numerical index is well determined in the case of C * -algebras, JB * -algebras, and non-necessarily commutative JB * -algebras. More concretely, a C * -algebra A satisfies that n(A) = 1 if A is commutative and n(A) = 1 2 otherwise [50], and the same conclusion holds when A is replaced by a (unital) non-commutative Jordan V -algebra [69,Theorem 26] or by a non-necessarily commutative JB * -algebra [34,51] (see also [19,Proposition 3.5.44 and comments in §2.1.47 and page 422]). The results in this section show that every commutative JB * -triple has numerical index one (see Lemma 3.1), while for a JBW * -triple M we show that n(M ) = 1 if M is commutative and 1 e ≤ n(M ) ≤ 1 2 otherwise (cf.…”

Section: The Numerical Index Of a Jb * -Triplementioning

confidence: 99%

“…Motivated by this characterization, it was conjectured that a JB * -algebra is associative if and only if it contains no non-trivial nilpotent elements. The conjecture was finally proved to be true by Iochum, Loupias and Rodríguez-Palacios in [51]. It is not clear how nilpotency can be applied in the setting of JB * -triples (specially because by the extended Gelfand-Naimark axiom {a, a, a} = a 3 for every element a in a JB * -triple E).…”

Section: Introductionmentioning

confidence: 99%

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Estimations of the numerical index of a JB$^*$-triple

Cabezas¹,

Peralta²

2023

Preprint

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We prove that every commutative JB * -triple has numerical index one. We also revisit the notion of commutativity in JB * -triples to show that a JBW * -triple M has numerical index one precisely when it is commutative, while e −1 ≤ n(M ) ≤ 2 −1 otherwise. Consequently, a JB * -triple E is commutative if and only if n(E * ) = 1 (equivalently, n(E * * ) = 1). In the general setting we prove that the numerical index of each JB * -triple E admitting a non-commutative element also satisfies e −1 ≤ n(M ) ≤ 2 −1 , and the same holds when the bidual of E contains a Cartan factor of rank ≥ 2 in its atomic part.

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“…Let A be an alternative C * -algebra with a unit 1. An element u in A is said to be unitary if the equalities Proposition 2.1 [22,Theorem 26] (see also [16,Theorem 4]). Let A be a nonzero, noncommutative JB * -algebra with a unit 1.…”

Section: Corollary 13 Let a Be A Nonzero C * -Algebra Then The Duality Mapping Of π(A) Is Norm-to-norm Upper Semi-continuous At P Amentioning

confidence: 99%

Geometrical Properties of the Product of a $C^*$-Algebra

Amin¹,

Campoy²,

Palacios³

2001

Rocky Mountain J. Math.

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The study of the geometry of norm-unital complex Banach algebras at their units [5], [6] takes its first impetus from the celebrated Bohnenblust-Karlin theorem [3] asserting that the unit of such an algebra A is a vertex of the closed unit ball of A. As observed in [5, pp. 33 34], the Bohnenblust-Karlin paper contains a stronger result, namely that, for such an algebra A, the inequality n(A, 1) ≥ (1/e) holds. Here 1 denotes the unit of A, and n(A, 1) is a suitably defined nonnegative real number which depends only on the Banach space of A and the norm-one distinguished element 1. As the main result, we prove in this paper that the product of every nonzero C * -algebra A is a vertex of the closed unit ball of the Banach space Π(A) of all continuous bilinear mappings from A × A into A. As in the above mentioned case, the vertex property follows from stronger "numerical" conditions. Indeed, if A is a nonzero C * -algebra, and if p A denotes the product of A, then n(Π(A), p A ) is equal to 1 or 1/2 depending on whether or not A is commutative (Theorem 1.1). We note that our main result improves the recent one in [24, Corollary 2.7] asserting that the product of every nonzero C * -algebra A is an extreme point of the closed unit ball of Π(A).In Section 2 we show that the main result remains true for the socalled alternative C * -algebras (Theorem 2.5). Alternative C * -algebras are defined by means of the Gelfand-Naimark abstract system of axioms but relaxing the familiar requirement of associativity to that of alternativity. Alternative C * -algebras arise in a natural way in functional analysis. Indeed, Gelfand-Naimark axioms on a general nonassociative unital algebra imply the alternativity [22, Theorem 14] (see also [9]) and the existence of alternative C * -algebras failing to be associative is well known (see [17, Example 13] and [8, Theorem 3.7]). Alternative C * -algebras are studied in detail in [20] and [8] and have shown

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Commutativity ofC-algebras and associativity ofJB-algebras

Abstract: A n.c.JB-algebra is associative and commutative if and only if it has no non-zero nilpotent elements. This generalizes the analogous theorem of Kaplansky forC-algebras. Different characterizations of associativity are given.

Cited by 8 publications

References 30 publications

Banach space characterizations of unitaries: A survey

Banach space characterizations of unitaries: A survey

Estimations of the numerical index of a JB$^*$-triple

Geometrical Properties of the Product of a $C^*$-Algebra

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Commutativity ofC*-algebras and associativity ofJB*-algebras

Abstract: A n.c.JB*-algebra is associative and commutative if and only if it has no non-zero nilpotent elements. This generalizes the analogous theorem of Kaplansky forC*-algebras. Different characterizations of associativity are given.

Cited by 8 publications

References 30 publications

Banach space characterizations of unitaries: A survey

Banach space characterizations of unitaries: A survey

Estimations of the numerical index of a JB$^*$-triple

Geometrical Properties of the Product of a $C^*$-Algebra

Contact Info

Product

Resources

About

Commutativity ofC-algebras and associativity ofJB-algebras

Abstract: A n.c.JB-algebra is associative and commutative if and only if it has no non-zero nilpotent elements. This generalizes the analogous theorem of Kaplansky forC-algebras. Different characterizations of associativity are given.