A Kaplansky theorem for JB*-triples

Fernández–Polo, Francisco J.; Garcés, Jorge J.; Peralta, Antonio M.

doi:10.1090/s0002-9939-2012-11157-8

Cited by 6 publications

(9 citation statements)

References 26 publications

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“…The proof of the following theorem combines the argument given by Mathieu [30] with a recent Kaplansky theorem for real and complex JB * -triples obtained in [16]. Given a real or complex JB * -triple E, a necessary and sufficient requirement for E to be reflexive is that E has the Radon-Nikodym property, or equivalently, E is isomorphic to a Hilbert space or E has finite rank ( When specialized to the setting of C * -algebras, the above result reads as follows:…”

Section: Weakly Compact Triple Homomorphismsmentioning

confidence: 95%

Weakly compact orthogonality preservers on C*-algebras

2010

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We establish a complete description of weakly compact orthogonality preserving operators from a C * -algebra A (or a JB * -algebra) to a JB * -triple E. We prove that any such operator is the sum of a series of mutually orthogonal triple homomorphisms from A * * to a certain Peirce-2 subspace of E * * multiplied by a mutually orthogonal norm-null sequence in E which enjoy certain operator commutativity relations.

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Section: Weakly Compact Triple Homomorphismsmentioning

confidence: 95%

Weakly compact orthogonality preservers on C*-algebras

2010

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“…A triple version of the celebrated Kaplansky-Cleveland theorem, established in [62], asserts that a non-necessarily continuous triple homomorphism from a JB * -triple to a normed Jordan triple has closed range whenever it is continuous [62, Corollary 18]. As a consequence of this result, the triple homomorphism has closed range.…”

Section: Proposition 310 Letmentioning

confidence: 99%

Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors

Friedman

Peralta

2022

Anal.Math.Phys.

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There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states $${\mathbf {P}}$$ P —i.e., the set of one-dimensional projections on a complex Hilbert space H– and the orthomodular lattice $${\mathbf {L}}$$ L of closed subspaces of H. These six groups are isomorphic when the dimension of H is $$\ge 3$$ ≥ 3 . The latter hypothesis is absolutely necessary in this identification. For example, the automorphisms group of all bijections preserving orthogonality and the order on $${\mathbf {L}}$$ L identifies with the bijections on $${\mathbf {P}}$$ P preserving transition probabilities only if dim$$(H)\ge 3$$ ( H ) ≥ 3 . Despite of the difficulties caused by $$M_2({\mathbb {C}})$$ M 2 ( C ) , rank two algebras are used for quantum mechanics description of the spin state of spin-$$\frac{1}{2}$$ 1 2 particles. However, there is a counterexample for Uhlhorn’s version of Wigner’s theorem for such state space. In this note we prove that in order that the description of the spin will be relativistic, it is not enough to preserve the projection lattice equipped with its natural partial order and orthogonality, but we also need to preserve the partial order set of all tripotents and orthogonality among them (a set which strictly enlarges the lattice of projections). Concretely, let M and N be two atomic JBW$$^*$$ ∗ -triples not containing rank–one Cartan factors, and let $${\mathcal {U}} (M)$$ U ( M ) and $${\mathcal {U}} (N)$$ U ( N ) denote the set of all tripotents in M and N, respectively. We show that each bijection $$\Phi : {\mathcal {U}} (M)\rightarrow {\mathcal {U}} (N)$$ Φ : U ( M ) → U ( N ) , preserving the partial ordering in both directions, orthogonality in one direction and satisfying some mild continuity hypothesis can be extended to a real linear triple automorphism. This, in particular, extends a result of Molnár to the wider setting of atomic JBW$$^*$$ ∗ -triples not containing rank–one Cartan factors, and provides new models to present quantum behavior.

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“…This tool has been applied by many authors in the study of automatic continuity of binary and ternary homomorphims, derivations and module homomorphisms (see, for example, [41,2,53,32,33,47,48,14] and [15], among others). These spaces also play an important role in the subsequent generalisations of Kaplansky's theorem (compare [12,26] and [18]).…”

Section: Triple Derivations and Triple Module Homomorphismsmentioning

confidence: 99%

“…Since A is abelian, L(a, b) = Q(a, b) in A sa , it follows from (17), that δQ(a, b) and Q(a, b)δ are continuous operators from A sa to X for every a ∈ Ann(σ X (δ)) and b ∈ A sa . This implies that (18) {a, x, b} = 0, for every a ∈ Ann(σ X (δ)), b ∈ A sa and x ∈ σ X (δ).…”

Section: Derivations On a C * -Algebramentioning

confidence: 99%

Automatic continuity of derivations onC⁎-algebras andJB⁎-triples

Peralta

Russo

2014

Journal of Algebra

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We introduce the notion of a Jordan triple module and determine the precise conditions under which every derivation from a JB * -triple E into a Banach (Jordan) triple E-module is continuous. In particular, every derivation from a real or complex JB * -triple into its dual space is automatically continuous. Among the consequences, we prove that every triple derivation from a C * -algebra A to a Banach triple A-module is continuous. In particular, every Jordan derivation from A to a Banach A-bimodule is a derivation, a result which complements a classical theorem due to B.E. Johnson and solves a problem which has remained open for over ten years.

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A Kaplansky theorem for JB*-triples

Cited by 6 publications

References 26 publications

Weakly compact orthogonality preservers on C*-algebras

Weakly compact orthogonality preservers on C*-algebras

Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors

Automatic continuity of derivations onC⁎-algebras andJB⁎-triples

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