Relatively weakly open sets in closed balls of Banach spaces, and real JB*-triples of finite rank

Guerrero, Julio Becerra; López-Pérez, Ginés; Peralta, Antonio M.; Rodríguez-Palacios, A.

doi:10.1007/s00208-004-0537-y

Cited by 33 publications

(2 citation statements)

References 28 publications

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“…A subset S of E is called orthogonal if 0 / ∈ S and x ⊥ y for every x = y in S. The minimal cardinal number r satisfying card(S) ≤ r for every orthogonal subset S ⊆ E is called the rank of E (cf. [28], [40] and [41] for basic results on the rank of a Cartan factor and a JBW * -triple and its relation with reflexivity). It is known that for each tripotent e in a Cartan factor C we have r (e) = r (C 2 (e)) (see, for example, [28, page 200]).…”

Section: (A)])mentioning

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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Section: (A)])mentioning

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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“…It follows that Γ is finite and therefore F a+I is finite dimensional. Since a+ I was arbitrary chosen, the statement of the lemma follows from [4,Theorem 3.8].…”

Section: Definition 5 a Jordan Banach Triple E Has Cohen's Factorisat...mentioning

confidence: 99%

Generalized Triple Homomorphisms and Derivations

Garcés¹,

Peralta²

2013

Can. j. math.

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Abstract. We introduce generalised triple homomorphism between Jordan Banach triple systems as a concept which extends the notion of generalised homomorphism between Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively. We prove that every generalised triple homomorphism between JB * -triples is automatically continuous. When particularised to C * -algebras, we rediscover one of the main theorems established by B.E. Johnson. We shall also consider generalised triple derivations from a Jordan Banach triple E into a Jordan Banach triple E-module, proving that every generalised triple derivation from a JB * -triple E into itself or into E * is automatically continuous.

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