Measures of weak non-compactness in preduals of von Neumann algebras and JBW⁎-triples

Hamhalter, Jan; Kalenda, Ondřej F. K.; Peralta, Antonio M.; Pfitzner, Hermann

doi:10.1016/j.jfa.2019.108300

Cited by 25 publications

(37 citation statements)

References 66 publications

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“…Building upon the relation "being orthogonal" we can define a canonical order ≤ on tripotents in E given by e ≤ u if and only if u − e is a tripotent and u − e ⊥ e. This partial order is precisely the order considered by L. Molnár in Theorem 1.2 and it provides an important tool in JB * -triples (see, for example, the recent papers [37][38][39] where it plays an important role). This relation enjoys several interesting properties, for example, e ≤ u if and only if e is a projection in the JB * -algebra E 2 (u) (cf.…”

Section: (A)])mentioning

confidence: 99%

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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%

“…Since, by (37), every element x in C decomposes uniquely in the form x = λu + P 1 (u)(x) + P 0 (u)(x) and P 0…”

Section: Type 2 Cartan Factors Not Admitting a Unitary Elementmentioning

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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“…For proofs see [36, Lemma 3.9], [24,Proposition 6.7] or [23,Lemma 2.1]. The induced partial order defined by this orthogonality on the set of tripotents is given by e ≤ u if u − e is a tripotent with u − e ⊥ e.…”

Section: 2mentioning

confidence: 99%

“…To prove the equivalence (1)⇔(2) observe that assertion (2) means that 1 = {u, u, 1}, i.e., 1 ∈ B 2 (u). It remains to use [24,Proposition 6.6].…”

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confidence: 99%

On optimality of constants in the Little Grothendieck Theorem

2022

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“…In [14] we studied two weaker preorders on tripotents (denoted by ≤ 2 and ≤ 0 ). The preorder ≤ 2 was used in [15] (without giving the notation) to study the strong * topology and is implicitly mentioned already in [19,Lemma 1.14(1)]. If A is a unital C * -algebra, then e ≤ 1 means that e is a projection in A and e ≤ 2 1 is valid for any partial isometry in A.…”

Section: Introductionmentioning

confidence: 99%

Order type relations on the set of tripotents in a JB$^*$-triple

Hamhalter¹,

Kalenda²,

Peralta³

2021

Preprint

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We introduce, investigate and compare several order type relations on the set of tripotents in a JB * -triple. The main two relations we address are ≤ h and ≤n. We say that u ≤ h e (or u ≤n e) if u is a self-adjoint (or normal) element of the Peirce-2 subspace associated to e considered as a unital JB *algebra with unit e. It turns out that these relations need not be transitive, so we consider their transitive hulls as well. Properties of these transitive hulls appear to be closely connected with types of von Neumann algebras, with the results on products of symmetries, with determinants in finite-dimensional Cartan factors, with finiteness and other structural properties of JBW * -triples.2010 Mathematics Subject Classification. 17C65, 17C27, 17C10, 06A99, 46L10. Key words and phrases. JB * -triple, JBW * -triple, relations on tripotents, von Neumann algebra, partial isometries, products of symmetries.

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Measures of weak non-compactness in preduals of von Neumann algebras and JBW⁎-triples

Cited by 25 publications

References 66 publications

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

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

On optimality of constants in the Little Grothendieck Theorem

Order type relations on the set of tripotents in a JB$^*$-triple

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