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1
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The alternative Dunford-Pettis Property in the predual of a von Neumann algebra

100%
Martín M., Peralta A.
Studia Mathematica
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2001
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tom 147
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nr 2
197-200
EN
Let A be a type II von Neumann algebra with predual A⁎. We prove that A⁎ does not have the alternative Dunford-Pettis property introduced by W. Freedman [7], i.e., there is a sequence (φₙ) converging weakly to φ in A⁎ with ||φₙ|| = ||φ|| = 1 for all n ∈ ℕ and a weakly null sequence (xₙ) in A such that φₙ(xₙ) ↛ 0. This answers a question posed in [7].
2
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The Kadec-Pełczyński-Rosenthal subsequence splitting lemma for JBW*-triple preduals

100%
Peralta A., Pfitzner H.
Studia Mathematica
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2015
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tom 227
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nr 1
77-95
EN
Any bounded sequence in an L¹-space admits a subsequence which can be written as the sum of a sequence of pairwise disjoint elements and a sequence which forms a uniformly integrable or equiintegrable (equivalently, a relatively weakly compact) set. This is known as the Kadec-Pełczyński-Rosenthal subsequence splitting lemma and has been generalized to preduals of von Neuman algebras and of JBW*-algebras. In this note we generalize it to JBW*-triple preduals.
3
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On weak sequential convergence in JB*-triple duals

100%
Bunce L., Peralta A.
Studia Mathematica
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2004
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tom 160
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nr 2
117-127
EN
We study various Banach space properties of the dual space E* of a homogeneous Banach space (alias, a JB*-triple) E. For example, if all primitive M-ideals of E are maximal, we show that E* has the Alternative Dunford-Pettis property (respectively, the Kadec-Klee property) if and only if all biholomorphic automorphisms of the open unit ball of E are sequentially weakly continuous (respectively, weakly continuous). Those E for which E* has the weak* Kadec-Klee property are characterised by a compactness condition on E. Whenever it exists, the predual of E is shown to have the Kadec-Klee property if and only if E is atomic with no infinite spin part.
4
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Factorizing multilinear operators on Banach spaces, C*-algebras and JB*-triples

81%
Palazuelos C., Peralta A., Villanueva I.
Studia Mathematica
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2009
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tom 192
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nr 2
129-146
EN
In recent papers, the Right and the Strong* topologies have been introduced and studied on general Banach spaces. We characterize different types of continuity for multilinear operators (joint, uniform, etc.) with respect to the above topologies. We also study the relations between them. Finally, in the last section, we relate the joint Strong*-to-norm continuity of a multilinear operator T defined on C*-algebras (respectively, JB*-triples) to C*-summability (respectively, JB*-triple-summability).
5
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Automatic continuity of biorthogonality preservers between weakly compact JB*-triples and atomic JBW*-triples

81%
Burgos M., Garcés J., Peralta A.
Studia Mathematica
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2011
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tom 204
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nr 2
97-121
EN
We prove that every biorthogonality preserving linear surjection from a weakly compact JB*-triple containing no infinite-dimensional rank-one summands onto another JB*-triple is automatically continuous. We also show that every biorthogonality preserving linear surjection between atomic JBW*-triples containing no infinite-dimensional rank-one summands is automatically continuous. Consequently, two atomic JBW*-triples containing no rank-one summands are isomorphic if and only if there exists a (not necessarily continuous) biorthogonality preserving linear surjection between them.
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