JB*-triples have Pełczynski’s property V

Abstract. The strong* topology s * (X) of a Banach space X is defined as the locally convex topology generated by the seminorms x → Sx for bounded linear maps S from X into Hilbert spaces. The w-right topology for X, ρ(X), is a stronger locally convex topology, which may be analogously characterised by taking reflexive Banach spaces in place of Hilbert spaces. For any Banach space Y , a linear map T : X → Y is known to be weakly compact precisely when T is continuous from the w-right topology to the norm topology of Y . The main results deal with conditions for, and consequences of, the coincidence of these two topologies on norm bounded sets. A large class of Banach spaces, including all C * -algebras, and more generally, all JB * -triples, exhibit this behaviour.

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“…Since every JB*-triple also satisfies property (V ) (see [6]), the next corollary now follows as a consequence of Theorem 5.7.…”

Section: Unconditionally Converging and Quasi Completely Continuous Omentioning

confidence: 80%

Weakly compact operators and the strong* topology for a Banach space

Peralta¹,

Villanueva²,

Wright³

et al. 2010

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…a) ⇒ b). Since JB * -triples have Pelczynski's Property (V) (compare [6]) there exist θ > 0, (ϕ n ) ⊂ K and a w.u.c. series n≥1 z n in E with z n ≤ 1, such that |ϕ n (z n )| > θ, ∀n ∈ N. We may assume that K is contained in the closed unit ball of E * .…”

Section: B ) There Exists a Subtriple C Of E Isometric To An Abelian mentioning

confidence: 99%

“…Reference [6] is a basic forerunner of the problem studied in this paper. Briefly speaking, we could say [6] contains a partial answer for our problem in terms of Pelczynski's Property (V).…”

Section: Introductionmentioning

confidence: 99%

“…Briefly speaking, we could say [6] contains a partial answer for our problem in terms of Pelczynski's Property (V). We recall that a series n≥1 z n in a Banach space X is called weakly unconditionally convergent (w.u.c.…”

Section: Introductionmentioning

confidence: 99%

“…series n x n in X such that sup ϕ∈X * |ϕ(x n )| does not converge to zero. It is established in [6] that every JB*-triple satisfies property (V ). We shall see later that every bounded sequence of mutually orthogonal elements in a JB*-triple defines a w.u.c.…”

Section: Introductionmentioning

confidence: 99%

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Weak compactness in the dual space of a JB*-triple is commutatively determined

Polo¹,

Peralta²

2009

MATH. SCAND.

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We prove the following criterium of weak compactness in the dual of a JB*-triple: a bounded set K in the dual of a JB*-triple E is not relatively weakly compact if and only if there exist a sequence of pairwise orthogonal elements (a n ) in the closed unit ball of E, a sequence (ϕ n ) in K, and ϑ > 0 satisfying that |ϕ n (a n )| > ϑ for all n ∈ N. This solves a question stimulated by the main result in [11] and posed in [9].

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Pełczyński's property () of order p and its quantification

Chen

Chávez-Domínguez

2017

Mathematische Nachrichten

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We introduce the concepts of Pełczyński's property ( ) of order and Pełczyński's property ( * ) of order . It is proved that, for each 1 < < ∞, the James -spaces enjoys Pełczyński's property ( * ) of order and the James * -spaces * (where * denotes the conjugate number of ) enjoys Pełczyński's property ( ) of order . We prove that both 1 ( ) ( a finite positive measure) and 1 enjoy a quantitative version of Pełczyński's property ( * ). K E Y W O R D SPełczyński's property ( ) of order , Pełczyński's property ( * ) of order , quantitative Pełczyński's property ( * ) of order M S C ( 2 0 1 0 ) 46-B

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JB*-triples have Pełczynski’s property V

Cited by 20 publications

References 14 publications

Weakly compact operators and the strong* topology for a Banach space

Weakly compact operators and the strong* topology for a Banach space

Weak compactness in the dual space of a JB*-triple is commutatively determined

Pełczyński's property () of order p and its quantification

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