Intersection properties of balls in spaces of compact operators

Lima, Åsvald

doi:10.5802/aif.700

Cited by 72 publications

(45 citation statements)

References 20 publications

(21 reference statements)

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“…Condition (ii) of the theorem is called E.P.I.P. in [10] and appeared in [11, §4]. In [10, Corollary 3.6],Å. Lima proved (ii) ⇒ (iii) for arbitrary real Banach spaces.…”

Section: Proof the Implication (Ii) ⇒ (I) Is Straightforward Just Umentioning

confidence: 94%

“…This notion was introduced byÅ. Lima [10], generalizing the concept of CLspace given by R. Fullerton [8] in 1960. Real and complex almost-CL-spaces have numerical index 1 (see [13, §4] or [1]), but it is not known whether the reciprocal result is true.…”

Section: N(x) = Max{k ≥ 0 : K T ≤ V(t ) ∀ T ∈ L(x)}mentioning

confidence: 99%

“…FollowingÅ. Lima [9,10], a closed subspace Y of a Banach space X is said to be a semi L-summand if for every x ∈ X there exists a unique y ∈ Y such that x − y = d(x, Y ), and moreover this y satisfies x = y + x − y . A Banach space X is said to be semi-nicely embedded in C b (Ω) if there exists a linear isometry J : X → C b (Ω) such that for all s ∈ Ω the following properties are satisfied:…”

Section: Corollary 2 Let ω Be a Hausdorff Topological Space And Let mentioning

confidence: 99%

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Banach spaces having the Radon-Nikodym property and numerical index 1

Martı́n

2003

Proc. Amer. Math. Soc.

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“…Condition (ii) of the theorem is called E.P.I.P. in [10] and appeared in [11, §4]. In [10, Corollary 3.6],Å. Lima proved (ii) ⇒ (iii) for arbitrary real Banach spaces.…”

Section: Proof the Implication (Ii) ⇒ (I) Is Straightforward Just Umentioning

confidence: 94%

Section: N(x) = Max{k ≥ 0 : K T ≤ V(t ) ∀ T ∈ L(x)}mentioning

confidence: 99%

Section: Corollary 2 Let ω Be a Hausdorff Topological Space And Let mentioning

confidence: 99%

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Banach spaces having the Radon-Nikodym property and numerical index 1

Martı́n

2003

Proc. Amer. Math. Soc.

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“…for every k, Y is a finitedimensional lush space and, therefore, n(Y ) = 1, implying by [11,Corollary 3.7] that Y is a CL-space. This means, by definition, that B Y is the absolutely convex hull of every maximal convex subset (maximal face) of S Y .…”

Section: Theorem 52mentioning

confidence: 99%

Properties of lush spaces and applications to Banach spaces with numerical index 1

et al. 2009

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Abstract. The concept of lushness was introduced recently as a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c 0 can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably determined, is stable under ultraproducts, and we characterize those spaces of the form X = (R n , · ) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.

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“…(1) Using the identification (X/M)* = M° and Proposition 1.5(2) of [14] one can also give a more direct proof of Theorem 3.4(b) using the intersection property of M-ideals in the form of [11].…”

Section: Propositionmentioning

confidence: 99%

Banach spaces which are 𝑀-ideals in their biduals

Harmand¹,

Lima²

1984

Trans. Amer. Math. Soc.

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Abstract. We investigate Banach spaces X such that X is an A/-ideal in X**. Subspaces, quotients and c0-sums of spaces which are M-ideals in their biduals are again of this type. A nonreflexive space X which is an M-ideal in X** contains a copy of c0. Recently Lima has shown that if K(X) is an A/-ideal in L( X) then X is an A/-ideal in X**. Here we show that if X is reflexive and K( X) is an A/-ideal in L(X), then K(X)** is isometric to L(X), i.e. K(X) is an M-ideal in its bidual.Moreover, for real such spaces, we show that K( X) contains a proper M-ideal if and only if X or X* contains a proper M-ideal.1. Introduction. The object of this paper is to investigate Banach spaces X, which are M-ideals in their bidual X**. Previous investigations of the question, whether the space of compact operators on X is an M-ideal in the space of all bounded operators, and the general study of the nature of the embedding of X into X** have led us to consider these spaces.In §2 we show some general results concerning Af-structure, which are perhaps folklore, but which do not appear in the literature.In §3 we start to study the properties of spaces which are M-ideals in their biduals. We show that subspaces and quotients of spaces, which are M-ideals in their biduals, have the same property. The proof of this result is based upon the fact that X is an A/-ideal in A'** if and only if the natural projection from X*** onto X* is an L-projection. Using local reflexivity we show that if X is an M-ideal in X** and Zis nonreflexive, then X contains almost isometric copies of c0. From this it follows that subspaces and quotients of such papers which are isomorphic to dual spaces are reflexive.

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Intersection properties of balls in spaces of compact operators

Cited by 72 publications

References 20 publications

Banach spaces having the Radon-Nikodym property and numerical index 1

Banach spaces having the Radon-Nikodym property and numerical index 1

Properties of lush spaces and applications to Banach spaces with numerical index 1

Banach spaces which are 𝑀-ideals in their biduals

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