Properties of Q-Reflexive Banach Spaces

Venkova, Milena

doi:10.1006/jmaa.2001.7644

Journal of Mathematical Analysis and Applications

2001

DOI: 10.1006/jmaa.2001.7644

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Properties of Q-Reflexive Banach Spaces

Milena Venkova

Abstract: We investigate whether the Q-reflexive Banach spaces have the following properties: containment of l p , reflexivity, Dunford-Pettis property, etc.  2001 Elsevier Science

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Cited by 5 publications

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Geometry of integral polynomials, M-ideals and unique norm preserving extensions

Dimant

Galicer

García

2012

Journal of Functional Analysis

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We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral polynomials over a real Banach space X is {±φ k : φ ∈ X * , φ = 1}. With this description we show that, for real Banach spaces X and Y , if X is a non trivial Mideal in Y , then k,s ε k,s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M -ideal in k,s ε k,s Y . This result marks up a difference with the behavior of non-symmetric tensors since, when X is an M -ideal in Y , it is known that k ε k X (the k-th tensor product of X endowed with the injective tensor norm) is an M -ideal in k ε k Y . Nevertheless, if X is Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. We explicitly describe this extension.We also give necessary and sufficient conditions (related with the continuity of the Aron-Berner extension morphism) for a fixed k-homogeneous polynomial P belonging to a maximal polynomial ideal Q( k X) to have a unique norm preserving extension to Q( k X * * ). To this end, we study the relationship between the bidual of the symmetric tensor product of a Banach space and the symmetric tensor product of its bidual and show (in the presence of the BAP) that both spaces have 'the same local structure'. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.2010 Mathematics Subject Classification. 46G25, 46M05, 46B28.

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Geometry of integral polynomials, M-ideals and unique norm preserving extensions

Dimant

Galicer

García

2012

Journal of Functional Analysis

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Duality for spaces of polynomials on tree spaces

Boyd

Poulios

Venkova

2023

Journal of Mathematical Analysis and Applications

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Scalar-valued dominated polynomials on Banach spaces

Botelho

Pellegrino

2005

Proc. Amer. Math. Soc.

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Abstract. It is well known that 2-homogeneous polynomials on L ∞ -spaces are 2-dominated. Motivated by the fact that related coincidence results are possible only for polynomials defined on symmetrically regular spaces, we investigate the situation in several classes of symmetrically regular spaces. We prove a number of non-coincidence results which makes us suspect that there is no infinite dimensional Banach space E such that every scalar-valued homogeneous polynomial on E is r-dominated for every r ≥ 1.

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Properties of Q-Reflexive Banach Spaces

Abstract: We investigate whether the Q-reflexive Banach spaces have the following properties: containment of l p , reflexivity, Dunford-Pettis property, etc.  2001 Elsevier Science

Cited by 5 publications

References 15 publications

Geometry of integral polynomials, M-ideals and unique norm preserving extensions

Geometry of integral polynomials, M-ideals and unique norm preserving extensions

Duality for spaces of polynomials on tree spaces

Scalar-valued dominated polynomials on Banach spaces

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