The Daugavet property and translation-invariant subspaces

Lücking, Simon

doi:10.4064/sm221-3-5

Cited by 3 publications

(2 citation statements)

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“…An example of a Banach space which satisfies the WODP Let us recall that a Banach space X is said to be L-embedded if X * * = X ⊕ 1 Z for some subspace Z of X * * . Examples of L-embedded spaces are L 1 (µ)-spaces, preduals of von Neumann algebras, preduals of real or complex JBW * -triples or the predual of the disk algebra, and quotients of L 1 by nicely placed subspaces (see [18]). We invite the reader to go to the reference [8] for plenty of examples of L-embedded spaces.…”

Section: The Resultsmentioning

confidence: 99%

Daugavet Points in Projective Tensor Products

Dantas¹,

Jung²,

Zoca³

2021

The Quarterly Journal of Mathematics

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In this paper, we are interested in studying when an element z in the projective tensor product $X {\widehat{\otimes}_\pi} Y$ turns out to be a Daugavet point. We prove first that, under some hypothesis, the assumption of $X {\widehat{\otimes}_\pi} Y$ having the Daugavet property implies the existence of a great amount of isometries from Y into X*. Having this in mind, we provide methods for constructing non-trivial Daugavet points in $X {\widehat{\otimes}_\pi} Y$. We show that C(K)-spaces are examples of Banach spaces such that the set of the Daugavet points in $C(K) {\widehat{\otimes}_\pi} Y$ is weakly dense when Y is a subspace of C(K)*. Finally, we present some natural results on when an elementary tensor $x \otimes y$ is a Daugavet point.

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Section: The Resultsmentioning

confidence: 99%

Daugavet Points in Projective Tensor Products

Dantas¹,

Jung²,

Zoca³

2021

The Quarterly Journal of Mathematics

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“…Now, if we take a δ-net A of S F , for δ > 0 small enough, a standard argument (see the proofs of [37,Proposition 2.3] or of [32,Lemma II.1.1]) provide a norm-one polynomial Q and α 1 such that (a) is satisfied and so that the inequality Using again that Y is a subspace, we routinely get the desired inequality in (b).…”

Section: Weak Operator Daugavet Property and Polynomial Weak Operator...mentioning

confidence: 99%

Daugavet property in projective symmetric tensor products of Banach spaces

Martín¹,

Zoca²

2020

Preprint

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We show that all the symmetric projective tensor products of a Banach space X have the Daugavet property provided X has the Daugavet property and either X is an L 1 -predual (i.e. X * is isometric to an L 1 -space) or X is a vector-valued L 1 -space. In the process of proving it, we get a number of results of independent interest. For instance, we characterise "localised" versions of the Daugavet property (i.e. Daugavet points and ∆-points introduced in [1]) for L 1 -preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real L 1preduals in terms of the absence of ∆-points and also to provide new examples of L 1 -preduals having the convex diametral local diameter two property. These results are also applied to nicely embedded Banach spaces (in the sense of [40]) so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for L 1 -preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results in [34] about the Daugavet property for projective tensor products is also got.

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