Two properties of Müntz spaces

Abrahamsen, Trond A.; Leraand, Aleksander; Martiny, André; Nygaard, Olav

doi:10.1515/dema-2017-0025

Cited by 4 publications

(2 citation statements)

References 9 publications

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“…Our initial motivation for doing this, was the known fact that such spaces X are isomorphic, even almost isometrically isomorphic in the case X = M 0 (Λ), to subspaces of c (see [17] and [16]). Based on this, the results from [3], and other results from [16] one could expect similar results for Müntz spaces as for c. And, indeed, this is the case, at least for X = M 0 (Λ) (see Theorem 3.13). In this class of Müntz spaces the ∆-and Daugavet-points are the same and the Daugavet-points are exactly the functions f ∈ S X for which f (1) = ±1.…”

Section: ∆And Daugavet-points For Different Classes Of Spacessupporting

confidence: 82%

Delta- and Daugavet points in Banach spaces

Abrahamsen¹,

Haller²,

Lima³

et al. 2020

Proceedings of the Edinburgh Mathematical Society

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A ∆-point x of a Banach space is a norm one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance 2 from x. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, x is a Daugavet-point. A Banach space X has the Daugavet property if and only if every norm one element is a Daugavet-point.We show that ∆-and Daugavet-points are the same in L 1spaces, L 1 -preduals, as well as in a big class of Müntz spaces. We also provide an example of a Banach space where all points on the unit sphere are ∆-points, but where none of them are Daugavetpoints.We also study the property that the unit ball is the closed convex hull of its ∆-points. This gives rise to a new diameter two property that we call the convex diametral diameter two property. We show that all C(K) spaces, K infinite compact Hausdorff, as well as all Müntz spaces have this property. Moreover, we show that this property is stable under absolute sums. 2010 Mathematics Subject Classification. Primary 46B20, 46B22, 46B04. Key words and phrases. Diametral diameter two property, Daugavet property, L 1 -space, L 1 -predual space, Müntz space. R. Haller and K. Pirk were partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research. 1 2 T. A. ABRAHAMSEN, R. HALLER, V. LIMA, AND K. PIRKWe will sometimes need to clarify which Banach space we are working with and write ∆ X ε (x) and ∆ X instead of ∆ ε (x) and ∆ respectively. The starting point of this research was the discovery that if a Banach space X satisfies B X = conv ∆, then X has the LD2P.We study spaces that satisfy the property B X = conv ∆ in Section 5. The case S X = ∆, i.e. x ∈ conv ∆ ε (x) for all x ∈ S X and ε > 0, has already appeared in the literature, but under different names: The diametral local diameter two property (DLD2P) ([5]), the LD2P+ ([1] and [2]), and space with bad projections ([13]). We will use the term DLD2P in this paper. From [18, Corollary 2.3 and (7) p. 95] and [13, Theorem 1.4] the following characterization is known. Proposition 1.1. Let X be a Banach space. The following assertions are equivalent:(1) X has the DLD2P;(2) for all x ∈ S X we have x ∈ conv ∆ ε (x) for all ε > 0;(3) for all projections P : X → X of rank-1 we have Id − P ≥ 2.Related to the DLD2P is the Daugavet property. We have (cf. [18, Corollary 2.3]): Proposition 1.2. Let X be a Banach space. The following assertions are equivalent:(1) X has the Daugavet property, i.e. for all bounded linear rank-1 operators T : X → X we have Id − T = 1 + T ; (2) for all x ∈ S X we have B X = conv ∆ ε (x) for all ε > 0.

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Section: ∆And Daugavet-points For Different Classes Of Spacessupporting

confidence: 82%

Delta- and Daugavet points in Banach spaces

Abrahamsen¹,

Haller²,

Lima³

et al. 2020

Proceedings of the Edinburgh Mathematical Society

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“…(a) Lindenstrauss spaces (this follows by inspecting the proof of Proposition 4.6 in [4]); (b) uniform algebras (see Theorem 4.2 in [3]); (c) ASQ-spaces, in particular, Banach spaces which are M-ideals in their bidual (see [1]); (d) Banach spaces with an infinite-dimensional centralizer (this follows by inspecting the proof of Proposition 3.3 in [6]); (e) somewhat regular linear subspaces of C 0 (L), whenever L is an infinite locally compact Hausdorff topological space [5]; (f) Müntz spaces (this follows by inspecting the proof of Theorem 2.5 in [2]).…”

Section: Introductionmentioning

confidence: 99%

Symmetric Strong Diameter Two Property

et al. 2019

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We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and Põldvere.The symmetric strong diameter two property is not just formally stronger than the strong diameter two property (finite convex combinations of slices have diameter 2). We show that the symmetric strong diameter two property is only preserved by ℓ ∞ -sums, and working with weak star slices we show that Lip 0 (M ) have the weak star version of the property for several classes of metric spaces M .2010 Mathematics Subject Classification. Primary 46B20, 46B22. Key words and phrases. strong diameter 2 property, almost square spaces, Lipschitz spaces.R. Haller, J. Langemets, and R. Nadel were partially supported by institutional research funding IUT20-57 of the Estonian Ministry of Education and Research.

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