0000000000425079
AUTHOR
Trond A. Abrahamsen
Strongly extreme points and approximation properties
We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned. In contrast to the above results we also construct a non-symmetric norm on $c_0$ for which all points on the unit sphere are strongly extreme, but …
ON λ-STRICT IDEALS IN BANACH SPACES
AbstractWe define and study λ-strict ideals in Banach spaces, which for λ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do λ-strict u-ideals in their biduals, at least for λ>1/2. An open question, posed by Godefroy et al. [‘Unconditional ideals in Banach spaces’, Studia Math.104 (1993), 13–59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when κu(X)=1; we prove that if κu(X)=1 then X is a strict u-ideal in Ba (X) , and we establish the converse in the separable case.
Relatively weakly open convex combinations of slices
We show that c 0 c_0 and, in fact, C ( K ) C(K) for any scattered compact Hausdorff space K K have the property that finite convex combinations of slices of the unit ball are relatively weakly open.
Diameter 2 properties and convexity
We present an equivalent midpoint locally uniformly rotund (MLUR) renorming $X$ of $C[0,1]$ on which every weakly compact projection $P$ satisfies the equation $\|I-P\| = 1+\|P\|$ ($I$ is the identity operator on $X$). As a consequence we obtain an MLUR space $X$ with the properties D2P, that every non-empty relatively weakly open subset of its unit ball $B_X$ has diameter 2, and the LD2P+, that for every slice of $B_X$ and every norm 1 element $x$ inside the slice there is another element $y$ inside the slice of distance as close to 2 from $x$ as desired. An example of an MLUR space with the D2P, the LD2P+, and with convex combinations of slices of arbitrary small diameter is also given.
Banach spaces where convex combinations of relatively weakly open subsets of the unit ball are relatively weakly open
We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such spaces are established, and we introduce and discuss a geometric condition---property (co)---on a Banach space. Property (co) essentially says that the operation of taking convex combinations of elements of the unit ball is, in a sense, an open map. We show that if a finite dimensional Banach space $X$ has property (co), then for any scattered locally compact Hausdorff space $K$, the space $C_0(K,X)$ of continuous $X$-valued functions vanishing at infinity has…
Daugavet- and delta-points in Banach spaces with unconditional bases
We study the existence of Daugavet- and delta-points in the unit sphere of Banach spaces with a 1 1 -unconditional basis. A norm one element x x in a Banach space is a Daugavet-point (resp. delta-point) if every element in the unit ball (resp. x x itself) is in the closed convex hull of unit ball elements that are almost at distance 2 2 from x x . A Banach space has the Daugavet property (resp. diametral local diameter two property) if and only if every norm one element is a Daugavet-point (resp. delta-point). It is well-known that a Banach space with the Daugavet property does not have an unconditional basis. Similarly spaces with the diametral local diameter two property do not have an un…
New applications of extremely regular function spaces
Let $L$ be an infinite locally compact Hausdorff topological space. We show that extremely regular subspaces of $C_0(L)$ have very strong diameter $2$ properties and, for every real number $\varepsilon$ with $0<\varepsilon<1$, contain an $\varepsilon$-isometric copy of $c_0$. If $L$ does not contain isolated points they even have the Daugavet property, and thus contain an asymptotically isometric copy of $\ell_1$.
Polyhedrality and decomposition
Abstract The aim of this note is to present two results that make the task of finding equivalent polyhedral norms on certain Banach spaces, having either a Schauder basis or an uncountable unconditional basis, easier and more transparent. The hypotheses of both results are based on decomposing the unit sphere of a Banach space into countably many pieces, such that each one satisfies certain properties. Some examples of spaces having equivalent polyhedral norms are given.
Delta- and Daugavet points in Banach spaces
AbstractA Δ-pointxof 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 fromx. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations,xis a Daugavet point. A Banach spaceXhas the Daugavet property if and only if every norm-one element is a Daugavet point. We show that Δ- and Daugavet points are the same inL1-spaces, inL1-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 Δ-points, but none of them are Daugavet points. We also study the property that the unit ball is the clo…
Asymptotic geometry and Delta-points
We study Daugavet- and $\Delta$-points in Banach spaces. A norm one element $x$ is a Daugavet-point (respectively a $\Delta$-point) if in every slice of the unit ball (respectively in every slice of the unit ball containing $x$) you can find another element of distance as close to $2$ from $x$ as desired. In this paper we look for criteria and properties ensuring that a norm one element is not a Daugavet- or $\Delta$-point. We show that asymptotically uniformly smooth spaces and reflexive asymptotically uniformly convex spaces do not contain $\Delta$-points. We also show that the same conclusion holds true for the James tree space as well as for its predual. Finally we prove that there exis…
Almost square Banach spaces
We single out and study a natural class of Banach spaces -- almost square Banach spaces. In an almost square space we can find, given a finite set $x_1,x_2,\ldots,x_N$ in the unit sphere, a unit vector $y$ such that $\|x_i-y\|$ is almost one. These spaces have duals that are octahedral and finite convex combinations of slices of the unit ball of an almost square space have diameter 2. We provide several examples and characterizations of almost square spaces. We prove that non-reflexive spaces which are M-ideals in their biduals are almost square. We show that every separable space containing a copy of $c_0$ can be renormed to be almost square. A local and a weak version of almost square spa…
Almost square dual Banach spaces
Abstract We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on l ∞ . As a consequence we get that every dual Banach space containing c 0 has an equivalent almost square dual norm. Finally we characterize separable real almost square spaces in terms of their position in their fourth duals.
Two properties of Müntz spaces
Abstract We show that Müntz spaces, as subspaces of C[0, 1], contain asymptotically isometric copies of c0 and that their dual spaces are octahedral.