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A Δ-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 Δ- and Daugavet points are the same in L1-spaces, in L1-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 closed convex hull of its Δ-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.
We prove that the class of reflexive asymptotic-
$c_{0}$
Banach spaces is coarsely rigid, meaning that if a Banach space
$X$
coarsely embeds into a reflexive asymptotic-
$c_{0}$
space
$Y$
, then
$X$
is also reflexive and asymptotic-
$c_{0}$
. In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-
$c_{0}$
space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.
In this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space
${\mathcal{J}}$
nor into its dual
${\mathcal{J}}^{\ast }$
. It is a particular case of a more general result on the non-equi-coarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic structure. This allows us to exhibit a coarse invariant for Banach spaces, namely the non-equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property
${\mathcal{Q}}$
of Kalton. We conclude with a remark on the coarse geometry of the James tree space
${\mathcal{J}}{\mathcal{T}}$
and of its predual.
The Chebyshev conjecture posits that Chebyshev subsets of a real Hilbert space
$X$
are convex. Works by Asplund, Ficken and Klee have uncovered an equivalent formulation of the Chebyshev conjecture in terms of uniquely remotal subsets of
$X$
. In this tradition, we develop another equivalent formulation in terms of Chebyshev subsets of the unit sphere of
$X\times \mathbb{R}$
. We characterise such sets in terms of the image under stereographic projection. Such sets have superior structure to Chebyshev sets and uniquely remotal sets.
We study the Daugavet property in tensor products of Banach spaces. We show that
$L_{1}(\unicode[STIX]{x1D707})\widehat{\otimes }_{\unicode[STIX]{x1D700}}L_{1}(\unicode[STIX]{x1D708})$
has the Daugavet property when
$\unicode[STIX]{x1D707}$
and
$\unicode[STIX]{x1D708}$
are purely non-atomic measures. Also, we show that
$X\widehat{\otimes }_{\unicode[STIX]{x1D70B}}Y$
has the Daugavet property provided
$X$
and
$Y$
are
$L_{1}$
-preduals with the Daugavet property, in particular, spaces of continuous functions with this property. With the same techniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.
We introduce a notion of modulated topological vector spaces, that generalises, among others, Banach and modular function spaces. As applications, we prove some results which extend Kirk’s and Browder’s fixed point theorems. The theory of modulated topological vector spaces provides a very minimalist framework, where powerful fixed point theorems are valid under a bare minimum of assumptions.
We prove that every separable Banach space containing an isomorphic copy of
$\ell _{1}$
can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math.95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.
We prove that if
$M$
is a
$\text{JBW}^{\ast }$
-triple and not a Cartan factor of rank two, then
$M$
satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of
$M$
onto the unit sphere of another real Banach space
$Y$
extends to a surjective real linear isometry from
$M$
onto
$Y$
.
We show that if
$(X,\Vert \cdot \Vert )$
is a Banach space that admits an equivalent locally uniformly rotund norm and the set of all norm-attaining functionals is residual then the dual norm
$\Vert \cdot \Vert ^{\ast }$
on
$X^{\ast }$
is Fréchet at the points of a dense subset of
$X^{\ast }$
. This answers the main open problem in a paper by Guirao, Montesinos and Zizler [‘Remarks on the set of norm-attaining functionals and differentiability’, Studia Math.241 (2018), 71–86].
(1) For any finite metric space
$M$
the Lipschitz-free space on
$M$
contains a large well-complemented subspace that is close to
$\ell _{1}^{n}$
.
(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to
$\ell _{1}^{n}$
of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.
Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.
We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces…which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for the numerical radius is given.
Given two (real) normed (linear) spaces
$X$
and
$Y$
, let
$X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$
, where
$\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$
. It is known that
$X\otimes _{1}Y$
is
$2$
-UR if and only if both
$X$
and
$Y$
are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if
$X$
is
$m$
-dimensional and
$Y$
is
$k$
-UR, then
$X\otimes _{1}Y$
is
$(m+k)$
-UR. In the other direction, we observe that if
$X\otimes _{1}Y$
is
$k$
-UR, then both
$X$
and
$Y$
are
$(k-1)$
-UR. Given a monotone norm
$\Vert \cdot \Vert _{E}$
on
$\mathbb{R}^{2}$
, we let
$X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$
where
$\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$
. It is known that if
$X$
is uniformly rotund in every direction,
$Y$
has the weak fixed point property for nonexpansive maps (WFPP) and
$\Vert \cdot \Vert _{E}$
is strictly monotone, then
$X\otimes _{E}Y$
has WFPP. Using the notion of
$k$
-uniform rotundity relative to every
$k$
-dimensional subspace we show that this result holds with a weaker condition on
$X$
.
A precise quantitative version of the following qualitative statement is proved: If a finite-dimensional normed space contains approximately Euclidean subspaces of all proportional dimensions, then every proportional dimensional quotient space has the same property.
We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space
$\text{Ces}_{\infty }$
and its sequence counterpart
$\text{ces}_{\infty }$
are isomorphic. This is rather surprising since
$\text{Ces}_{\infty }$
(like Talagrand’s example) has no natural lattice predual. We prove that
$\text{ces}_{\infty }$
is not isomorphic to
$\ell _{\infty }$
nor is
$\text{Ces}_{\infty }$
isomorphic to the Tandori space
$\widetilde{L_{1}}$
with the norm
$\Vert f\Vert _{\widetilde{L_{1}}}=\Vert \widetilde{f}\Vert _{L_{1}}$
, where
$\widetilde{f}(t):=\text{ess}\,\sup _{s\geqslant t}|f(s)|$
. Our investigation also involves an examination of the Schur and Dunford–Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro–Marcinkiewicz and Cesàro–Lorentz spaces have the latter property.
We study approximation of operators between Banach spaces
$X$
and
$Y$
that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair
$(X,Y)$
has the pointwise Bishop–Phelps–Bollobás property (pointwise BPB property for short). In this paper we mostly concentrate on those
$X$
, called universal pointwise BPB domain spaces, such that
$(X,Y)$
possesses pointwise BPB property for every
$Y$
, and on those
$Y$
, called universal pointwise BPB range spaces, such that
$(X,Y)$
enjoys pointwise BPB property for every uniformly smooth
$X$
. We show that every universal pointwise BPB domain space is uniformly convex and that
$L_{p}(\unicode[STIX]{x1D707})$
spaces fail to have this property when
$p>2$
. No universal pointwise BPB range space can be simultaneously uniformly convex and uniformly smooth unless its dimension is one. We also discuss a version of the pointwise BPB property for compact operators.
In this paper, we give a complete description of left symmetric points for
Birkhoff orthogonality in the preduals of von Neumann algebras. As a
consequence, except for
$\mathbb{C}$
,
$\ell _{\infty }^{2}$
and
$M_{2}(\mathbb{C})$
, there are no von Neumann algebras whose preduals have
nonzero left symmetric points for Birkhoff orthogonality.
Our main result states that whenever we have a non-Euclidean norm
$\Vert \cdot \Vert$
on a two-dimensional vector space
$X$
, there exists some
$x\neq 0$
such that for every
$\unicode[STIX]{x1D706}\neq 1$
,
$\unicode[STIX]{x1D706}>0$
, there exist
$y,z\in X$
satisfying
$\Vert y\Vert =\unicode[STIX]{x1D706}\Vert x\Vert$
,
$z\neq 0$
and
$z$
belongs to the bisectors
$B(-x,x)$
and
$B(-y,y)$
. We also give several results about the geometry of the unit sphere of strictly convex planes.
Extending recent results by Cascales et al. [‘Plasticity of the unit ball of a strictly convex Banach space’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.110(2) (2016), 723–727], we demonstrate that for every Banach space
$X$
and every collection
$Z_{i},i\in I$
, of strictly convex Banach spaces, every nonexpansive bijection from the unit ball of
$X$
to the unit ball of the sum of
$Z_{i}$
by
$\ell _{1}$
is an isometry.
Given a Banach space X and a real number α ≥ 1, we write: (1) D(X) ≤ α if, for any locally finite metric space A, all finite subsets of which admit bilipschitz embeddings into X with distortions ≤ C, the space A itself admits a bilipschitz embedding into X with distortion ≤ α ⋅ C; (2) D(X) = α+ if, for every ϵ > 0, the condition D(X) ≤ α + ϵ holds, while D(X) ≤ α does not; (3) D(X) ≤ α+ if D(X) = α+ or D(X) ≤ α. It is known that D(X) is bounded by a universal constant, but the available estimates for this constant are rather large. The following results have been proved in this work: (1) D((⊕n=1∞Xn)p) ≤ 1+ for every nested family of finite-dimensional Banach spaces {Xn}n=1∞ and every 1 ≤ p ≤ ∞. (2) D((⊕n=1∞ ℓ∞n)p) = 1+ for 1 < p < ∞. (3) D(X) ≤ 4+ for every Banach space X with no nontrivial cotype. Statement (3) is a strengthening of the Baudier–Lancien result (2008).