Gonzalo Martínez-Cervantes

University of Murcia | UM

We prove that, given two Banach spaces X and Y and bounded, closed convex sets \(C\subseteq X\) and \(D\subseteq Y\), if a nonzero element \(z\in {\overline{{{\,\textrm{co}\,}}}}(C\otimes D)\subseteq X\widehat{\otimes }_\pi Y\) is a preserved extreme point then \(z=x_0\otimes y_0\) for some preserved extreme points \(x_0\in C\) and \(y_0\in D\), wh...

The main goal of this article is to show that for every (reflexive) infinite-dimensional Banach space $X$ there exists a reflexive Banach space $Y$ and $T, R \in \mathcal{L}(X,Y)$ such that $R$ is a rank-one operator, $\|T+R\|>\|T\|$ but $T+R$ does not attain its norm. This answers a question posed by S.~Dantas and the first two authors. Furthermor...

We prove that, given two Banach spaces X and Y and bounded, closed convex sets C ⊆ X and D ⊆ Y , if a nonzero element z ∈ co(C ⊗D) ⊆ X ⊗ π Y is a preserved extreme point then z = x0 ⊗y0 for some preserved extreme points x0 ∈ C and y0 ∈ D, whenever K(X, Y *) separates points of X ⊗ π Y (in particular, whenever X or Y has the compact approximation pr...

We study the class of compact spaces that appear as structure spaces of separable Banach lattices. In other words, we analyze what C ( K ) spaces appear as principal ideals of separable Banach lattices. Among other things, it is shown that every such compactum K admits a strictly positive regular Borel measure of countable type that is analytic, an...

A well-known classical result states that c0 is linearly embeddable in a Banach lattice if and only if it is lattice embeddable. Improving results of H.P. Lotz, H.P. Rosenthal and N. Ghoussoub, we prove that C[0,1] shares this property with c0. Furthermore, we show that any infinite-dimensional closed sublattice of C[0,1] is either lattice isomorph...

Given two Banach spaces X and Y, we analyze when the projective tensor product X⊗ˆπY has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) X⊗ˆπY is WLD if and only if both X and Y are WLD and all operators from...

Given a Banach space $X$, we say that a sequence $\{x_n\}$ in the unit ball of $X$ is $L$-orthogonal if $\Vert x+x_n\Vert \rightarrow 1+\Vert x\Vert $ for every $x\in X$. On the other hand, an element $x^{**}$ in the bidual sphere is said to be $L$-orthogonal (to $X$) if $\|x+x^{**}\|= 1+\Vert x\Vert $ for every $x\in X$. The aim of this paper is t...

We prove that if $M$ is an infinite complete metric space then the set of strongly norm-attaining Lipschitz functions $\SA(M)$ contains a linear subspace isomorphic to $c_0$. This solves an open question posed by V. Kadets and O. Rold\'an.

Let X be a Banach space and $$Y \subseteq X$$ Y ⊆ X be a closed subspace. We prove that if the quotient X / Y is weakly Lindelöf determined or weak Asplund, then for every $$w^*$$ w ∗ -convergent sequence $$(y_n^*)_{n\in \mathbb N}$$ ( y n ∗ ) n ∈ N in $$Y^*$$ Y ∗ there exist a subsequence $$(y_{n_k}^*)_{k\in \mathbb N}$$ ( y n k ∗ ) k ∈ N and a $$...

It follows from a theorem of Rosenthal that a compact space is ccc if and only if its every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces $${\mathcal {C}}$$ C we define its orthogonal $${\mathcal {C}}^\perp $$ C ⊥ as the class of all compact spaces for which every continuous image in $${\mathcal {C...

Given two Banach spaces X and Y , we analyze when the projec-tive tensor product X ⊗ π Y has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) X ⊗ π Y is WLD if and only if both X and Y are WLD and all operators...

Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Banach lattices and lattice homomorphisms. Namely, a Banach lattice X satisfies the lattice-lifting property if every lattice homomorphism to X having a bounded...

A recent result of T. Abrahamsen, P. Hájek and S. Troyanski states that a separable Banach space is almost square if and only if there exists h∈SX⁎⁎⁎⁎ such that ‖x+h‖=max⁡{‖x‖,1} for all x∈X. The proof passes through a sequential version of being almost square which we call being sequentially almost square. In this article we study these conditions...

A recent result of T.~Abrahamsen, P.~H\'ajek and S.~Troyanski states that a separable Banach space is almost square if and only if there exists $h\in S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in X$. The proof passes through a sequential version of being almost square which we call being \textit{sequentially almost square}. In thi...

A well-known classical result states that $c_0$ is linearly embeddable in a Banach lattice if and only if it is lattice embeddable. Improving results of H.P.~Lotz, H.P.~Rosenthal and N.~Ghoussoub, we prove that $C[0,1]$ shares this property with $c_0$. Furthermore, we show that any infinite-dimensional sublattice of $C[0,1]$ is either lattice isomo...

We study how local complementation behaves under several free constructions in Functional Analysis, including free Lipschitz spaces and free Banach lattices.

In this article we deal with the free Banach lattice generated by a lattice and its behavior with respect to subspaces. In general, any lattice embedding $i\colon \mathbb{L} \longrightarrow \mathbb{M}$ between two lattices $\mathbb{L} \subseteq \mathbb{M}$ induces a Banach lattice homomorphism $\hat \imath\colon FBL \langle \mathbb{L} \rangle \long...

In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure...

A pair of Banach spaces (E,F) is said to have the weak maximizing property (WMP, for short) if for every bounded linear operator T from E into F, the existence of a non-weakly null maximizing sequence for T implies that T attains its norm. This property was recently introduced in a paper by R. Aron, D. García, D. Pelegrino and E. Teixeira, raising...

A result of G. Godefroy asserts that a Banach space $X$ contains an isomorphic copy of $\ell_1$ if and only if there is an equivalent norm $|||\cdot|||$ such that, for every finite-dimensional subspace $Y$ of $X$ and every $\varepsilon>0$, there exists $x\in S_X$ so that $|||y+r x|||\geq (1-\varepsilon)(|||y|||+\vert r\vert)$ for every $y\in Y$ and...

It follows from a theorem of Rosenthal that a compact space is $ccc$ if and only if every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces $\mathcal{C}$ we define its orthogonal $\mathcal{C}^\perp$ as the class of all compact spaces for which every continuous image in $\mathcal{C}$ is metrizable. We s...

Given a Banach space $X$, we say that a sequence $\{x_n\}$ in the unit ball of $X$ is $L$-orthogonal if $\Vert x+x_n\Vert\rightarrow 1+\Vert x\Vert$ for every $x\in X$. On the other hand, an element $x^{**}$ in the bidual sphere is said to be $L$-orthogonal (to $X$) if $\|x+x^{**}\|= 1+\Vert x\Vert$ for every $x\in X$. A result of V. Kadets, V. She...

A pair of Banach spaces (E, F) is said to have the weak maximizing property (WMP, for short) if for every bounded linear operator T from E into F , the existence of a non-weakly null maximizing sequence for T implies that T attains its norm. This property was recently introduced in an article by R. Aron, D. García, D. Pelegrino and E. Teixeira, rai...

In this article we deal with the free Banach lattice $FBL\langle \mathbb{L} \rangle$ generated by a lattice $\mathbb{L}$. We prove that if $FBL\langle \mathbb{L} \rangle$ is projective then $\mathbb{L}$ has a maximum and a minimum. On the other hand, we show that if $\mathbb{L}$ has maximum and minimum then $FBL\langle \mathbb{L} \rangle$ is $2$-la...

Let $X$ be a Banach space and $Y \subseteq X$ be a closed subspace. We prove that if the quotient $X/Y$ is weakly Lindel\"{o}f determined or weak Asplund, then for every $w^*$-convergent sequence $(y_n^*)_{n\in \mathbb N}$ in $Y^*$ there exist a subsequence $(y_{n_k}^*)_{k\in \mathbb N}$ and a $w^*$-convergent sequence $(x_k^*)_{k\in \mathbb N}$ in...

In this paper we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in L(E, F). By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of L(E, F) (in the weak operator topology) such that 0 is an element of its closure (in the weak operator topology)...

Motivated by the Lipschitz-lifting property of Banach spaces introduced by Godefroy and Kalton, we consider the lattice-lifting property, which is an analogous notion within the category of Banach lattices and lattice homomorphisms. Namely, a Banach lattice $X$ satisfies the lattice-lifting property if every lattice homomorphism to $X$ having a bou...

In this paper, we study octahedral norms in free Banach lattices FBL[E] generated by a Banach space E. We prove that if E is an L1(μ)-space, a predual of von Neumann algebra, a predual of a JBW∗-triple, the dual of an M-embedded Banach space, the disc algebra or the projective tensor product under some hypothesis, then the norm of FBL[E] is octahed...

In this paper we study the structure of the set $Hom(X,\mathbb{R})$ of all lattice homomorphisms from a Banach lattice $X$ into $\mathbb{R}$. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice $FBL(A)$ generated by a set $A$ contains a disjoint family of cardinality $2...

In this paper, we study octahedral norms in free Banach lattices $FBL[E]$ generated by a Banach space $E$. We prove that if $E$ is an $L_1(\mu)$-space, a predual of von Neumann algebra, a predual of a JBW$^*$-triple, the dual of an $M$-embedded Banach space, the disc algebra or the projective tensor product under some hypothesis, then the norm of $...

We show that if a Banach lattice is projective, then every bounded sequence that can be mapped by a homomorphism onto the basis of c0 must contain an ℓ1-subsequence. As a consequence, if Banach lattices ℓp or FBL[E] are projective, then p=1 or E has the Schur property, respectively. On the other hand, we show that C(K) is projective whenever K is a...

We show that if a Banach lattice is projective, then every bounded sequence that can be mapped by a homomorphism onto the basis of $c_0$ must contain an $\ell_1$-subsequence. As a consequence, if Banach lattices $\ell_p$ or $FBL[E]$ are projective, then $p=1$ or $E$ has the Schur property, respectively. On the other hand, we show that $C(K)$ is pro...

We show that \(c_0\) is not a projective Banach lattice, answering a question of B. de Pagter and A. Wickstead. .On the other hand, we show that \(c_0\) is complemented in the free Banach lattice generated by itself (seen as a Banach space). As a consequence, the free Banach lattice generated by \(c_0\) is not projective.

We show that $c_0$ is not a projective Banach lattice, answering a question of B. de Pagter and A. Wickstead. On the other hand, we show that $c_0$ is complemented in the free Banach lattice generated by itself (seen as a Banach space). As a consequence, the free Banach lattice generated by $c_0$ is not projective.

We study the class of Banach spaces X such that the locally convex space (X,μ(X,Y)) is complete for every norming and norm-closed subspace Y⊂X⁎, where μ(X,Y) denotes the Mackey topology on X associated to the dual pair 〈X,Y〉. Such Banach spaces are called fully Mackey complete. We show that fully Mackey completeness is implied by Efremov's property...

We prove that if a compact line is fragmentable, then it is a Radon–Nikodým compact space.

We study the class of Banach spaces $X$ such that the locally convex space $(X,\mu(X,Y))$ is complete for every norming and norm-closed subspace $Y \subset X^*$, where $\mu(X,Y)$ denotes the Mackey topology on $X$ associated to the dual pair $\langle X,Y\rangle$. Such Banach spaces are called fully Mackey complete. We show that fully Mackey complet...

A metric space M is length when every two points have an approximate midpoint, while M has property (Z) when for every x,y∈M there exists z such that d(x,z)+d(z,y)−d(x,y) is arbitrarily smaller than d(x,z) and d(z,y). Answering a problem posed by García-Lirola, Procházka and Rueda Zoca, we prove that every complete metric space with property (Z) is...

A Banach space X is said to have Efremov's property (E) if every element of the weak⁎-closure of a convex bounded set C⊆X⁎ is the weak⁎-limit of a sequence in C. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of N for which the corresponding Johnson–Lindenstrauss spaces enjoy (re...

We prove that every complete metric space with property (Z) is a length space. These answers questions posed by Garc\'{i}a-Lirola, Proch\'{a}zka and Rueda Zoca, and by Becerra Guerrero, L\'{o}pez-P\'{e}rez and Rueda Zoca, related to the structure of Lipschitz-free Banach spaces of metric spaces.

We prove that if a compact line is fragmentable, then it is a Radon-Nikod\'ym compact space.

A Banach space $X$ is said to have Efremov's property ($\mathcal{E}$) if every element of the weak$^*$-closure of a convex bounded set $C \subseteq X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of $\mathbb{N}$ for which the corres...

We show that for every $d\ge 1$, if $L_1,\ldots, L_d$ are linearly ordered compact spaces and there is a continuous surjection \[ L_1\times L_2\times \dots\times L_d\to K_1\times K_2\times\ldots\times K_{d}\times K_{d+1},\] where all the spaces $K_i$ are infinite, then $K_i, K_j$ are metrizable for some $1\le i<j\le d+1$. This answers a problem pos...

A topological space is said to be sequential if every sequentially closed subspace is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if $X$ is a Banach space with weak*-sequentially compact dual ball and $Y \subset X$ is a subspace such that $Y$ and $X/Y$ have weak*-sequential dual ball, then $X$ has...

A compact space is said to be weakly Radon-Nikod\'{y}m (WRN) if it can be weak*-embedded into the dual of a Banach space not containing $\ell_1$. We investigate WRN Boolean algebras, i.e. algebras whose Stone space is WRN compact. We show that the class of WRN algebras and the class of minimally generated algebras are incomparable. In particular, w...

In this paper we focus on the relation between Riemann integrability and weak continuity. A Banach space $X$ is said to have the weak Lebesgue property if every Riemann integrable function from $[0,1]$ into $X$ is weakly continuous almost everywhere. We prove that the weak Lebesgue property is stable under $\ell_1$-sums and obtain new examples of B...

A compact space is said to be weakly Radon-Nikod\'ym if it is homeomorphic to a weak*-compact subset of the dual of a Banach space not containing an isomorphic copy of $\ell_1$. In this work we provide an example of a continuous image of a Radon-Nikod\'ym compact space which is not weakly Radon-Nikod\'ym. Moreover, we define a superclass of the con...