Related papers: How many miles from $L_\infty$ to $\ell_\infty$?

We investigate the numerical radius of weighted shifts on $\ell^{2}$ as a norm on $\ell^{\infty}$. Along the way, we prove that the largest value a Banach limit can attain on the sequence of absolute values of a bounded sequences is larger…

In a companion paper (Studia Math., 2023), we proved for every $\lambda\in(1,2]$ the existence of a $(\lambda^+)$-injective renorming of $\ell_\infty$ that is not $\lambda$-injective, thereby establishing a~forgotten theorem of…

We investigate a complex analogue of Spencer's Six Standard Deviations Theorem. Specifically, we propose the following conjecture: for any dimension $n \geq 2$, given vectors $a_1, \ldots, a_n \in \mathbb{C}^n$ satisfying $\|a_i\|_{\infty}…

In this working note we study the one-sided positive Banach-Mazur distance between some pairs of $C(K)$ Banach spaces. Building on methods developed in [4], we solve, in particular, one of the problems posed in [2].

We address the following question: what is the class of Banach spaces isomorphic to subspaces of indecomposable Banach spaces? We show that this class includes all Banach spaces of density not bigger than the continuum which do not admit…

We use the Gowers block Ramsey theorem to characterize Banach spaces containing isomorphs of $\ell_p$ (for some $1 \leq p < \infty$) or $c_0$.

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 $\ell_\infty.$ As a consequence we get that every dual Banach space containing…

We formulate general conditions which imply that $L(X,Y)$, the space of operators from a Banach space $X$ to a Banach space $Y$, has $2^{\mathfrak c}$ closed ideals where $\mathfrak c$ is the cardinality of the continuum. These results are…

We prove that a non ergodic Banach space must be near Hilbert. In particular, $\ell_p$ ($2<p<\infty$) is ergodic. This reinforces the conjecture that $\ell_2$ is the only non ergodic Banach space. As an application of our criterion for…

We investigate the geometry of $C(K,X)$ and $\ell_{\infty}(X)$ spaces through complemented subspaces of the form $\left(\bigoplus_{i\in \varGamma}X_i\right)_{c_0}$. Concerning the geometry of $C(K,X)$ spaces we extend some results of D.…

This paper is concerned with the isomorphic structure of the Banach space $\ell_\infty/c_0$ and how it depends on combinatorial tools whose existence is consistent but not provable from the usual axioms of ZFC. Our main global result is…

It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space $\ell_1$, then every non-expansive bijection $F: B_M \to B_M$ is an isometry. We extend these results to non-expansive bijections $F: B_E…

We construct a hereditarily indecomposable Banach space with dual isomorphic to $\ell_1$. Every bounded linear operator on this space has the form $\lambda I+K$ with $\lambda$ a scalar and $K$ compact.

We prove that the outer $L^p_\mu(\ell^r)$ spaces, introduced by Do and Thiele, are isomorphic to Banach spaces, and we show the expected duality properties between them for $1 < p \leq \infty, 1 \leq r < \infty$ or $p=r \in \{ 1, \infty \}$…

It is shown that for any W weakly compact set of a real Banach space X, the set $L_\infty(\mu ,W)$ is N-simultaneously proximinal in $L_\infty(\mu ,X)$ for arbitrary monotonous norm N in $\mathbb{R}^n$.

In this paper, we improve the lower estimate of multidimensional Bohr radius for unit ball of $\ell^n_q$-spaces ($1\leq q\leq \infty$) for bounded holomorphic functions with values in finite dimensional complex Banach spaces. The new…

We prove that the spaces $\ell_p$, $1<p<\infty, p\ne 2$, and all infinite-dimensional subspaces of their quotient spaces do not admit equivalent almost transitive renormings. This is a step towards the solution of the Banach-Mazur rotation…

The main result is that there are infinitely many; in fact, a continuum; of closed ideals in the Banach algebra $L(L_1)$ of bounded linear operators on $L_1(0,1)$. This answers a question from A. Pietsch's 1978 book "Operator Ideals". The…

Let $E$ be one of the spaces $C(K)$ and $L_1$, $F$ be an arbitrary Banach space, $p>1,$ and $(X,\sigma)$ be a space with a finite measure. We prove that $E$ is isometric to a subspace of the Lebesgue-Bochner space $L_p(X;F)$ only if $E$ is…

Nonatomic bounded sequences in $\ell_1$, that is, those giving rise to nonatomic submeasures on $\mathbb N$ are introduced and shown to form a closed subspace nonat$(\ell_1)$ of $\ell_\infty(\ell_1)$, and some spaces of relevant operators…