Related papers: (Non-)amenability of $\mathcal B(E)$ and Banach sp…

Continuing with the study of Approximately ultrahomogeneous and Fra\"iss\'e Banach spaces introduced by V. Ferenczi, J. L\'opez-Abad, B. Mbombo and S. Todorcevic, we define formally weaker and in some aspects more natural properties of…

Read produced the first example of a Banach space $E_{\text{R}}$ such that the associated Banach algebra $\mathscr{B}(E_{\text{R}})$ of bounded operators admits a discontinuous derivation (J. London Math. Soc. 1989). We generalise Read's…

Let $G$ be a locally compact abelian group, and let $\omega:G \to [1,\infty)$ be a measurable weight, i.e., $\omega$ is measurable, and $\omega(s+t)\leq \omega(s)\omega(t)$ for all $s, t \in G$. Let $\mathcal{A}$ be a semisimple commutative…

In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibit a striking resemblance to the geometry of James' space. Further, we show that the averaging…

For a Banach space $B$ of functions which satisfies for some $m>0$ $$ \max(\|F+G\|_B,\|F-G\|_B) \ge (\|F\|^s_B + m\|G\|^s_B)^{1/s}, \forall F,G\in B \ (*) $$ a significant improvement for lower estimates of the moduli of smoothness…

We present some basic elements of the theory of generalised Br\`{e}gman relative entropies over nonreflexive Banach spaces. Using nonlinear embeddings of Banach spaces together with the Euler--Legendre functions, this approach unifies two…

Given a map $f \colon E \longrightarrow F$ between Banach spaces (or Banach lattices), a set $A$ of $E$-valued bounded sequences, ${\bf x} \in A$ and a vector topology $\tau$ on $F$, we investigate the existence of an infinite dimensional…

In this paper, we study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotically uniformly smoothness and convexity, and several distinct Banach-Saks-like properties. Among…

We prove a refinement of the inequality by Hoffmann-Jorgensen that is significant for three reasons. First, our result improves on the state-of-the-art even for real-valued random variables. Second, the result unifies several versions in…

In this note, we prove that a semigroup $S$ is left amenable if and only if every two nonzero elements of $\ell^1_+(S)$ have a common nonzero right multiple, where $\ell^1_+(S)$ is the positive part of the Banach algebra $\ell^1(S)$, or…

We modify the very well known theory of normed spaces $(E, \norm)$ within functional analysis by considering a sequence $(\norm_n : n\in\N)$ of norms, where $\norm_n$ is defined on the product space $E^n$ for each $n\in\N$. Our theory is…

Tsirelson's space $\mathcal{T}$ made its appearance in Banach space theory in 1974 soon to become one of the most significant counterexamples in the theory. Its structure broke the ideal pattern that analysts had conceived for a generic…

We investigate Borel reducibility between equivalence relations $E(X,p)=X^{\Bbb N}/\ell_p(X)$'s where $X$ is a separable Banach space. We show that this reducibility is related to the so called H\"older$(\alpha)$ embeddability between…

We prove that if $ C $ is a family of separable Banach spaces which is analytic with respect to the Effros-Borel structure and none member of $ C $ is isometrically universal for all separable Banach spaces, then there exists a separable…

In this paper we survey known results of characterizations of reflexive Banach spaces, which are based on convergence of usual and generalized arithmetic mean (or Ces\`aro sum), weakly compact subsets, affine sets in a Banach space or its…

The standard theory of Banach spaces is built upon the notions of vector space, triangle inequality and Cauchy completeness. Here we propose a `hyperbolic' variant of this `elliptic' framework where general linear combinations are replaced…

Let $1\leq p\leq q\leq\infty.$ Being motivated by the classical notions of limited, $p$-limited and coarse $p$-limited subsets of a Banach space, we introduce and study $(p,q)$-limited subsets and their equicontinuous versions and coarse…

We start the systematic study of Fr\'{e}chet spaces which are $\aleph$-spaces in the weak topology. A topological space $X$ is an $\aleph_0$-space or an $\aleph$-space if $X$ has a countable $k$-network or a $\sigma$-locally finite…

Let $ H $ be a compact subgroup of a locally compact group $G$. In this paper we define a convolution on $ M(G/H) $, the space of all complex bounded Radon measures on the homogeneous space G/H. Then we prove that the measure space $ M(G/H,…

Let $A$ be a Banach algebra and $A^{**}$ be the second dual of it. We show that by some new conditions, $A$ is weakly amenable whenever $A^{**}$ is weakly amenable. We will study this problem under generalization, that is, if $(n+2)-th$…