William B. Johnson
If $X$ is a closed subspace of a Banach space $L$ which embeds into a Banach lattice not containing $\ell_\infty^n$'s uniformly and $L/X$ contains $\ell_\infty^n$'s uniformly, then $X$ cannot have local unconditional structure in the sense of Gordon-Lewis (GL-{\sl l.u.st.}).
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Several new characterizations of Banach spaces containing a subspace isomorphic to $\ell^1$, are obtained. These are applied to the question of when $\ell^1$ embeds in the injective tensor product of two Banach spaces.
A well known argument of James yields that if a Banach space $X$ contains $\ell_1^n$'s uniformly then $X$ contains $\ell_1^n$'s almost isometrically. In the first half of the paper we extend this idea to the ordinal $\ell_1$-indices of Bourgain. In the second half we use our results to calculate the $\ell_1$-index of certain Banach spaces. Furthermore we show that the $\ell_1$-index of a separable Banach space not containing $\ell_1$ must be of the form $ω^α$ for some countable ordinal $α$.
A Hereditarily Indecomposable asymptotic $\ell_2$ Banach space is constructed. The existence of such a space answers a question of B. Maurey and verifies a conjecture of W.T. Gowers.
The class of countably intersected families of sets is defined. For any such family we define a Banach space not containing $\ell^{1}(\NN )$. Thus we obtain counterexamples to certain questions related to the heredity problem for W.C.G. Banach spaces. Among them we give a subspace of a W.C.G. Banach space not containing $\ell^{1}(\NN )$ and not being itself a W.C.G. space.
We introduce and investigate a class of non-separable tree-like Banach spaces. As a consequence, we prove that we can not achieve a satisfactory extension of Rosenthal's $\ell_1$-theorem to spaces of the type $\ell_1(κ)$, for $κ$ an uncountable cardinal.
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 in $\ell_1$ are considered.
We prove that, when $2<p<\infty$, in the free Banach lattice generated by $\ell_p$ (respectively by $c_0$), the absolute values of the canonical basis form an $\ell_r$-sequence, where $\frac{1}{r} = \frac{1}{2} + \frac{1}{p}$ (respectively an $\ell_2$-sequence). In particular, in any Banach lattice, the absolute values of any $\ell_p$ sequence always have an upper $\ell_r$-estimate. Quite surprisingly, this implies that the free Banach lattices generated by the nonseparable $\ell_p(Γ)$ for $2<p<\infty$, as well as $c_0(Γ)$, are weakly compactly generated whereas this is not the case for $1\leq…
We present condition on higher order asymptotic behaviour of basic sequences in a Banach space ensuring the existence of bounded non-compact strictly singular operator on a subspace. We apply it in asymptotic $\ell_p$ spaces, $1\leq p<\infty$, in particular in convexified mixed Tsirelson spaces and related asymptotic $\ell_p$ HI spaces.
In this note we study Banach spaces of traces of real polynomials on $\mathbb R^n$ to compact subsets equipped with supremum norms from the point of view of Geometric Functional Analysis.
In this paper we define the module extension dual Banach algebras and we use this Banach algebras to finding the relationship between $weak^*-$continuous homomorphisms of dual Banach algebras and Connes-amenability. So we study the $weak^*-$continuous derivations on module extension dual Banach algebras.
Two examples of asymptotic $\ell_{1}$ Banach spaces are given. The first, $X_{u}$, has an unconditional basis and is arbitrarily distortable. The second, $X$, does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson. We thus answer a question raised by W.T.Gowers.
See title. (A Banach space is said to be L-embedded if it is complemented in its bidual such that the norm between the two complementary subspaces is additive.)
We give sufficient conditions on a Banach space $X$ which ensure that $\ell_{\infty}$ embeds in $\mathcal{L}(X)$, the space of all operators on $X$. We say that a basic sequence $(e_n)$ is quasisubsymmetric if for any two increasing sequences $(k_n)$ and $(\ell_n)$ of positive integers with $k_n \leq \ell_n$ for all $n$, we have that $(e_{k_n})$ dominates $(e_{\ell_n})$. We prove that if a Banach space $X$ has a seminormalized quasisubsymmetric basis then $\ell_{\infty}$ embeds in $\mathcal{L}(X)$.
The main purpose of this paper is to study some geometric and topological properties on $ c_0$ sum of the finite dimensional Banach lattice $\ell_2^n$, its dual and its bidual. Among other results, we show that the Banach lattices $c_0(\ell_2^n)$ has the strong Gelfand-Philips property, but does not have the positive Grothendieck property. We also prove that the closed unit ball of $l_{\infty}(\ell_2^n)$ is an almost limited set.
Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space $X$ and every collection $Z_i, i\in I$ of strictly convex Banach spaces every non-expansive bijection from the unit ball of $X$ to the unit ball of sum of $Z_i$ by $\ell_1$ is an isometry.
In this paper we first show that if $X$ is a Banach space and $α$ is a left invariant crossnorm on $\ell_\infty\otimes X$, then there is a Banach lattice $L$ and an isometric embedding $J$ of $X$ into $L$, so that $I\otimes J$ becomes an isometry of $\ell_\infty\otimes_αX$ onto $\ell_\infty\otimes_m J(X)$. Here $I$ denotes the identity operator on $\ell_\infty$ and $\ell_\infty\otimes_m J(X)$ the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to characterize the Gordon-Lewis property $\GL$ in terms of emb…
We study the Bishop-Phelps-Bollobás property for operators from $\ell_\infty ^4 $ to a Banach space. For this reason we introduce an appropiate geometric property, namely the AHSp-$\ell_\infty ^4$. We prove that spaces $Y$satisfying AHSp-$\ell_\infty ^4$ are precisely those spaces $Y$ such that $(\ell_\infty^4,Y)$ has the Bishop-Phelps-Bollobás property. We also provide classes of Banach spaces satisfying this condition. For instance, finite-dimensional spaces, uniformly convex spaces, $C_0(L)$ and $L_1 (μ)$ satisfy AHSp-$\ell_\infty ^4 $.
The notion of linear Hahn-Banach extension operator was first studied in detail by Heinrich and Mankiewicz (1982). Previously, J. Lindenstrauss (1966) studied similar versions of this notion in the context of non separable reflexive Banach spaces. Subsequently, Sims and Yost (1989) proved the existence of linear Hahn-Banach extension operators via interspersing subspaces in a purely Banach space theoretic set up. In this paper, we study similar questions in the context of Banach modules and module homomorphisms, in particular, Banach algebras of operators on Banach spaces. Based on Dales, Kani…
We construct a class of minimal trees and use these trees to establish a number of coloring theorems on general trees. Among the applications of these trees and coloring theorems are quantification of the Bourgain $\ell_p$ and $c_0$ indices, dualization of the Bourgain $c_0$ index, establishing sharp positive and negative results for constant reduction, and estimating the Bourgain $\ell_p$ index of an arbitrary Banach space $X$ in terms of a subspace $Y$ and the quotient $X/Y$.
Using upper $\ell_p$-estimates for normalized weakly null sequence images, we describe a new family of operator ideals $\mathcal{WD}_{\ell_p}^{(\infty,ξ)}$ with parameters $1\leq p\leq\infty$ and $1\leqξ\leqω_1$. These classes contain the completely continuous operators, and are distinct for all choices $1\leq p\leq\infty$ and, when $p\neq 1$, for all choices $ξ\neqω_1$. For the case $ξ=1$, there exists an ideal norm $\|\cdot\|_{(p,1)}$ on the class $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ under which it forms a Banach ideal.
We classify the closed ideals of bounded operators acting on the Banach spaces $\left(\bigoplus_{n \in \mathbb{N}} \ell_2^n\right)_{c_0} \oplus c_0(Γ)$ and $\left(\bigoplus_{n \in \mathbb{N}} \ell_2^n\right)_{\ell_1} \oplus \ell_1(Γ)$ for every uncountable cardinal $Γ$.
We characterize $k-$smoothness of an element on the unit sphere of a finite-dimensional polyhedral Banach space. Then we study $k-$smoothness of an operator $T \in \mathbb{L}(\ell_{\infty}^n,\mathbb{Y}),$ where $\mathbb{Y}$ is a two-dimensional Banach space with the additional condition that $T$ attains norm at each extreme point of $B_{\ell_{\infty}^{n}}.$ We also characterize $k-$smoothness of an operator defined between $\ell_{\infty}^3$ and $\ell_{1}^3.$
We introduce a notion of p-orthogonality in a general Banach space $1 \le p \le \infty$. We use this concept to characterize $\ell_p$-spaces among Banach spaces and also among complete order smooth p-normed spaces. We further introduce a notion of $p$-orthogonal decomposition in order smooth p-normed spaces. We prove that if the $\infty$-orthogonal decomposition holds in an order smooth $\infty$-normed space, then the 1-orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique.
In this article we shall focus on the derivations on module extension of Banach algebras and determine the general structure of them. Then we obtain some results concerning the automatic continuity of these mappings.
We prove that an onto isometry between unit spheres of finite-dimensional polyhedral Banach spaces extends to a linear isometry of the corresponding spaces.
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 ergodicity, we prove that there is no separable Banach space which is complementably universal for the class of all subspaces of $\ell_p$, for $1\leq p <2$. This solves a question left open by W. B. Johnson and A. Szankowski in 1976.
Let $\mathcal A$, $\mathcal B$ be Banach $\mathfrak A$-modules with compatible actions and $\mathcal M$ be a left Banach $\mathcal A$-$\mathfrak A$-module and a right Banach $\mathcal B$-$\mathfrak A$-module. In the current paper, we study module amenability, $n$-weak module amenability and module Arens regularity of the triangular Banach algebra $\mathcal T=[ {cc} \mathcal A & \mathcal M & \mathcal B ]$ (as an $\mathfrak T:=\Big{[ {cc} α& & α ] | α\in\mathfrak A\Big}$-module). We employ these results to prove that for an inverse semigroup $S$ with subsemigroup $E$ of idempotents, the triangul…
We show that the dual of every infinite-dimensional Lipschitz-free Banach space contains an isometric copy of $\ell_\infty$ and that it is often the case that a Lipschitz-free Banach space contains a $1$-complemented subspace isometric to $\ell_1$. Even though we do not know whether the latter is true for every infinite-dimensional Lipschitz-free Banach space, we show that the space is never rotund. Further, in the last section we survey the relations between "isometric embedding of~$\ell_\infty$ into the dual" and "containing as good copy of~$\ell_1$ as possible" in a general Banach space.
Isometric embeddings of separable Banach spaces into $(\ell^\infty \setminus c)\cup\{0\}$
该研究证明可分Banach空间可等距嵌入到与收敛序列空间$c$仅交于原点的$\ell^\infty$子集中，且相关可分子空间可保避开$c$的性质扩展为包含任意可分Banach空间等距副本的子空间，推广了经典嵌入结论。
Notes on Bernstein spaces, sampling, Boas interpolation formulas and their extensions to Banach spaces
本文综述了调和分析中经典结果及其在巴纳赫空间中的扩展，涵盖伯恩斯坦空间、香农采样和博斯插值公式等核心内容。
We shall develop two notions of pointwise amenability, namely pointwise Connes amenability and pointwise $w^*$-approximate Connes amenability, for dual Banach algebras which take the $w^*$-topology into account. We shall study these concepts for the Banach sequence algebras $\ell^1(ω)$ and the weighted semigroup algebras $ \ell^{1}(\mathbb{N}_{\wedge},ω)$. For a weight $ω$ on a discrete semigroup $S$, we shall investigate pointwise amenability/Connes amenability of $\ell^1(S,ω)$ in terms of diagonals.
We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize $c_0$ in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace.
On the numerical radius of weighted shifts on $\ell^{2}$ as a norm on $\ell^{\infty}$ and connections to Banach limits
本文研究了 $\ell^{2}$ 上加权移位的数值半径作为 $\ell^{\infty}$ 上范数的性质，证明了其与 Banach 极限、谱半径间的取值关系，并比较了不同算子范数的异同。
We construct for each $0<p\le 1$ an infinite collection of subspaces of $\ell_p$ that extend the example from [J. Lindenstrauss, On a certain subspace of $\ell_{1}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 539-542] of a subspace of $\ell_{1}$ with no unconditional basis. The structure of this new class of $p$-Banach spaces is analyzed and some applications to the general theory of $\mathcal{L}_{p}$-spaces for $0<p<1$ are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in $p$-Banach spaces for $p<1$…
On the complete separation of unique $\ell_{1}$ spreading models and the Lebesgue property of Banach spaces
本文构造了一个反射Banach空间，其扩散模型与$\ell_1$的单位向量基均匀等价，但其所有无限维闭子空间均不满足Lebesgue性质。
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 problem, which asks whether a separable Banach space with a transitive norm has to be isometric or isomorphic to a Hilbert space. We obtain this as a consequence of a new property of almost transitive spaces with a Schauder basis, namely we prove that in such spaces the unit vector basis of $\ell_2^2$ belongs to the two-dimensional asymptotic structure and w…
Let $A$ and $X$ be Banach algebras and let $X$ be an algebraic Banach $A-$module. Then the $\ell^1-$direct sum $A\times X$ equipped with the multiplication $$(a,x)(b,y)=(ab, ay+xb+xy)\quad (a,b\in A, x,y\in X)$$ is a Banach algebra, denoted by $A\bowtie X$, which will be called "\textit{a generalized module extension Banach algebra}". Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. We characterize the structure of $n-$dual valued ($n\in\mathbb N)$) derivations on $A\bowtie X$ from which we investigate the $n-$we…
Let $X$ be a Banach space with a separable dual $X^{*}$. Let $Y\subset X$ be a closed subspace, and $f:Y\to\mathbb{R}$ a $C^{1}$-smooth function. Then we show there is a $C^{1}$ extension of $f$ to $X$.
In 1989, G. Godefroy proved that a Banach space contains an isomorphic copy of $\ell_1$ if and only if it can be equivalently renormed to be octahedral. It is known that octahedral norms can be characterized by means of covering the unit sphere by a finite number of balls. This observation allows us to connect the theory of octahedral norms with ball-covering properties of Banach spaces introduced by L. Cheng in 2006. Following this idea, we extend G. Godefroy's result to higher cardinalities. We prove that, for an infinite cardinal $κ$, a Banach space $X$ contains an isomorphic copy of $\ell_…
In this paper we define module biprojctivity and module biflatness for a Banach algebra which is a Banach module over another Banach algebra with compatible actions, and find their relation to classical biprojectivity and biflatness. As a typical example, We show that for an inverse semigroup $S$ with an upward directed set of idempotents $E$, the semigroup algebra $ \ell ^{1}(S)$, as an $\ell ^{1}(E)$-module, is module biprojective if and only if an appropriate group homomorphic image of $S$ is finite. Also we show that $\ell ^{1}(S)$ is module biflat if and only if $S$ is amenable.
Corrigendum to "On Smooth extensions of vector-valued functions defined on closed subsets of Banach spaces" [Math. Ann. (2013) 355: 1201--1219]
本文是对2013年发表于Math. Ann.关于Banach空间闭子集上向量值函数光滑延拓论文的勘误说明。
We show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing the harmonic extension we prove existence and uniqueness of a bounded solution (i.e. of the harmonic extension).
In the paper is considered two problems on extension of operators whose range space for the first problem (or domain space for the second one) belongs to the fixed class of finite equivalence, which is generated by a given Banach space $X$. Both problems are considered in two variants: isometric and isomorphic ones. Received a full solution of the first problem and the solution, close to final one for the second problem.
Targeting at sparse multi-task learning, we consider regularization models with an $\ell^1$ penalty on the coefficients of kernel functions. In order to provide a kernel method for this model, we construct a class of vector-valued reproducing kernel Banach spaces with the $\ell^1$ norm. The notion of multi-task admissible kernels is proposed so that the constructed spaces could have desirable properties including the crucial linear representer theorem. Such kernels are related to bounded Lebesgue constants of a kernel interpolation question. We study the Lebesgue constant of multi-task kernels…
In the present paper, the concepts of module (uniform) approximate amenability and contractibility of Banach algebras that are modules over another Banach algebra, are introduced. The general theory is developed and some hereditary properties are given. In analogy with the Banach algebraic approximate amenability, it is shown that module approximate amenability and contractibility are the same properties. It is also shown that module uniform approximate (contractibility) amenability and module (contractibility, respectively) amenability for commutative Banach modules are equivalent. Applying t…
Plasticity of the unit ball of the real Banach space $\ell_\infty$
研究实巴拿赫空间$\ell_\infty$单位球的可塑性问题，通过证明单位球上非扩张双射将极点映射到极点，利用半径为一的球的基本覆盖及Fakhoury定理得出单位球是可塑的。
Krivine and Maurey proved in 1981 that every stable Banach space contains almost isometric copies of $\ell_p$, for some $p\in[1,\infty)$. In 1983, Raynaud showed that if a Banach space uniformly embeds into a superstable Banach space, then $X$ must contain an isomorphic copy of $\ell_p$, for some $p\in[1,\infty)$. In these notes, we show that if a Banach space coarsely embeds into a superstable Banach space, then $X$ has a spreading model isomorphic to $\ell_p$, for some $p\in[1,\infty)$. In particular, we obtain that there exist reflexive Banach spaces which do not coarsely embed into any sup…
We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $\ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $\ell_1$ and that operators which do not fix copies of $\ell_1$ on a space with the alternati…
We characterise those Banach spaces $X$ which satisfy that $L(Y,X)$ is octahedral for every non-zero Banach space $Y$. They are those satisfying that, for every finite dimensional subspace $Z$, $\ell_\infty$ can be finitely-representable in a part of $X$ kind of $\ell_1$-orthogonal to $Z$. We also prove that $L(Y,X)$ is octahedral for every $Y$ if, and only if, $L(\ell_p^n,X)$ is octahedral for every $n\in\mathbb N$ and $1<p<\infty$. Finally, we find examples of Banach spaces satisfying the above conditions like $\Lip(M)$ spaces with octahedral norms or $L_1$-preduals with the Daugavet propert…
In the paper is considered two problems on extension of operators whose range space for the first problem (or domain space for the second one) belongs to the fixed class of finite equivalence, which is generated by a given Banach space $X$. Both problems are considered in two variants: isometric and isomorphic ones. Received a full solution of the first problem and the solution, close to final one for the second problem.
In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In this paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions on the singular values for compact Hilbert space operators …
We study spaces with directionally asymptotically controlled ellipsoids approximating the unit ball in finite-dimensions. These ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids with a given functional of the dual space. The term asymptotical refers to the fact that we take '$\limsup$' over finite-dimensional subspaces. This leads to some isomorphic and isometric characterizations of Hilbert spaces. An application involving Mazur's rotation problem…
We show that there exists a Banach space $E$ with the following properties: the Banach algebra $\mathscr{B}(E)$ of bounded, linear operators on $E$ has a singular extension which splits algebraically, but it does not split strongly, and the homological bidimension of $\mathscr{B}(E)$ is at least two. The first of these conclusions solves a natural problem left open by Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), while the second answers a question of Helemskii. The Banach space $E$ that we use was originally introduced by Read (J. London Math. Soc. 1989).
We give a simple example of a countable metric space $M$ that does not embed bi-Lipschitz with distortion strictly less than 2 into any Asplund space. Actually, if $M$ embeds with distortion strictly less than 2 to a Banach space $X$, then $X$ contains an isomorphic copy of $\ell_1$. We also show that the space $M$ does not embed with distortion strictly less than $2$ into $\ell_1$ itself but it does embed isometrically into a space that is isomorphic to $\ell_1$.
Weak minimizing property on pairs of classical Banach spaces
本文研究了经典巴拿赫空间对的弱最小性质，证明了特定类型的对满足该性质，而其他类型的对则不满足。
In this paper we deal with those Banach spaces $Z$ which satisfy the Mazur--Ulam property, namely that every surjective isometry $Δ$ from the unit sphere of $Z$ to the unit sphere of any Banach space $Y$ admits an unique extension to a surjective real-linear isometry from $Z$ to $Y$. We prove that for every countable set $Γ$ with $\vert Γ\vert \geq 2$, the Banach space $\bigoplus_{γ\in Γ}^{c_0} X_γ$ satisfies the Mazur--Ulam property, whenever the Banach space $X_γ$ is strictly convex with dim$((X_γ)_{\mathbb{R}})\geq 2$ for every $γ$. Moreover we prove that the Banach space $C_0(K,X)$ satisfi…
A recent result of Leung (Proceedings of the American Mathematical Society, to appear) states that the Banach algebra $\mathscr{B}(X)$ of bounded, linear operators on the Banach space $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^n\bigr)_{\ell_1}$ contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^n\bigr)_{\ell_p}$ and $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_1^n\bigr)_{\ell_p}$ whenever $p\in(1,\infty)$.
In this paper, first we study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$ be the topological centers of the left module action $π_\ell:~A\times B\rightarrow B$ and the right module action $π_r:~B\times A\rightarrow B$, respectively. We investigate some relationships between topological center of $A^{**}$, ${Z}_1({A^{**}})$ with respect to the first Arens product and topological centers of module actions ${Z}^\ell_{B^{**}}(A^{**})$ and ${Z}^\ell_{A^{**}}(B^{**})$. On the other hand, if $…
The maximum of the Banach-Mazur distance $d_{BM}^M(X,\ell_\infty^n)$, where $X$ ranges over the set of all $n$-dimensional real Banach spaces, is difficult to compute. In fact, it is already not easy to get the maximum of $d_{BM}^M(\ell_p^n,\ell_\infty^n)$ for all $p\in [1,\infty]$. We prove that $d_{BM}^M(\ell_p^3,\ell_\infty^3)\leq 9/5,~\forall p\in[1,\infty]$. As an application, the following result related to Borsuk's partition problem in Banach spaces is obtained: any subset $A$ of $\ell_p^3$ having diameter $1$ is the union of $8$ subsets of $A$ whose diameters are at most $0.9$.
We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in $\ell_1(\mathbb{N}_0)$ is open. By appealing to ultraproduct techniques, we demonstrate that neither in $\ell_1(\mathbb{Z})$ nor in $\ell_1(\mathbb Q)$ convolution is uniform…
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