Let G be a locally compact group. We use the canonical operator space structure on the spaces $L^p\left(G\right)$ for p ∈ [1,∞] introduced by G. Pisier to define operator space analogues $O{A}_p\left(G\right)$ of the classical Figà-Talamanca-Herz algebras $A_p\left(G\right)$. If p ∈ (1,∞) is arbitrary, then $A_p\left(G\right)\subset O{A}_p\left(G\right)$ and the inclusion is a contraction; if p = 2, then OA₂(G) ≅ A(G) as Banach spaces, but not necessarily as operator spaces. We show that $O{A}_p\left(G\right)$ is a completely contractive Banach algebra for each p ∈ (1,∞), and that $O{A}_q\left(G\right)\subset O{A}_p\left(G\right)$ completely contractively...

A Banach space $X$ has Pełczyński’s property (V) if for every Banach space $Y$ every unconditionally converging operator $T:X\to Y$ is weakly compact. H. Pfitzner proved that $C^{*}$-algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that $C\left(K\right)$ spaces for a compact Hausdorff space $K$ enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover,...

In this paper, we define the direct sum ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the H-property if and only if each $X_i$ has the H-property, and ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the Schur property if and only if each $X_i$ has the Schur property. Moreover, we also show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ is rotund if and only if each $X_i$ is rotund.

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

We introduce and study two new classes of Banach spaces, the so-called sequentially Right Banach spaces of order $p$, and those defined by the dual property, the sequentially Right${}^{*}$ Banach spaces of order $p$ for $1\le p\le \infty$. These classes of Banach spaces are characterized by the notions of $L_p$-limited sets in the corresponding dual space and $R_p^{*}$ subsets of the involved Banach space, respectively. In particular, we investigate whether the injective tensor product of a Banach space $X$ and a reflexive Banach...

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies ${\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\partial }_j{f}_j{\left(\epsilon \right)\left|\right|}^{p}d\mu \left(\epsilon \right){\le }^p{\int }_{-1,1ⁿ}{\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\delta }_j\Delta f_j\left(\epsilon \right){\left|\right|}^{p}d\mu \left(\epsilon \right)d\mu \left(\delta \right)$, where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${{\partial }_j}_{j=1}^n$ and $\Delta ={\sum }_{j=1}^n{\partial }_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by ${ⁿ}_p\left(X\right)$, we show that ${ⁿ}_p\left(X\right)\le {\sum }_{k=1}^{n}1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

In this paper we investigate $\Delta$-weak character amenability of certain Banach algebras such as projective tensor product $A\widehat{\otimes }B$ and Lau product $A{\times }_{\theta }B$, where $A$ and $B$ are two arbitrary Banach algebras and $\theta \in \Delta \left(B\right)$, the character space of $B$. We also investigate $\Delta$-weak character amenability of abstract Segal algebras and module extension Banach algebras.

By using the concepts of limited $p$-converging operators between two Banach spaces $X$ and $Y$, $L_p$-sets and $L_p$-limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as $*$-Dunford–Pettis property of order $p$ and Pelczyński’s property of order $p$, $1\le p<\infty$.

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

Suppose that A and B are unital Banach algebras with units $1_A$ and $1_B$, respectively, M is a unital Banach A,B-module, $=\left[\begin{array}{cc}A& M\\ 0& B\end{array}\right]$ is the triangular Banach algebra, X is a unital -bimodule, $X_{AA}=1_{A}X{1}_A$, $X_{BB}=1_{B}X{1}_B$, $X_{AB}=1_{A}X{1}_B$ and $X_{BA}=1_{B}X{1}_A$. Applying two nice long exact sequences related to A, B, , X, $X_{AA}$, $X_{BB}$, $X_{AB}$ and $X_{BA}$ we establish some results on (co)homology of triangular Banach algebras.

We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f:{[0,1]}^m\to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f:{[0,1]}^m\to X$ with respect to any norming subset there exists a separately increasing function $g:{[0,1]}^m\to ℝ$ such that the sets of...

A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_k}$ for which $su{p}_{f\in S_{X*}}limsu{p}_{k\to \infty }f\left(x_{n_k}\right)$ is attained at some f in the dual unit sphere $S_{X*}$. A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f\in S_{X*}$, there exists $g\in S_{X*}$ such that $limsu{p}_{n\to \infty }f\left(xₙ\right)<limin{f}_{n\to \infty }g\left(xₙ\right)$. Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

The generalized notion of weak amenability, namely $(\varphi ,\psi )$-weak amenability, where $\varphi ,\psi$ are continuous homomorphisms on a Banach algebra $𝒜$, was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the $(\varphi ,\psi )$-weak amenability on the measure algebra $M\left(G\right)$, the group algebra $L^1\left(G\right)$ and the Segal algebra $S^1\left(G\right)$, where $G$ is a locally compact group, are studied. As a typical example, the $(\varphi ,\psi )$-weak amenability of a special semigroup algebra is shown as well.

We give a full solution of the following problems concerning the spaces $M_{\phi }\left(X⃗\right)$: (i) to what extent two functions φ and ψ should be different in order to ensure that $M_{\phi }\left(X⃗\right)\ne M_{\psi }\left(X⃗\right)$ for any nontrivial Banach couple X⃗; (ii) when an embedding $M_{\phi }\left(X⃗\right)⊊M_{\psi }\left(X⃗\right)$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_{\phi }\left(X⃗\right)$-space for some (unknown beforehand) Banach couple X⃗.

Let A be a unital Banach function algebra with character space ${\Phi }_A$. For $x\in {\Phi }_A$, let $M_x$ and $J_x$ be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of $J_x$ is connected, and that if x does not belong to the Shilov boundary of A then the set $y\in {\Phi }_A:M_x\supseteq J_y$ has an infinite connected subset. Various related results are given.

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection $a_{\alpha }:\alpha \in \subset A$ there exist α ≠ β ∈ such that $a_{\alpha }\in a_{\beta }{A}^{}$ is isomorphic to ${⨁}_{i=1}^r(ℂ\left[X\right]/X^{d_i}ℂ\left[X\right])\oplus E$, where $d₁,...,d_r\in \mathbb{N}$, and E is either $Xℂ\left[X\right]/X^{d₀}ℂ\left[X\right]$ for some d₀ ∈ ℕ or a 1-dimensional ${⨁}_{i=1}^rℂ\left[X\right]/X^{d_i}ℂ\left[X\right]$-bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.

Let G be a locally compact group, let (φ,ψ) be a complementary pair of Young functions, and let $L^{\phi }\left(G\right)$ and $L^{\psi }\left(G\right)$ be the corresponding Orlicz spaces. Under some conditions on φ, we will show that for a Banach $L^{\phi }\left(G\right)$-submodule X of $L^{\psi }\left(G\right)$, the multiplier space $Ho{m}_{L^{\phi }\left(G\right)}(L^{\phi }\left(G\right),X*)$ is a dual Banach space with predual $L^{\phi }\left(G\right)\bullet X:=\overline{span}ux:u\in L^{\phi }\left(G\right),x\in X$, where the closure is taken in the dual space of $Ho{m}_{L^{\phi }\left(G\right)}(L^{\phi }\left(G\right),X*)$. We also prove that if $\phi$ is a Δ₂-regular N-function, then $C{v}_{\phi }\left(G\right)$, the space of convolutors of $M^{\phi }\left(G\right)$, is identified with the dual of a Banach algebra of functions on G...

We investigate the amenability and its related homological notions for a class of $I\times I$-upper triangular matrix algebra, say $\mathrm{UP}(I,A)$, where $A$ is a Banach algebra equipped with a nonzero character. We show that $\mathrm{UP}(I,A)$ is pseudo-contractible (amenable) if and only if $I$ is singleton and $A$ is pseudo-contractible (amenable), respectively. We also study pseudo-amenability and approximate biprojectivity of $\mathrm{UP}(I,A)$.

Let ${\beta \left(n\right)}_{n=0}^{\infty }$ be a sequence of positive numbers and 1 ≤ p < ∞. We consider the space ${\ell }^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\infty }f̂\left(n\right)zⁿ$ such that ${\sum }_{n=0}^{\infty }{\left|f̂\left(n\right)\right|}^p{\left|\beta \left(n\right)\right|}^p<\infty$. We give some sufficient conditions for the multiplication operator, $M_z$, to be unicellular on the Banach space ${\ell }^p\left(\beta \right)$. This generalizes the main results obtained by Lu Fang [1].

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

It is known that $ℬ\left({\ell }^p\right)$ is not amenable for p = 1,2,∞, but whether or not $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞), then so are ${\ell }^{\infty }\left(ℬ\left({\ell }^p\right)\right)$ and ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$. Moreover, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable so is ${\ell }^{\infty }(,\left(E\right))$ for any index set and for any infinite-dimensional ${ℒ}^p$-space E; in particular, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable for p ∈ (1,∞), then so is ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$. We show that ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter over...

We study weakly precompact sets and operators. We show that an operator is weakly precompact if and only if its adjoint is pseudo weakly compact. We study Banach spaces with the $p$-$L$-limited${}^{*}$ and the $p$-(SR${}^{*}$) properties and characterize these classes of Banach spaces in terms of $p$-$L$-limited${}^{*}$ and $p$-Right${}^{*}$ subsets. The $p$-$L$-limited${}^{*}$ property is studied in some spaces of operators.

Let 1 ≤ p < ∞, $={\left(Xₙ\right)}_{n\in \mathbb{N}}$ be a sequence of Banach spaces and $l_p\left(\right)$ the coresponding vector valued sequence space. Let $={\left(Xₙ\right)}_{n\in \mathbb{N}}$, $={\left(Yₙ\right)}_{n\in \mathbb{N}}$ be two sequences of Banach spaces, $={\left(Vₙ\right)}_{n\in \mathbb{N}}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{}:l_p\left(\right)\to l_q\left(\right)$ by $M_{}\left({\left(xₙ\right)}_{n\in \mathbb{N}}\right):={\left(Vₙ\left(xₙ\right)\right)}_{n\in \mathbb{N}}$. We give necessary and sufficient conditions for $M_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞. ...