Weakly amenable groups and the RNP for some Banach algebras related to the Fourier algebra

Edmond E. Granirer

Displaying similar documents to “Weakly amenable groups and the RNP for some Banach algebras related to the Fourier algebra”

On projectional skeletons in Vašák spaces

Ondřej F. K. Kalenda (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We provide an alternative proof of the theorem saying that any Vašák (or, weakly countably determined) Banach space admits a full $1$ -projectional skeleton. The proof is done with the use of the method of elementary submodels and is comparably simple as the proof given by W. Kubiś (2009) in case of weakly compactly generated spaces.

(Non-)amenability of ℬ(E)

Volker Runde (2010)

Banach Center Publications

Similarity:

In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that $ℬ (E) = (E) + ℂ i d_{E}$ . Still, ℬ(ℓ²) is...

Operator Segal algebras in Fourier algebras

Brian E. Forrest, Nico Spronk, Peter J. Wood (2007)

Studia Mathematica

Similarity:

Let G be a locally compact group, A(G) its Fourier algebra and L¹(G) the space of Haar integrable functions on G. We study the Segal algebra S¹A(G) = A(G) ∩ L¹(G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of S¹A(G). We use it to show that the restriction operator ${u \mapsto u |}_{H} : S ¹ A (G) \to A (H)$ , for some non-open closed subgroups H, is a surjective complete quotient map. We also show that if N is a non-compact closed subgroup,...

On the compact approximation property

Vegard Lima, Åsvald Lima, Olav Nygaard (2004)

Studia Mathematica

Similarity:

We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space = S ∘ T: S compact operator on X is an ideal in = span(,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net $(S_{γ})$ of compact operators on X such that $s u p_{γ} | | S_{γ} T | | \leq | | T | |$ and $S_{γ} \to I_{X}$ in the strong operator topology. Similar results for dual spaces are also proved.

Geometry of Banach spaces and biorthogonal systems

S. Dilworth, Maria Girardi, W. Johnson (2000)

Studia Mathematica

Similarity:

A separable Banach space X contains $ℓ_{1}$ isomorphically if and only if X has a bounded fundamental total $w c_{0} *$ -stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total $w c_{0} *$ -biorthogonal system.

Factorization of vector measures and their integration operators

José Rodríguez (2016)

Colloquium Mathematicae

Similarity:

Let X be a Banach space and ν a countably additive X-valued measure defined on a σ-algebra. We discuss some generation properties of the Banach space L¹(ν) and its connection with uniform Eberlein compacta. In this way, we provide a new proof that L¹(ν) is weakly compactly generated and embeds isomorphically into a Hilbert generated Banach space. The Davis-Figiel-Johnson-Pełczyński factorization of the integration operator $I_{ν} : L ¹ (ν) \to X$ is also analyzed. As a result, we prove that if $I_{ν}$ is both completely...

Some results on $A_{p} (G)$ submodules of $P M_{p} (G)$

Edmond E. Granirer (1987)

Colloquium Mathematicae

Similarity:

On the existence of non-linear frames

Shah Jahan, Varinder Kumar, S.K. Kaushik (2017)

Archivum Mathematicum

Similarity:

A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if $𝒳$ is a Banach space such that $𝒳^{*}$ has a SRBF, then $𝒳$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $𝒳$ has an approximative Schauder frame, then $𝒳^{*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.

On Banach spaces C(K) isomorphic to c₀(Γ)

Witold Marciszewski (2003)

Studia Mathematica

Similarity:

We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight $ω_{ω}$ and with the third derived set $K^{(3)}$ empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and $c ₀ (ω_{ω})$ are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

Linearization of isometric embedding on Banach spaces

Yu Zhou, Zihou Zhang, Chunyan Liu (2015)

Studia Mathematica

Similarity:

Let X,Y be Banach spaces, f: X → Y be an isometry with f(0) = 0, and $T : \bar{s p a n} (f (X)) \to X$ be the Figiel operator with $T \circ f = I d_{X}$ and ||T|| = 1. We present a sufficient and necessary condition for the Figiel operator T to admit a linear isometric right inverse. We also prove that such a right inverse exists when $\bar{s p a n} (f (X))$ is weakly nearly strictly convex.

On asymptotically symmetric Banach spaces

M. Junge, D. Kutzarova, E. Odell (2006)

Studia Mathematica

Similarity:

A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences ${(x_{j}^{i})}_{j = 1}^{\infty} \subseteq X$ , 1 ≤ i ≤ m, for all permutations σ of 1,...,m and all ultrafilters ₁,...,ₘ on ℕ, $l i m_{n ₁, ₁} . . . l i m_{n ₘ, ₘ} | | \sum_{i = 1}^{m} x_{n_{i}}^{i} | | \leq C l i m_{n_{σ (1)},_{σ (1)}} . . . l i m_{n_{σ (m)},_{σ (m)}} | | \sum_{i = 1}^{m} x_{n_{i}}^{i} | |$ . We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences ${(x_{j}^{i})}_{j = 1}^{\infty}$ . Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then...

A note on a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

Colloquium Mathematicae

Similarity:

This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F (x) : = F ̃ x^{(2 n + 1)}$ where $F ̃ : X^{2 n + 1} \to Y$ is a continuous (2n+1)-linear operator.

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

Similarity:

Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and $ℓ^{\infty}$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then $ℓ^{\infty}$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X \otimes_{γ} Y *$ . Applications to embeddings of c₀ in various spaces of operators are given.

On diffeomorphisms deleting weak compacta in Banach spaces

Daniel Azagra, Alejandro Montesinos (2004)

Studia Mathematica

Similarity:

We prove that if X is an infinite-dimensional Banach space with $C^{p}$ smooth partitions of unity then X and X∖ K are $C^{p}$ diffeomorphic for every weakly compact set K ⊂ X.

Estimation of the Szlenk index of Banach spaces via Schreier spaces

Ryan Causey (2013)

Studia Mathematica

Similarity:

For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index $ω^{α + 1}$ which is universal for the class of separable Banach spaces with Szlenk index not exceeding $ω^{α}$ . Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

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...

Order-bounded operators from vector-valued function spaces to Banach spaces

Marian Nowak (2005)

Banach Center Publications

Similarity:

Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space $(X, | | \cdot | |_{X})$ let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function ${| | f (\cdot) | |}_{X}$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_{u} (= {f \in E (X) : | | f (\cdot) | |}_{X} \leq u)$ stand for the order interval in E(X). For a real Banach space $(Y, | | \cdot | |_{Y})$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈...

The Lindelöf property in Banach spaces

B. Cascales, I. Namioka, J. Orihuela (2003)

Studia Mathematica

Similarity:

A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^{D}$ the following four conditions are equivalent: (i) K is fragmented by $d_{D}$ , where, for each S ⊂ D, $d_{S} (x, y) = s u p ϱ (x (t), y (t)) : t \in S$ . (ii) For each countable subset...

Application of $(L)$ sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H&#039;michane, Aziz Elbour (2016)

Mathematica Bohemica

Similarity:

The paper contains some applications of the notion of $Ł$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $(L)$ -Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $(L)$ sets. As a sequence characterization of such operators, we see that an operator $T : X \to E$ from a Banach space into a Banach lattice is order $Ł$ -Dunford-Pettis, if and only if $| T (x_{n}) | \to 0$ for $σ (E, E^{'})$ for every...

Operator Figà-Talamanca-Herz algebras

Volker Runde (2003)

Studia Mathematica

Similarity:

Let G be a locally compact group. We use the canonical operator space structure on the spaces $L^{p} (G)$ for p ∈ [1,∞] introduced by G. Pisier to define operator space analogues $O A_{p} (G)$ of the classical Figà-Talamanca-Herz algebras $A_{p} (G)$ . If p ∈ (1,∞) is arbitrary, then $A_{p} (G) \subset O A_{p} (G)$ 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 $O A_{p} (G)$ is a completely contractive Banach algebra for each p ∈ (1,∞), and that $O A_{q} (G) \subset O A_{p} (G)$ completely contractively...

Banach spaces of bounded Szlenk index II

D. Freeman, E. Odell, Th. Schlumprecht, A. Zsák (2009)

Fundamenta Mathematicae

Similarity:

For every α < ω₁ we establish the existence of a separable Banach space whose Szlenk index is $ω^{α ω + 1}$ and which is universal for all separable Banach spaces whose Szlenk index does not exceed $ω^{α ω}$ . In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with Tsirelson type upper estimates.

James boundaries and σ-fragmented selectors

B. Cascales, M. Muñoz, J. Orihuela (2008)

Studia Mathematica

Similarity:

We study the boundary structure for w*-compact subsets of dual Banach spaces. To be more precise, for a Banach space X, 0 < ϵ < 1 and a subset T of the dual space X* such that ⋃ B(t,ϵ): t ∈ T contains a James boundary for $B_{X *}$ we study different kinds of conditions on T, besides T being countable, which ensure that $X * = {\bar{s p a n T}}^{| | \cdot | |}$ . (SP) We analyze two different non-separable cases where the equality (SP) holds: (a) if $J : X \to 2^{B_{X *}}$ is the duality mapping and there exists a σ-fragmented map f: X → X* such that...

Reflexivity and approximate fixed points

Eva Matoušková, Simeon Reich (2003)

Studia Mathematica

Similarity:

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 $s u p_{f \in S_{X *}} l i m s u p_{k \to \infty} f (x_{n_{k}})$ 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 $l i m s u p_{n \to \infty} f (x ₙ) < l i m i n f_{n \to \infty} g (x ₙ)$ . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

The Dual of a Non-reflexive L-embedded Banach Space Contains $l^{\infty}$ Isometrically

Hermann Pfitzner (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

A Banach space is said to be L-embedded if it is complemented in its bidual in such a way that the norm between the two complementary subspaces is additive. We prove that the dual of a non-reflexive L-embedded Banach space contains $l^{\infty}$ isometrically.

Extension and lifting of weakly continuous polynomials

Raffaella Cilia, Joaquín M. Gutiérrez (2005)

Studia Mathematica

Similarity:

We show that a Banach space X is an ℒ₁-space (respectively, an $ℒ_{\infty}$ -space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space $_{w b} (^{m} X)$ of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an $ℒ_{\infty}$ -space.

Weakly null sequences with upper estimates

Daniel Freeman (2008)

Studia Mathematica

Similarity:

We prove that if $(v_{i})$ is a seminormalized basic sequence and X is a Banach space such that every normalized weakly null sequence in X has a subsequence that is dominated by $(v_{i})$ , then there exists a uniform constant C ≥ 1 such that every normalized weakly null sequence in X has a subsequence that is C-dominated by $(v_{i})$ . This extends a result of Knaust and Odell, who proved this for the cases in which $(v_{i})$ is the standard basis for $ℓ_{p}$ or c₀.

On (Co)homology of triangular Banach algebras

Mohammad Sal Moslehian (2005)

Banach Center Publications

Similarity:

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, $= [\begin{matrix} A & M \\ 0 & B \end{matrix}]$ is the triangular Banach algebra, X is a unital -bimodule, $X_{A A} = 1_{A} X 1_{A}$ , $X_{B B} = 1_{B} X 1_{B}$ , $X_{A B} = 1_{A} X 1_{B}$ and $X_{B A} = 1_{B} X 1_{A}$ . Applying two nice long exact sequences related to A, B, , X, $X_{A A}$ , $X_{B B}$ , $X_{A B}$ and $X_{B A}$ we establish some results on (co)homology of triangular Banach algebras.

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

Similarity:

A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{\infty} : L^{\infty} (X) \to L ¹ (X)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T \circ i_{\infty} : L^{\infty} (X) \to Y$ is a weakly compact operator. Moreover, we obtain that...

Estimation of the Szlenk index of reflexive Banach spaces using generalized Baernstein spaces

Ryan Causey (2015)

Fundamenta Mathematicae

Similarity:

For each ordinal α < ω₁, we prove the existence of a separable, reflexive Banach space W with a basis so that Sz(W), $S z (W *) \leq ω^{α + 1}$ which is universal for the class of separable, reflexive Banach spaces X satisfying Sz(X), $S z (X *) \leq ω^{α}$ .

On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity

Artur Michalak (2003)

Studia Mathematica

Similarity:

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. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\bar{l i n f ([0, 1])}$ contains an order isomorphic copy of D(0,1), (2) $\bar{l i n f (Q)}$ contains an isomorphic copy of C([0,1]), (3) $\bar{l i n f ([0, 1])} / \bar{l i n f (Q)}$ contains an isomorphic copy of c₀(Γ) for some uncountable...

The extension of the Krein-Šmulian theorem for order-continuous Banach lattices

Antonio S. Granero, Marcos Sánchez (2008)

Banach Center Publications

Similarity:

If X is a Banach space and C ⊂ X a convex subset, for x** ∈ X** and A ⊂ X** let d(x**,C) = inf||x**-x||: x ∈ C be the distance from x** to C and d̂(A,C) = supd(a,C): a ∈ A. Among other things, we prove that if X is an order-continuous Banach lattice and K is a w*-compact subset of X** we have: (i) $d ̂ ({\bar{c o}}^{w *} (K), X) \leq 2 d ̂ (K, X)$ and, if K ∩ X is w*-dense in K, then $d ̂ ({\bar{c o}}^{w *} (K), X) = d ̂ (K, X)$ ; (ii) if X fails to have a copy of ℓ₁(ℵ₁), then $d ̂ ({\bar{c o}}^{w *} (K), X) = d ̂ (K, X)$ ; (iii) if X has a 1-symmetric basis, then $d ̂ ({\bar{c o}}^{w *} (K), X) = d ̂ (K, X)$ .

On some properties of generalized Marcinkiewicz spaces

Evgeniy Pustylnik (2001)

Studia Mathematica

Similarity:

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

Essentially Incomparable Banach Spaces of Continuous Functions

Rogério Augusto dos Santos Fajardo (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We construct, under Axiom ♢, a family ${(C (K_{ξ}))}_{ξ < 2^{(2^{ω})}}$ of indecomposable Banach spaces with few operators such that every operator from $C (K_{ξ})$ into $C (K_{η})$ is weakly compact, for all ξ ≠ η. In particular, these spaces are pairwise essentially incomparable. Assuming no additional set-theoretic axiom, we obtain this result with size $2^{ω}$ instead of $2^{(2^{ω})}$ .

Approximate amenability for Banach sequence algebras

H. G. Dales, R. J. Loy, Y. Zhang (2006)

Studia Mathematica

Similarity:

We consider when certain Banach sequence algebras A on the set ℕ are approximately amenable. Some general results are obtained, and we resolve the special cases where $A = ℓ^{p}$ for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras $ℓ^{p} (ω)$ .

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

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.

On the mutually non isomorphic $l_{p} (l_{q})$

Pilar Cembranos, Jose Mendoza (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Similarity:

In this note we survey the partial results needed to show the following general theorem: $l_{p} (l_{q}) : 1 \leq p, q \leq + \infty$ is a family of mutually non isomorphic Banach spaces. We also comment some related facts and open problems.

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Pisier's inequality revisited

Tuomas Hytönen, Assaf Naor (2013)

Studia Mathematica

Similarity:

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 $\int_{- 1, 1 ⁿ} | | \sum_{j = 1}^{n} \partial_{j} f_{j} {(ε) | |}^{p} d μ (ε) \leq^{p} \int_{- 1, 1 ⁿ} \int_{- 1, 1 ⁿ} | | \sum_{j = 1}^{n} δ_{j} Δ f_{j} (ε) {| |}^{p} d μ (ε) d μ (δ)$ , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${\partial_{j}}_{j = 1}^{n}$ and $Δ = \sum_{j = 1}^{n} \partial_{j}$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by $ⁿ_{p} (X)$ , we show that $ⁿ_{p} (X) \leq \sum_{k = 1}^{n} 1 / k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

Arens regularity of module actions

M. Eshaghi Gordji, M. Filali (2007)

Studia Mathematica

Similarity:

We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if has a brai (blai), then the right (left) module action of on * is Arens regular if and only if is reflexive. We find that Arens regularity is implied by the factorization of * or ** when is a left or a right ideal in **. The Arens regularity and strong irregularity of are related to those of the module actions of on the nth dual $^{(n)}$ of . Banach algebras for which Z( **)...

On complemented copies of c₀(ω₁) in C(Kⁿ) spaces

Leandro Candido, Piotr Koszmider (2016)

Studia Mathematica

Similarity:

Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $\otimes ̂_{ε}^{n} C (K)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $\otimes ̂_{ε}^{n} C (K)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X \otimes ̂_{ε} Y$ contains a complemented copy of c₀ if one of the infinite-dimensional...

Weak precompactness and property (V*) in spaces of compact operators

Ioana Ghenciu (2015)

Colloquium Mathematicae

Similarity:

We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w *} (E *, F)$ (resp. $K_{w *} (E *, F)$ ) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of $K_{w *} (E *, F)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then $K_{w *} (E *, F)$ has property (wV*). Suppose...

On the number of non-isomorphic subspaces of a Banach space

Valentin Ferenczi, Christian Rosendal (2005)

Studia Mathematica

Similarity:

We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let be a Banach space with an unconditional basis ${(e_{i})}_{i \in ℕ}$ ; then either there exists a perfect set P of infinite subsets of ℕ such that for any two distinct A,B ∈ P, ${[e_{i}]}_{i \in A} ≇ {[e_{i}]}_{i \in B}$ , or for a residual set of infinite subsets A of ℕ, ${[e_{i}]}_{i \in A}$ is isomorphic to , and in that case, is isomorphic to its square, to its hyperplanes, uniformly isomorphic to $\oplus {[e_{i}]}_{i \in D}$ for any D ⊂ ℕ, and isomorphic to a denumerable Schauder...

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

Similarity:

Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets $_{ν} (X)$ and $_{ν} (X)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

On the vector-valued Fourier transform and compatibility of operators

In Sook Park (2005)

Studia Mathematica

Similarity:

Let be a locally compact abelian group and let 1 < p ≤ 2. ’ is the dual group of , and p’ the conjugate exponent of p. An operator T between Banach spaces X and Y is said to be compatible with the Fourier transform $F^{}$ if $F^{} \otimes T : L_{p} () \otimes X \to L_{p^{'}} (^{'}) \otimes Y$ admits a continuous extension $[F^{}, T] : [L_{p} (), X] \to [L_{p^{'}} (^{'}), Y]$ . Let $ℱ T_{p}^{}$ denote the collection of such T’s. We show that $ℱ T_{p}^{ℝ \times} = ℱ T_{p}^{ℤ \times} = ℱ T_{p}^{ℤ ⁿ \times}$ for any and positive integer n. Moreover, if the factor group of by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then...