Compact Approximation Property Research Articles

Bounded compact and dual compact approximation properties of Hardy spaces: New results and open problems

The aim of the paper is to highlight some open problems concerning approximation properties of Hardy spaces. We also present some results on the bounded compact and the dual compact approximation properties (shortly, BCAP and DCAP) of such spaces, to provide background for the open problems. Namely, we consider abstract Hardy spaces H[X(w)] built upon translation-invariant Banach function spaces X with weights w such that w∈X and w−1∈X′, where X′ is the associate space of X. We prove that if X is separable, then H[X(w)] has the BCAP with the approximation constant M(H[X(w)])≤2. Moreover, if X is reflexive, then H[X(w)] has the BCAP and the DCAP with the approximation constants M(H[X(w)])≤2 and M∗(H[X(w)])≤2, respectively. In the case of classical weighted Hardy space Hp(w)=H[Lp(w)] with 1<p<∞, one has a sharper result: M(Hp(w))≤2|1−2/p| and M∗(Hp(w))≤2|1−2/p|.

Extremal Structure of Projective Tensor Products

We prove that, given two Banach spaces X and Y and bounded, closed convex sets Csubseteq X and Dsubseteq Y, if a nonzero element zin {overline{{{,textrm{co},}}}}(Cotimes D)subseteq Xwidehat{otimes }_pi Y is a preserved extreme point then z=x_0otimes y_0 for some preserved extreme points x_0in C and y_0in D, whenever K(X,Y^*) separates points of X widehat{otimes }_pi Y (in particular, whenever X or Y has the compact approximation property). Moreover, we prove that if x_0in C and y_0in D are weak-strongly exposed points then x_0otimes y_0 is weak-strongly exposed in {overline{{{,textrm{co},}}}}(Cotimes D) whenever x_0otimes y_0 has a neighbourhood system for the weak topology defined by compact operators. Furthermore, we find a Banach space X isomorphic to ell _2 with a weak-strongly exposed point x_0in B_X such that x_0otimes x_0 is not a weak-strongly exposed point of the unit ball of Xwidehat{otimes }_pi X.

A note on numerical radius attaining mappings

We prove that if every bounded linear operator (or N N -homogeneous polynomials) on a Banach space X X with the compact approximation property attains its numerical radius, then X X is a finite dimensional space. Moreover, we present an improvement of the polynomial James’ theorem for numerical radius proved by Acosta, Becerra Guerrero and Galán [Q. J. Math. 54 (2003), pp. 1–10]. Finally, the denseness of weakly (uniformly) continuous 2 2 -homogeneous polynomials on a Banach space whose Aron-Berner extensions attain their numerical radii is obtained.

Quantitative Characterizations of Bounded Compact Approximation Property by Semi-Fredholm Operators

Quantitative Characterizations of Bounded Compact Approximation Property by Semi-Fredholm Operators

Sharp estimates for conditionally centered moments and for compact operators on Lp$L^p$ spaces

AbstractLet be a probability space, ξ be a random variable on , be a sub‐σ‐algebra of , and let be the corresponding conditional expectation operator. We obtain sharp estimates for the moments of in terms of the moments of ξ. This allows us to find the optimal constant in the bounded compact approximation property of , .

Characterizations of bounded compact approximation property by Calkin representations

In this note we characterize the bounded compact approximation property via Calkin representations for Banach spaces.

EQUIVALENCE OF SEMI-NORMS RELATED TO SUPER WEAKLY COMPACT OPERATORS

AbstractWe study super weakly compact operators through a quantitative method. We introduce a semi-norm $\sigma (T)$ of an operator $T:X\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.

On Absolute Uniform Retracts, Uniform Approximation Property and Super Weakly Compact Sets of Banach Spaces

In this paper, we show that every super weakly compact convex subset of a Banach space is an absolute uniform retract, and that it also admits the uniform compact approximation property. These can be regarded as extensions of Lindenstrauss and Kalton’s corresponding results.

The compact approximation property for spaces of holomorphic mappings on Fréchet spaces

In this paper, the compact approximation property on Frechet spaces is characterized in terms of holomorphic mappings. We show that a Frechet space E has the compact approximation property if and only if every holomorphic mapping on a balanced open subset $$U\subset E$$ with values in a Frechet space can be approximated uniformly on compact subsets of U by compact holomorphic mappings. This extends the well-known linear characterization to the holomorphic setting. We also give characterizations of the compact approximation property in terms of bounded holomorphic mappings on Banach spaces.

Strongly Extreme Points and Approximation Properties

AbstractWe show that if x is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at x, then x is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the suõcient conditions mentioned.

Homological properties of the algebra of compact operators on a Banach space

The conditions on a Banach space $E$ under which the algebra $\mathcal {K}(E)$ of compact operators on $E$ is right flat or homologically unital are investigated. These homological properties are related to factorization in the algebra, and, it is shown that, for $\mathcal {K}(E)$, they are closely associated with the approximation property for $E$. The class of spaces $E$ such that $\mathcal {K}(E)$ is known to be right flat and homologically unital is extended to include spaces which do not have the bounded compact approximation property.

Weak sequential completeness of spaces of homogeneous polynomials

Let Pw(En;F) be the space of all continuous n-homogeneous polynomials from a Banach space E into another F, that are weakly continuous on bounded sets. We give sufficient conditions for the weak sequential completeness of Pw(En;F). These sufficient conditions are also necessary if both E⁎ and F have the bounded compact approximation property. We also show that the weak sequential completeness and the reflexivity of Pw(En;F) are equivalent whenever both E and F are reflexive.

On mean ergodic convergence in the Calkin algebras

In this paper, we give a geometric characterization of mean ergodic convergence in the Calkin algebras for Banach spaces that have the bounded compact approximation property.

Weak Amenability of Locally Compact Quantum Groups and Approximation Properties of Extended Quantum SU(1, 1)

We study weak amenability for locally compact quantum groups in the sense of Kustermans and Vaes. In particular, we focus on non-discrete examples. We prove that a coamenable quantum group is weakly amenable if there exists a net of positive, scaling invariant elements in the Fourier algebra A(G) whose representing multipliers form an approximate identity in C_0(G) that is bounded in the M_0A(G) norm; the bound being an upper estimate for the associated Cowling-Haagerup constant. As an application, we find the appropriate approximation properties of the extended quantum SU(1,1) group and its dual. That is, we prove that it is weakly amenable and coamenable. Furthermore, it has the Haagerup property in the quantum group sense, introduced by Daws, Fima, Skalski and White.

Weak Sequential Completeness of 𝑲(X,Y)

AbstractFor Banach spaces X and Y, we show that if X* and Y are weakly sequentially complete and every weakly compact operator from X to Y is compact, then the space of all compact operators from X to Y is weakly sequentially complete. The converse is also true if, in addition, either X* or Y has the bounded compact approximation property.

The Banach ideal of [formula omitted]-compact operators and related approximation properties

We use the notion of A-compact sets (determined by an operator ideal A), introduced by Carl and Stephani (1984), to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework. For Banach operator ideals A, we introduce a way to measure the size of A-compact sets and use it to give a norm on KA, the ideal of A-compact operators. Then, we study two types of approximation properties determined by A-compact sets. We focus our attention on an approximation property which makes use of the norm defined on KA. This notion fits the definition of the A-approximation property, recently introduced by Oja (2012), with KA instead of A. We exemplify the power of the Carl–Stephani theory and the geometric structure introduced here by appealing to some recent developments on p-compactness.

Three-space problems for the bounded compact approximation property

National Natural Science Foundation of China [10526034, 10701063]; Fundamental Research Funds for the Central Universities [2011121039]; NSF [DMS-0800061, DMS-1068838]

The weakly compact approximation of the projective tensor product of Banach spaces

We show that X ^ ›Y , the projective tensor product of Banach spaces X and Y , has the (bounded) compact approximation property if and only if both X and Y have the same property. We also show that X ^ ›Y has the weakly compact approximation property (W.A.P.) if both X and Y has the W.A.P. and either (i) every bounded linear operator from X (resp. from Y ) to Y ⁄ (resp. to X ⁄ ) is completely continuous, or (ii) one of X and Y has the Dunford-Pettis property. As a consequence, we show that if K is scattered and Y has the W.A.P., then C(K) ⁄^ ›Y has the W.A.P.

Erratum to: Ideals of Homogeneous Polynomials and Weakly Compact Approximation Property in Banach Spaces

In the proof of the implication (c) =⇒ (a) of Theorem 11 we state that P ∈ PW( E; F ) since P = Q ◦ R with R ∈ W(E; G), Q ∈ P(G; F ) and G Banach. But since PL[W] is not contained in PW (see, e.g., example 27 in G. Botelho: Ideals of polynomials generated by weakly compact operators, Note Mat. 25 (2006), 69–102) this conclusion is not true in general. So the given proof of this implication is not true. The proof of Theorem 11 can be completed as follows: (c) =⇒ (a): By following the pattern of the proof of Lemma 9, having the equality P[W]( E; F ) = PL[W]( E; F ) in mind, one can prove that if P(E; F ) = P[W](E; F ) τc for every Banach space F and for some m ∈ N then, L(E; F ) = W(E; F ) τc for every Banach space F . This shows that E has the WCAP.

On commuting approximation properties of Banach spaces

Let a, c ≥ 0 and let B be a compact set of scalars. We show that if X is a Banach space such that the canonical projection π from X*** onto X* satisfies the inequalityand 1 ≤ λ &lt; max |B| + c, then every λ-commuting bounded compact approximation of the identity of X is shrinking. This generalizes a theorem by Godefroy and Saphar from 1988. As an application, we show that under the conditions described above both X and X* have the metric compact approximation property (MCAP). Relying on the Willis construction, we show that the commuting MCAP does not imply the approximation property.