Bounded Approximation Property Research Articles

Retractions and the bounded approximation property in Banach spaces

In the present paper, we prove that a necessary condition for a Banach space X to admit a generating compact Lipschitz retract K, which satisfies an additional mild assumption on its shape, is that X enjoys the Bounded Approximation Property. This is a partial solution to a question raised by Godefroy and Ozawa.

Operators with the Lipschitz bounded approximation property

Operators with the Lipschitz bounded approximation property

Translation and modulation invariant Banach spaces of tempered distributions satisfy the metric approximation property

In this paper, we establish the validity of the so-called Bounded Approximation Property (BAP) for a comprehensive class of translation and modulation invariant Banach spaces (B, ∥ · ∥B) of tempered distributions on the Euclidean space [Formula: see text]. In fact, such spaces have a double module structure, over some Beurling algebra with respect to convolution, and with respect to pointwise multiplication over some Fourier Beurling algebra. Combining this double module structure with functional analytic arguments which describe the approximation of convolution operators by discrete convolutions we are able to verify the BAP, in fact, for most cases even the Metric Approximation Property (MAP) for such Banach space. The family of spaces under consideration is very rich and contains virtually all the classical function spaces relevant for mathematical analysis, as long as the Schwartz space [Formula: see text] is dense in (B, ∥ · ∥B). In particular, all the reflexive spaces in this family are included. Moreover, this family of Banach spaces is closed with respect to intersections, sums and various interpolation methods.

The Lipschitz bounded approximation property for operator ideals

In this article, we introduce the Lipschitz bounded approximation property for operator ideals. With this notion, we extend the original work done by Godefroy and Kalton and give some partial answers on the equivalence between the bounded approximation property and the Lipschitz bounded approximation property based on an arbitrary operator ideal. Furthermore, we investigate the three space problems on the preceding bounded approximation properties.

On the completely bounded approximation property of crossed products

On the completely bounded approximation property of crossed products

Strongly zero product determined Banach algebras

C⁎-algebras, group algebras, and the algebra A(X) of approximable operators on a Banach space X having the bounded approximation property are known to be zero product determined. In this paper we give a quantitative estimate of this property by showing that, for the Banach algebra A, there exists a constant α with the property that for every continuous bilinear functional φ:A×A→C there exists a continuous linear functional ξ on A such thatsup‖a‖=‖b‖=1⁡|φ(a,b)−ξ(ab)|≤αsup‖a‖=‖b‖=1,ab=0⁡|φ(a,b)| in each of the following cases: (i) A is a C⁎-algebra, in which case α=8; (ii) A=L1(G) for a locally compact group G, in which case α=60271+sin⁡π101−2sin⁡π10; (iii) A=A(X) for a Banach space X having property (A) (which is a rather strong approximation property for X), in which case α=60271+sin⁡π101−2sin⁡π10C2, where C is a constant associated with the property (A) that we require for X.

A Schauder basis for $$L_2$$ consisting of non-negative functions

We prove that \(L_2(\mathbb {R})\) contains a Schauder basis of non-negative functions. Similarly, for all \(1<p<\infty \), \(L_{p}(\mathbb {R})\) contains a Schauder basic sequence of non-negative functions such that \(L_p(\mathbb {R})\) embeds into the closed span of the sequence. We prove as well that if X is a separable Banach space with the bounded approximation property, then any set in X with dense span contains a quasi-basis (Schauder frame) for X.

Almost everywhere convergent sequences of weak∗‐to‐norm continuous operators

Let $X$ and $Y$ be Banach spaces, and $T:X^*\to Y$ be an operator. We prove that if $X$ is Asplund and $Y$ has the approximation property, then for each Radon probability $\mu$ on $(B_{X^*},w^*)$ there is a sequence of $w^*$-to-norm continuous operators $T_n:X^*\to Y$ such that $\|T_n(x^*)-T(x^*)\| \to 0$ for $\mu$-a.e. $x^*\in B_{X^*}$; if $Y$ has the $\lambda$-bounded approximation property for some $\lambda\geq 1$, then the sequence can be chosen in such a way that $\|T_n\|\leq \lambda\|T\|$ for all $n\in \mathbb{N}$. The same conclusions hold if $X$ contains no subspace isomorphic to $\ell_1$, $Y$ has the approximation property (resp., $\lambda$-bounded approximation property) and $T$ has separable range. This extends to the non-separable setting a result by Mercourakis and Stamati.

Quasi-greedy bases in ℓp (0 < p < 1) are democratic

The list of known Banach spaces whose linear geometry determines the (nonlinear) democracy functions of their quasi-greedy bases to the extent that they end up being democratic, reduces to c0, ℓ2, and all separable L1-spaces. Oddly enough, these are the only Banach spaces that, when they have an unconditional basis, it is unique. Our aim in this paper is to study the connection between quasi-greediness and democracy of bases in non-locally convex spaces. We prove that all quasi-greedy bases in ℓp for 0<p<1 (which also has a unique unconditional basis) are democratic with fundamental function of the same order as (m1/p)m=1∞. The methods we develop allow us to obtain even more, namely that the same occurs in any separable Lp-space, 0<p<1, with the bounded approximation property.

The Grauert–Oka principle in a Banach space

Let M be a Stein Banach manifold modeled on a separable Banach space with the bounded approximation property, Γ→M a (locally trivial) holomorphic Banach Lie group bundle over M, and OΓ→M(CΓ→M) the sheaf of germs of holomorphic (continuous) sections of Γ. We show that the natural map H1(M,OΓ)→H1(M,CΓ) of first cohomology sets is bijective.

E-approximation properties

In this paper, we introduce the E-approximation property and the $$E^u$$ -approximation property which generalize Sinha and Karn’s p-approximation property. Characterizations of these properties are given parallel to Lima and Oja’s results on the strong approximation property and the weak bounded approximation property. Representation theorems for the dual of $${\mathcal {L}} (X, Y)$$ under the topology of uniform convergence on E-compact sets and $$E^u$$ -compact sets are also provided. As an application, the representations are used to generalize the main theorem of Choi and Kim in [1]. These results build up fundamental bases for future investigations of these properties.

History, Developments and Open Problems on Approximation Properties

In this paper, we give a comprehensive review of the classical approximation property. Then, we present some important results on modern variants, such as the weak bounded approximation property, the strong approximation property and p-approximation property. Most recent progress on E-approximation property and open problems are discussed at the end.

A Banach Lattice Having the Approximation Property, but not Having the Bounded Approximation Property

The first example of a Banach space with the approximation property but without the bounded approximation property was given by Figiel and Johnson in 1973. We give the first example of a Banach lattice with the approximation property but without the bounded approximation property. As a consequence, we prove the existence of an integral operator (in the sense of Grothendieck) on a Banach lattice which is not strictly integral.

Copies of $c_0(\tau )$ spaces in projective tensor products

This paper is dedicated to the memory of our friend Eve Oja. Abstract. Let X and Y be Banach spaces and consider the projective tensor product X ⊗ π Y. Suppose that τ is an infinite cardinal, X has the bounded approximation property and the density character of X is strictly smaller than the cofinality of τ. We prove the following c 0 (τ) generalizations of classical c 0 results due to Oja (1991) and Kwapien (1974) respectively: (i) If c 0 (τ) is isomorphic to a complemented subspace of X ⊗ π Y , then c 0 (τ) is isomorphic to a complemented subspace of Y. (ii) If c 0 (τ) is isomorphic to a subspace of X ⊗ π Y , then c 0 (τ) is isomorphic to a subspace of Y. We also show that the result (i) is optimal for regular cardinals τ and Banach spaces X without copies of c 0 (τ). In order to do so, we provide a c 0 (τ) extension of a classical c 0 result due to Emmanuele (1988) concerning the c 0 (τ) complemented subspaces of Lp(D τ , Y) spaces, 1 ≤ p ≤ ∞, where D τ is the Cantor cube. Finally, as a consequence of (i) we conclude that under the continuum hypothesis, the space c 0 (ℵα), with α > 1, is not isomorphic to a complemented subspace of l∞ ⊗ π l∞(ℵα).

A remark on Defant-Floret-Lima-Oja’s conjecture

Let $$\lambda \ge 1$$. We prove that the $$\lambda $$-bounded approximation property and the weak $$\lambda $$-bounded approximation property are equivalent for every Banach space if they are equivalent for every separable Banach space.

On pathological properties of fixed point algebras in Kirchberg algebras

AbstractWe investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

On the bounded approximation property on subspaces of ℓ p when 0 &lt; p &lt; 1 and related issues

Abstract This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator ℓ p → X {\ell_{p}\to X} has the BAP when X has it and 0 &lt; p ≤ 1 {0&lt;p\leq 1} , which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec–Pełczyński–Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.

Orthogonally additive polynomials on the algebras of approximable operators

Let X and Y be Banach spaces, let stands for the algebra of approximable operators on X, and let be an orthogonally additive, continuous n-homogeneous polynomial. If has the bounded approximation property, then we show that there exists a unique continuous linear map such that for each .

The bounded approximation property of variable Lebesgue spaces and nuclearity

In this paper we prove the bounded approximation property for variable exponent Lebesgue spaces, study the concept of nuclearity on such spaces and apply it to trace formulae such as the Grothendieck-Lidskii formula. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb{R}^n$ in terms of global symbols.

Cluster values for algebras of analytic functions

The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space X, we study the Cluster Value Problem for the ball algebra Au(BX), the Banach algebra of all uniformly continuous holomorphic functions on the unit ball BX; and also for the Fréchet algebra Hb(X) of holomorphic functions of bounded type on X (more generally, for Hb(U), the algebra of holomorphic functions of bounded type on a given balanced open subset U⊂X). We show that Cluster Value Theorems hold for all of these algebras whenever the dual of X has the bounded approximation property. These results are an important advance in this problem, since the validity of these theorems was known only for trivial cases (where the spectrum is formed only by evaluation functionals) and for the infinite dimensional Hilbert space.As a consequence, we obtain weak analytic Nullstellensatz theorems and several structural results for the spectrum of these algebras.