Bounded Approximation Properties Research Articles

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.

An Ando–Choi–Effros lifting theorem respecting subspaces

We prove a version of the Ando–Choi–Effros lifting theorem respecting subspaces, which in turn relies on Oja's principle of local reflexivity respecting subspaces. To achieve this, we first develop a theory of pairs of M-ideals. As a first consequence we get a version respecting subspaces of the Michael–Pełczyński extension theorem. Other applications are related to linear and Lipschitz bounded approximation properties (BAPs) for a pair consisting of a Banach space and a subspace. We show that in the separable case, the BAP for such a pair is equivalent to the simultaneous splitting of an associated pair of short exact sequences given by a construction of Lusky. We define a Lipschitz version of the BAP for pairs, and study its relationship to the (linear) BAP for pairs. The two properties are not equivalent in general, but they are when the pair has an additional Lipschitz-lifting property in the style of Godefroy and Kalton. We also characterize, in the separable case, those pairs of a metric space and a subset whose corresponding pair of Lipschitz-free spaces has the BAP.

A characterization of bounded convex approximation properties

The characterization of bounded approximation properties defined by arbitrary operator ideals due to Oja is extended to bounded convex approximation properties. As an application, it is shown that the unique extension property of a Banach space X enables to lift the metric convex approximation property from a Banach space X to its dual space X*.

Lifting convex approximation properties from Banach spaces to their dual spaces and the related local reflexivity

Our main result demonstrates that rather weak forms of the extendable local reflexivity and of the principle of local reflexivity are needed for the lifting of bounded convex approximation properties from Banach spaces to their dual spaces. This provides a unified approach to the lifting of various bounded approximation properties, including, besides the classical ones, the approximation property for pairs of Banach spaces, and the positive approximation property of Banach lattices.

THE SEPARABLE WEAK BOUNDED APPROXIMATION PROPERTY

In this paper we introduce and study the separable weak bounded approximation properties which is strictly stronger than the approximation property and but weaker than the bounded approximation property. It provides new sufficient conditions for the metric approximation property for a dual Banach space.

Principle of local reflexivity respecting subspaces

We obtain a strengthening of the principle of local reflexivity in a general form. The added strength makes local reflexivity operators respect given subspaces. Applications are given to bounded approximation properties of pairs, consisting of a Banach space and its subspace.

Some duality results on bounded approximation properties of pairs

The main result is as follows. Let $X$ be a Banach space and let $Y$ be a closed subspace of $X$. Assume that the pair $(X^{*}, Y^{\perp })$ has the $\lambda $-bounded approximation property. Then there exists a net $( S_\alpha )$ of finite-rank oper

A bounded approximation of weakly compact operators

We study bounded approximation properties of operators. It is shown that for every weakly compact operator T on a Banach space X, if T is uniformly approximated on each compact subset of X by finite rank operators, then T2 can be approximated in the same topology by finite rank operators with the operator norm less than or equal to ‖T‖2.

Bounded approximation properties in terms of $C[0,1

Let $X$ be a Banach space and let $\mathcal I$ be the Banach operator ideal of integral operators. We prove that $X$ has the $\lambda$-bounded approximation property ($\lambda$-BAP) if and only if for every operator $T\in \mathcal I(X,C[0,1]^*)$ there exists a net $(S_\alpha)$ of finite-rank operators on $X$ such that $S_\alpha\to I_X$ pointwise and 26767 \limsup_\alpha\|TS_\alpha\|_{\mathcal I}\leq\lambda\|T\|_{\mathcal I}. 26767 We also prove that replacing $\mathcal I$ by the ideal $\mathcal N$ of nuclear operators yields a condition which is equivalent to the weak $\lambda$-BAP.

Bounded approximation properties in non‐archimedean Banach spaces

AbstractSome non‐archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non‐spherically complete base fields, to the following problem, posed in 13, p. 95: Does there exist an absolutely convex edged set B in a non‐archimedean locally convex space such that its closure \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\overline{B}$\end{document} is not edged?

New Criteria of Some Bounded Approximation Properties

The bounded (resp. bounded compact) approximation property is well known in the theory of Banach spaces. This paper is concerned with some bounded approximation properties in the more general setting. We establish various new criteria of bounded approximation properties.

On bounded approximation properties of Banach spaces

On bounded approximation properties of Banach spaces

Bounded approximation properties via integral and nuclear operators

Published version of an article in the journal:Proceedings of the American Mathematical Society. Also available from the publisher, Open Access

CHARACTERIZATIONS OF BOUNDED APPROXIMATION PROPERTIES

New necessary and sufficient conditions for Banach spaces to have bounded approximation properties are established, which are easier to check than known ones. Also using these it is shown that for a Banach space $X$, the dual $X^{*}$ has the bounded approximation property if and only if $X$ has the bounded approximation property approximated by the weak adjoint operator topology, and if $X^{*}$ has the bounded weak approximation property, then $X$ has the bounded weak approximation property approximated by the weak adjoint operator topology.

Lifting bounded approximation properties from Banach spaces to their dual spaces

Based on a new reformulation of the bounded approximation property, we develop a unified approach to the lifting of bounded approximation properties from a Banach space X to its dual X * . This encompasses cases when X has the unique extension property or X is extendably locally reflexive. In particular, it is shown that the unique extension property of X permits to lift the metric A -approximation property from X to X * , for any operator ideal A , and that there exists a Banach space X such that X , X * * , … are extendably locally reflexive, but X * , X * * * , … are not.

Three-space problems and bounded approximation properties

Let fRng 1=1 be a commuting approximating sequence of the Banach space X leaving the closed subspace A X invariant. Then we prove three-space results of the following kind: If the operators Rn induce basis projections on X=A, and X or A is anLp-space, then both X and A have bases. We apply these results to show that the spaces C = spanfz k : k2 g C(T) and L = spanfz k : k2 g L1(T) have bases whenever Z and Zn is a Sidon set.

Approximate Identities in Banach Algebras of Compact Operators

AbstractLet X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.

On bounded approximation properties of spaces of holomorphic functions on certain open subsets of strong duals of nuclear Fr�chet spaces

On bounded approximation properties of spaces of holomorphic functions on certain open subsets of strong duals of nuclear Fr�chet spaces