On commuting approximation properties of Banach spaces

2009 ◽  
Vol 139 (3) ◽  
pp. 551-565 ◽  
Author(s):  
Eve Oja ◽  
Indrek Zolk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Approximation Property ◽  
Canonical Projection ◽  
Compact Set ◽  
Approximation Properties ◽  
Compact Approximation ◽  
Image Position ◽  
Approximation Of The Identity

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 < 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.

scholarly journals Strongly Extreme Points and Approximation Properties

2018 ◽  
Vol 61 (3) ◽  
pp. 449-457
Author(s):  
Trond A. Abrahamsen ◽  
Petr Hájek ◽  
Olav Nygaard ◽  
Stanimir L. Troyanski
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Extreme Point ◽  
Convex Subset ◽  
Approximation Property ◽  
Extreme Points ◽  
Local Approximation ◽  
Approximation Properties ◽  
Compact Approximation ◽  
Approximation Of The Identity

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.

scholarly journals Self-induced compactness in Banach spaces

1996 ◽  
Vol 126 (2) ◽  
pp. 355-362 ◽  
Author(s):  
P. G. Casazza ◽  
H. Jarchow
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Approximation Property ◽  
Negative Answer ◽  
Affirmative Answer ◽  
Compact Set ◽  
Large Classes ◽  
Compact Approximation

We consider the question: is every compact set in a Banach space X contained in the closed unit range of a compact (or even approximable) operator on X? We give large classes of spaces where the question has an affirmative answer, but observe that it has a negative answer, in general, for approximable operators. We further construct a Banach space failing the bounded compact approximation property, though all of its duals have the metric compact approximation property.

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

scholarly journals Approximation properties of tensor norms and operator ideals for Banach spaces

2020 ◽  
Vol 18 (1) ◽  
pp. 1698-1708
Author(s):  
Ju Myung Kim
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Approximation Property ◽  
Approximation Properties ◽  
Operator Ideals ◽  
Finitely Generated ◽  
Tensor Norm ◽  
Tensor Norms

Abstract For a finitely generated tensor norm α \alpha , we investigate the α \alpha -approximation property ( α \alpha -AP) and the bounded α \alpha -approximation property (bounded α \alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ \lambda -bounded α p , q {\alpha }_{p,q} -AP ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ) (1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the λ \lambda -bounded g p {g}_{p} -AP. As a consequence, it follows that if a Banach space X has the λ \lambda -bounded g p {g}_{p} -AP, then X has the λ \lambda -bounded w p {w}_{p} -AP.

scholarly journals Ideals of compact operators

2004 ◽  
Vol 77 (1) ◽  
pp. 91-110 ◽  
Author(s):  
Åsvald Lima ◽  
Eve Oja
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Approximation Property ◽  
Closed Subspace ◽  
Reflexive Banach Space ◽  
Compact Operators ◽  
Compact Approximation

AbstractWe give an example of a Banach space X such that K (X, X) is not an ideal in K (X, X**). We prove that if z* is a weak* denting point in the unit ball of Z* and if X is a closed subspace of a Banach space Y, then the set of norm-preserving extensions H B(x* ⊗ z*) ⊆ (Z*, Y)* of a functional x* ⊗ Z* ∈ (Z ⊗ X)* is equal to the set H B(x*) ⊗ {z*}. Using this result, we show that if X is an M-ideal in Y and Z is a reflexive Banach space, then K (Z, X) is an M-ideal in K(Z, Y) whenever K (Z, X) is an ideal in K (Z, Y). We also show that K (Z, X) is an ideal (respectively, an M-ideal) in K (Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, an M-ideal) in Y and X * has the compact approximation property with conjugate operators.

scholarly journals Approximation properties for dual spaces

2003 ◽  
Vol 93 (2) ◽  
pp. 297 ◽  
Author(s):  
Vegard Lima
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Approximation Property ◽  
Extension Operator ◽  
Approximation Properties ◽  
Dual Spaces ◽  
Metric Approximation

We prove that a Banach space $X$ has the metric approximation property if and only if $\mathcal F(Y,X)$ is an ideal in $\mathcal L(Y,X^{**})$ for all Banach spaces $Y$. Furthermore, $X^*$ has the metric approximation property if and only if for all Banach spaces $Y$ and all Hahn-Banach extension operators $\phi : X^* \rightarrow X^{***}$ there exists a Hahn-Banach extension operator $\Phi : {\mathcal F(Y,X)}^* \rightarrow {\mathcal L(Y,X^{**})}^*$ such that $\Phi(x^* \otimes y^{**}) = (\phi x^*) \otimes y^{**}$ for all $x^* \in X^*$ and all $y^{**} \in Y^{**}$. We also prove that $X^*$ has the approximation property if and only if for all Banach spaces $Y$ and all Hahn-Banach extension operators $\phi : X^* \rightarrow X^{***}$ there exists a Hahn-Banach extension operator $\Phi : {\mathcal F(Y,X)}^* \rightarrow {\mathcal W(Y,X^{**})}^*$ such that $\Phi(x^* \otimes y^{**}) = (\phi x^*) \otimes y^{**}$ for all $x^* \in X^*$ and all $y^{**} \in Y^{**}$, which in turn is equivalent to $\mathcal F(Y,\hat{X})$ being an ideal in $\mathcal W(Y,\hat{X}^{**})$ for all Banach spaces $Y$ and all equivalent renormings $\hat{X}$ of $X$.

SOME RESULTS OF THE -APPROXIMATION PROPERTY FOR BANACH SPACES

2018 ◽  
Vol 61 (03) ◽  
pp. 545-555 ◽  
Author(s):  
JU MYUNG KIM
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Approximation Property ◽  
Compact Operators ◽  
Operator Ideal ◽  
Formula Group ◽  
Image Position ◽  
The Ideal

AbstractGiven a Banach operator ideal $\mathcal A$, we investigate the approximation property related to the ideal of $\mathcal A$-compact operators, $\mathcal K_{\mathcal A}$-AP. We prove that a Banach space X has the $\mathcal K_{\mathcal A}$-AP if and only if there exists a λ ≥ 1 such that for every Banach space Y and every R ∈ $\mathcal K_{\mathcal A}$(Y, X), $$ \begin{equation} R \in \overline {\{SR : S \in \mathcal F(X, X), \|SR\|_{\mathcal K_{\mathcal A}} \leq \lambda \|R\|_{\mathcal K_{\mathcal A}}\}}^{\tau_{c}}. \end{equation} $$ For a surjective, maximal and right-accessible Banach operator ideal $\mathcal A$, we prove that a Banach space X has the $\mathcal K_{(\mathcal A^{{\rm adj}})^{{\rm dual}}}$-AP if the dual space of X has the $\mathcal K_{\mathcal A}$-AP.

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

scholarly journals Homological characterizations of the approximation property for Banach spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 229-239 ◽  
Author(s):  
Yu. V. Selivanov
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Cohomology Group ◽  
Approximation Property ◽  
Three Dimensional ◽  
Nuclear Operators ◽  
Finite Dimensionality

Let E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:(i) biprojectivity of N(E);(ii) biflatness of N(E);(iii) homological finite-dimensionality of N(E);(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).

Sign in / Sign up

Export Citation Format

Share Document