A unified approach to the strong approximation property and the weak bounded approximation property of Banach spaces

2012 ◽  
Vol 211 (3) ◽  
pp. 199-214 ◽  
Author(s):  
Aleksei Lissitsin
Keyword(s):  
Banach Spaces ◽  
Strong Approximation ◽  
Approximation Property ◽  
Unified Approach

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)).

scholarly journals On the uniqueness of cellular injectives

2018 ◽  
Vol 167 (3) ◽  
pp. 489-504 ◽  
Author(s):  
J. ROSICKÝ
Keyword(s):  
Banach Spaces ◽  
Category Theory ◽  
Existence And Uniqueness ◽  
Boolean Algebras ◽  
Small Object ◽  
Unified Approach ◽  
Basic Tool ◽  
Saturated Models ◽  
Small Object Argument

AbstractA. Avilés and C. Brech proved an intriguing result about the existence and uniqueness of certain injective Boolean algebras or Banach spaces. Their result refines the standard existence and uniqueness of saturated models. They express a wish to obtain a unified approach in the context of category theory. We provide this in the framework of weak factorisation systems. Our basic tool is the fat small object argument.

scholarly journals THE QUOTIENT ALGEBRA OF COMPACT-BY-APPROXIMABLE OPERATORS ON BANACH SPACES FAILING THE APPROXIMATION PROPERTY

2019 ◽  
pp. 1-23
Author(s):  
HANS-OLAV TYLLI ◽  
HENRIK WIRZENIUS
Keyword(s):  
Banach Spaces ◽  
Structural Properties ◽  
Approximation Property ◽  
Quotient Algebra ◽  
Isomorphic Embedding ◽  
Compact Approximation

We initiate a study of structural properties of the quotient algebra ${\mathcal{K}}(X)/{\mathcal{A}}(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_{0}$ into ${\mathcal{K}}(Z)/{\mathcal{A}}(Z)$ , where $Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space $c_{0}(\unicode[STIX]{x1D6E4})$ into ${\mathcal{K}}(Z_{FJ})/{\mathcal{A}}(Z_{FJ})$ , where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel.

scholarly journals On the bounded approximation property in Banach spaces

2013 ◽  
Vol 198 (1) ◽  
pp. 243-259 ◽  
Author(s):  
Jesús M. F. Castillo ◽  
Yolanda Moreno
Keyword(s):  
Banach Spaces ◽  
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>

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