Homological Methods in Banach Space Theory

✍ Scribed by Félix Cabello Sánchez, Jesús M. F. Castillo

• Publisher: Cambridge University Press
• Year: 2023
• Tongue: English
• Leaves: 559
• Series: Cambridge Studies in Advanced Mathematics 203
• Category: Library

✦ Synopsis

Many researchers in geometric functional analysis are unaware of algebraic aspects of the subject and the advances they have permitted in the last half century. This book, written by two world experts on homological methods in Banach space theory, gives functional analysts a new perspective on their field and new tools to tackle its problems. All techniques and constructions from homological algebra and category theory are introduced from scratch and illustrated with concrete examples at varying levels of sophistication. These techniques are then used to present both important classical results and powerful advances from recent years. Finally, the authors apply them to solve many old and new problems in the theory of (quasi-) Banach spaces and outline new lines of research. Containing a lot of material unavailable elsewhere in the literature, this book is the definitive resource for functional analysts who want to know what homological algebra can do for them.

✦ Table of Contents

Contents
Preface
Preliminaries
1. Complemented Subspaces of Banach Spaces
1.1 Banach and Quasi-Banach Spaces
1.2 Complemented Subspaces
1.3 Uncomplemented Subspaces
1.4 Local Properties and Techniques
1.5 The Dunford–Pettis, Grothendieck, Pełczyński and Rosenthal Properties
1.6 C(K)-Spaces and Their Complemented Subspaces
1.7 Sobczyk’s Theorem and Its Derivatives
1.8 Notes and Remarks
2. The Language of Homology
2.1 Exact Sequences of Quasi-Banach Spaces
2.2 Basic Examples of Exact Sequences
2.3 Topologically Exact Sequences
2.4 Categorical Constructions for Absolute Beginners
2.5 Pullback and Pushout
2.6 Pushout and Exact Sequences
2.7 Projective Presentations: the Universal Property of 𝓁ₚ
2.8 Pullbacks and Exact Sequences
2.9 Injective Presentations: the Universal Property of 𝓁_∞
2.10 All about That Pullback/Pushout Diagram
2.11 Diagonal and Parallel Principles
2.12 Homological Constructions Appearing in Nature
2.13 The Device
2.14 Extension and Lifting of Operators
2.15 Notes and Remarks
3. Quasilinear Maps
3.1 An Introduction to Quasilinear Maps
3.2 Quasilinear Maps in Action
3.3 Quasilinear Maps versus Exact Sequences
3.4 Local Convexity of Twisted Sums and 𝒦-Spaces
3.5 The Pullback and Pushout in Quasilinear Terms
3.6 Spaces of Quasilinear Maps
3.7 Homological Properties of 𝓁ₚ and Lₚ When 0 < p ≤ 1
3.8 Exact Sequences of Banach Spaces and Duality
3.9 Different Versions of a Quasilinear Map
3.10 Linearisation of Quasilinear Maps
3.11 The Type of Twisted Sums
3.12 A Glimpse of Centralizers
3.13 Notes and Remarks
4. The Functor Ext and the Homology Sequences
4.1 The Functor Ext
4.2 The Homology Sequences
4.3 Homology in Quasilinear Terms
4.4 Alternative Constructions of Ext
4.5 Topological Aspects of Ext
4.6 Notes and Remarks
5. Local Methods in the Theory of Twisted Sums
5.1 Local Splitting
5.2 Uniform Boundedness Principles for Exact Sequences
5.3 The Mysterious Role of the BAP
5.4 Notes and Remarks
6 Fraïssé Limits by the Pound
6.1 Fraïssé Classes and Fraïssé Sequences
6.2 Almost Universal Disposition
6.3 Almost Universal Complemented Disposition
6.4 A Universal Operator on Gₚ
6.5 Notes and Remarks
7. Extension of Operators, Isomorphisms and Isometries
7.1 Operators: Extensible and UFO Spaces
7.2 Isomorphisms: the Automorphic Space Problem
7.3 Isometries: Universal Disposition
7.4 Positions in Banach Spaces
7.5 Notes and Remarks
8. Extension of C(K)-Valued Operators
8.1 Zippin Selectors
8.2 The Lindenstrauss–Pełczyński Theorem
8.3 Kalton’s Approach to the C -Extension Property
8.4 Sequence Spaces with the C -Extension Property
8.5 C-Extensible Spaces
8.6 The Dark Side of the Johnson–Zippin Theorem
8.7 The Astounding Story behind the CCKY Problem
8.8 Notes and Remarks
9. Singular Exact Sequences
9.1 Basic Properties and Techniques
9.2 Singular Quasilinear Maps
9.3 Amalgamation Techniques
9.4 Notes and Remarks
10. Back to Banach Space Theory
10.1 Vector-Valued Versions of Sobczyk’s Theorem
10.2 Polyhedral L_∞-Spaces
10.3 Lipschitz and Uniformly Homeomorphic L_∞-Spaces
10.4 Properties of Kernels of Quotient Maps on L₁ Spaces
10.5 3-Space Problems
10.6 Extension of L_∞-Valued Operators
10.7 Kadec Spaces
10.8 The Kalton–Peck Spaces
10.9 The Properties of Z₂ Explained by Itself
Bibliography
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Index
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