Softcover ISBN:  9780821807101 
Product Code:  CBMS/60 
List Price:  $35.00 
Individual Price:  $28.00 
eBook ISBN:  9781470424213 
Product Code:  CBMS/60.E 
List Price:  $33.00 
Individual Price:  $26.40 
Softcover ISBN:  9780821807101 
eBook: ISBN:  9781470424213 
Product Code:  CBMS/60.B 
List Price:  $68.00 $51.50 
Softcover ISBN:  9780821807101 
Product Code:  CBMS/60 
List Price:  $35.00 
Individual Price:  $28.00 
eBook ISBN:  9781470424213 
Product Code:  CBMS/60.E 
List Price:  $33.00 
Individual Price:  $26.40 
Softcover ISBN:  9780821807101 
eBook ISBN:  9781470424213 
Product Code:  CBMS/60.B 
List Price:  $68.00 $51.50 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 60; 1986; 154 ppMSC: Primary 47; Secondary 46
This book surveys the considerable progress made in Banach space theory as a result of Grothendieck's fundamental paper Reśumé de la théorie métrique des produits tensoriels topologiques. The author examines the central question of which Banach spaces \(X\) and \(Y\) have the property that every bounded operator from \(X\) to \(Y\) factors through a Hilbert space, in particular when the operators are defined on a Banach lattice, a \(C^*\)algebra or the disc algebra and \(H^\infty\). He reviews the six problems posed at the end of Grothendieck's paper, which have now all been solved (except perhaps the exact value of Grothendieck's constant), and includes the various results which led to their solution. The last chapter contains the author's construction of several Banach spaces such that the injective and projective tensor products coincide; this gives a negative solution to Grothendieck's sixth problem.
Although the book is aimed at mathematicians working in functional analysis, harmonic analysis and operator algebras, its detailed and selfcontained treatment makes the material accessible to nonspecialists with a grounding in basic functional analysis. In fact, the author is particularly concerned to develop very recent results in the geometry of Banach spaces in a form that emphasizes how they may be applied in other fields, such as harmonic analysis and \(C^*\)algebras.

Table of Contents

Chapters

Chapter 0. Preliminary Results and Background

Chapter 1. Absolutely Summing Operators and Basic Applications

Chapter 2. Factorization through a Hilbert Space

Chapter 3. Type and Cotype. Kwapień’s Theorem

Chapter 4. The “Abstract” Version of Grothendieck’s Theorem

Chapter 5. Grothendieck’s Theorem

Chapter 6. Banach Spaces Satisfying Grothendieck’s Theorem

Chapter 7. Applications of the Volume Ratio Method

Chapter 8. Banach Lattices

Chapter 9. $C^*$Algebras

Chapter 10. Counterexamples to Grothendieck’s Conjecture


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This book surveys the considerable progress made in Banach space theory as a result of Grothendieck's fundamental paper Reśumé de la théorie métrique des produits tensoriels topologiques. The author examines the central question of which Banach spaces \(X\) and \(Y\) have the property that every bounded operator from \(X\) to \(Y\) factors through a Hilbert space, in particular when the operators are defined on a Banach lattice, a \(C^*\)algebra or the disc algebra and \(H^\infty\). He reviews the six problems posed at the end of Grothendieck's paper, which have now all been solved (except perhaps the exact value of Grothendieck's constant), and includes the various results which led to their solution. The last chapter contains the author's construction of several Banach spaces such that the injective and projective tensor products coincide; this gives a negative solution to Grothendieck's sixth problem.
Although the book is aimed at mathematicians working in functional analysis, harmonic analysis and operator algebras, its detailed and selfcontained treatment makes the material accessible to nonspecialists with a grounding in basic functional analysis. In fact, the author is particularly concerned to develop very recent results in the geometry of Banach spaces in a form that emphasizes how they may be applied in other fields, such as harmonic analysis and \(C^*\)algebras.

Chapters

Chapter 0. Preliminary Results and Background

Chapter 1. Absolutely Summing Operators and Basic Applications

Chapter 2. Factorization through a Hilbert Space

Chapter 3. Type and Cotype. Kwapień’s Theorem

Chapter 4. The “Abstract” Version of Grothendieck’s Theorem

Chapter 5. Grothendieck’s Theorem

Chapter 6. Banach Spaces Satisfying Grothendieck’s Theorem

Chapter 7. Applications of the Volume Ratio Method

Chapter 8. Banach Lattices

Chapter 9. $C^*$Algebras

Chapter 10. Counterexamples to Grothendieck’s Conjecture