VI

CONTENTS

Chapter 5. Grothendieck's Theorem 5 3

a. Preliminaries. Localization techniques. J^-spaces 5 3

b. Operators on C( K )-spaces 5 4

c. Operators on Lj-space s 5 7

d. Cotype 2 spaces and absolutely summing operators 6 2

e. Bes t constants. Krivine's proof 6 4

f. A proof o f G.T. using harmonic analysis 6 8

Notes and reference s 6 9

Chapter 6. Banach Spaces Satisfying Grothendieck's Theorem 7 1

a. G.T. spaces 7

b. G.T. spaces of cotype 2 7 3

c. Quotients of L

l

b y a reflexive subspac e 7 8

d. Bourgain's theorem on L

x/Hl

8 3

Notes and reference s 8 6

Chapter 7. Applications of the Volume Ratio Method 8 9

Notes and reference s 9 5

Chapter 8. Banach Lattices 9 7

a. The Banach lattice version of G.T. 9 7

b. Ultraproducts. Factorization throug h an L^-spac e 101

c. Loca l unconditional structure. The Gordon-Lewis property 104

d. Examples of Banach spaces without l.u.st. 108

e. Finite-dimensiona l spaces with extreme l.u.st. constants 113

f. G.T . spaces with unconditional basis 114

g. Infinite-dimensional Kasi n decompositions 114

Notes and reference s 18

Chapter 9. C *-Algebras 19

a. The noncommutative version of G.T. 119

b. Applications 130

Notes and reference s 132

Chapter 10. Counterexamples to Grothendieck's Conjecture 135

a. Outline of the construction 135

b. Extensions of a Banach space 137

c. The construction 140

d. Particular cases of the conjectures 145

e. Some open problems 146

Notes and reference s 147

References 149