Contents
Preface v
Preface t o th e origina l IMP A editio n vi i
Introduction 1
Chapter 1. Th e Ramsey-Dvoretzky-Milma n phenomeno n 11
1.1. Finit e oscillatio n stabilit y 1
1.2. Firs t example : th e spher e ° 18
1.3. Concentratio n o f measure o n sphere s 2
1.4. Dvoretzky' s theore m 2 7
1.5. Secon d example : finite Ramse y theore m 3 3
1.6. Counterexample : ordere d pair s 3 7
Chapter 2 . Th e fixed poin t o n compac t a property 3 9
2.1. Extremel y amenabl e group s 3 9
2.2. Example : th e unitar y grou p 4 3
2.3. Example : th e grou p Au t (Q, ) 4 5
2.4. Counterexample : th e infinit e symmetri c grou p 4 6
Chapter 3 . Th e concentratio n propert y 4 9
3.1. Equivarian t compactificatio n 4 9
3.2. Essentia l set s an d th e concentratio n propert y 5 1
3.3. Veec h theorem 5 5
3.4. Bi g sets 5 8
3.5. Concentratio n propert y o f §°° 6 2
3.6. Group s o f operators wit h unifor m topolog y 7 0
Chapter 4 . Lev y group s 7 5
4.1. Infinit e unitar y grou p 7 5
4.2. Grou p o f measurable map s 7 8
4.3. Martingal e techniqu e 8 0
86
89
4.6.T7nl Ful l group s 9 4
4.3 . Martingal e techniqu e
4.4. Unitar y group s o f operator algebra s
4.5. Grou p o f measure-preserving transformation s
A f\ l r r v m m o
Chapter 5 . Urysoh n metri c spac e an d it s grou p o f isometries 103
5.1. Urysoh n universa l metri c spac e 103
5.2. Isometr y grou p 14
5.3. Approximatio n b y finite group s 17
5.4. Kirchberg' s propert y 122
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