CONTENTS

1 Introduction 1

2 Preliminaries 3

3 Unitary realization of ot(y,x) 5

4 Construction of M v 6

5 Coordinate representation of elements of M 7

6 The expectation E 10

7 Coordinates in M

v

11

8 The expectation E' 14

9 Tomita-Takesaki theory for M and M v 15

10 /(M)-automorphisms of M 22

11 Flows of automorphisms 26

12 The Feldman-Moore-type structure

theorem 32

12.1 Structure of operators in N 41

12.2 The maps ax{y^z) and ^

z )

41

12.3 Coordinate representation of elements of MQ 45

13 Isomorphisms of crossed products 52

13.1 /(M)-isomorphisms of crossed products 52

13.2 /-isomorphisms of crossed products 56

14 Bimodules and subalgebras of M 62

15 Spectral theorem for bimodules 70

16 Analytic algebra of a flow of automorphisms 81

17 Properties of M 83

18 Hyperfiniteness and dilations 85

19 The construction of Yamanouchi 93

vn