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

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