Introduction to Subfactors

Subfactors have been a subject of considerable research activity for about fifteen years and are known to have significant relations with other fields such as low dimensional topology and algebraic quantum field theory. These notes give an introduction to the subject suitable for a student who has only a little familiarity with the theory of Hilbert space. A new pictorial approach to subfactors is presented in a late chapter.

Preface

ix

(3)

Acknowledgement

xii

1 Factors

1

(18)

1.1 von Neumann algebras and factors

1

(3)

1.2 The standard form

4

(3)

1.3 Discrete crossed products

7

(3)

1.4 Examples of factors

10

(9)

1.4.1 Group von Neumann algebras

10

(2)

1.4.2 Crossed products of commutative von Neumann algebras

12

(5)

1.4.3 Infinite tensor products

17

(2)

2 Subfactors and index

19

(14)

2.1 The classification of modules

19

(4)

2.2 dimMH

23

(5)

2.3 Subfactors and index

28

(5)

3 Some basic facts

33

(14)

3.1 The basic construction

33

(3)

3.2 Finite-dimensional inclusions

36

(5)

3.3 The projections en and the tower

41

(6)

4 The principal and dual graphs

47

(18)

4.1 More on bimodules

47

(3)

4.2 The principal graphs

50

(4)

4.3 Bases

54

(5)

4.4 Relative commutants vs intertwiners

59

(6)

5 Commuting squares

65

(36)

5.1 The Pimsner-Popa inequality

65

(5)

5.2 Examples of commuting squares

70

(2)

5.2.1 The braid group example

70

(1)

5.2.2 Spin models

71

(1)

5.2.3 Vertex models

71

(1)

5.3 Basic construction in finite dimensions

72

(5)

5.4 Path algebras

77

(3)

5.5 The biunitarity condition

80

(9)

5.6 Canonical commuting squares

89

(3)

5.7 Ocneanu compactness

92

(9)

6 Vertex and spin models

101

(32)

6.1 Computing higher relative commutants

101

(10)

6.2 Some examples

111

(4)

6.3 On permutation vertex models

115

(7)

6.4 A diagrammatic formulation

122

(11)

Appendix

133

(18)

A.1 Concrete and abstract von Neumann algebras

133

(1)

A.2 Separable pre-duals, Tomita-Takesaki theorem

134

(1)

A.3 Simplicity of factors

135

(1)

A.4 Subgroups and subfactors

136

(5)

A.5 From subfactors to knots

141

(10)

Bibliography

151

(5)

Bibliographical Remarks

156

(4)

Index

160

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Introduction to Subfactors

Summary

Table of Contents

Supplemental Materials

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