Subfactors, a subject of considerable research activity for about fifteen years, are known to have significant relations with other fields such as low dimensional topology and algebraic quantum field theory. The authors present a new pictorial approach to subfactors in addition to discussions of basic principles. This book provides an important introduction to the subject and is aimed at students with only a little familiarity with the theory of Hilbert space and other newcomers to the field.
Author(s): V. Jones, V. S. Sunder
Series: London Mathematical Society Lecture Note Series
Publisher: Cambridge University Press
Year: 1997
Language: English
Pages: 174
1.1 von Neumann algebras and factors ......Page 12
1.2 The standard form ......Page 15
1.3 Discrete crossed products ......Page 18
1.4.1 Group von Neumann algebras ......Page 21
1.4.2 Crossed products of commutative von Neumann algebras ......Page 23
1.4.3 Infinite tensor products ......Page 28
2.1 The classification of modules ......Page 30
2.2 dim_M(H) ......Page 34
2.3 Subfactors and index ......Page 39
3.1 The basic construction ......Page 44
3.2 Finite-dimensional inclusions ......Page 47
3.3 The projections e, and the tower ......Page 52
4.1 More on bimodules ......Page 58
4.2 The principal graphs ......Page 61
4.3 Bases ......Page 65
4.4 Relative commutants vs intertwiners ......Page 70
5.1 The Pimsner-Popa inequality ......Page 76
5.2.1 The braid group example ......Page 81
5.2.3 Vertex models ......Page 82
5.3 Basic construction in finite dimensions ......Page 83
5.4 Path algebras ......Page 88
5.5 The biunitarity condition ......Page 91
5.6 Canonical commuting squares ......Page 100
5.7 Ocneanu compactness ......Page 103
6.1 Computing higher relative commutants ......Page 112
6.2 Some examples ......Page 122
6.3 On permutation vertex models ......Page 126
6.4 A diagrammatic formulation ......Page 133
A.1 Concrete and abstract von Neumann algebras ......Page 144
A.2 Separable pre-duals, Tomita-Takesaki theorem ......Page 145
A.3 Simplicity of factors ......Page 146
A.4 Subgroups and subfactors ......Page 147
A.5 From subfactors to knots ......Page 152
Bibliography ......Page 162
Bibliographical Remarks ......Page 167
Index ......Page 171
