Table Of ContentRecent Titles in This Series
86 Sori n Popa, Classification of subfactors and their endomorphisms, 1995
85 Michi o Jimbo and Tetsuji Miwa, Algebraic analysis of solvable lattice models, 1994
84 Hug h L. Montgomery, Ten lectures on the interface between analytic number theory and
harmonic analysis, 1994
83 Carlo s E. Kenig, Harmonic analysis techniques for second order elliptic boundary value
problems, 1994
82 Susa n Montgomery, Hopf algebras and their actions on rings, 1993
81 Steve n G. Krantz, Geometric analysis and function spaces, 1993
80 Vaugha n F. R. Jones, Subfactors and knots, 1991
79 Michae l Frazier, Bjorn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study
of function spaces, 1991
78 Edwar d Formanek, The polynomial identities and variants of n x n matrices, 1991
77 Michae l Christ, Lectures on singular integral operators, 1990
76 Klau s Schmidt, Algebraic ideas in ergodic theory, 1990
75 F . Thomas Farrell and L. Edwin Jones, Classical aspherical manifolds, 1990
74 Lawrenc e C. Evans, Weak convergence methods for nonlinear partial differential equations,
1990
73 Walte r A. Strauss, Nonlinear wave equations, 1989
72 Pete r Orlik, Introduction to arrangements, 1989
71 Harr y Dym, / contractiv e matrix functions, reproducing kernel Hilbert spaces and
interpolation, 1989
70 Richar d F. Gundy, Some topics in probability and analysis, 1989
69 Fran k D. Grosshans, Gian-Carlo Rota, and Joel A. Stein, Invariant theory and
superalgebras, 1987
68 J . William Helton, Joseph A. Ball, Charles R. Johnson, and John N. Palmer,
Operator theory, analytic functions, matrices, and electrical engineering, 1987
67 Haral d Upmeier, Jordan algebras in analysis, operator theory, and quantum mechanics,
1987
66 G . Andrews, ^-Series: Their development and application in analysis, number theory,
combinatorics, physics and computer algebra, 1986
65 Pau l H. Rabinowitz, Minimax methods in critical point theory with applications to
differential equations, 1986
64 Donal d S. Passman, Group rings, crossed products and Galois theory, 1986
63 Walte r Rudin, New constructions of functions holomorphic in the unit ball of Cn, 198 6
62 Bel a Bollobas, Extremal graph theory with emphasis on probabilistic methods, 1986
61 Mogen s Flensted-Jensen, Analysis on non-Riemannian symmetric spaces, 1986
60 Gille s Pisier, Factorization of linear operators and geometry of Banach spaces, 1986
59 Roge r Howe and Allen Moy, Harish-Chandra homomorphisms for p-adic groups, 1985
58 H . Blaine Lawson, Jr., The theory of gauge fields in four dimensions, 1985
57 Jerr y L. Kazdan, Prescribing the curvature of a Riemannian manifold, 1985
56 Har i Bercovici, Ciprian Foiag, and Carl Pearcy, Dual algebras with applications to invariant
subspaces and dilation theory, 1985
55 Willia m Arveson, Ten lectures on operator algebras, 1984
54 Willia m Fulton, Introduction to intersection theory in algebraic geometry, 1984
53 Wilhel m Klingenberg, Closed geodesies on Riemannian manifolds, 1983
52 Tsit-Yue n Lam, Orderings, valuations and quadratic forms, 1983
51 Masamich i Takesaki, Structure of factors and automorphism groups, 1983
(See the AMS catalog for earlier titles)
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Classification of Subfactors
and Their Endomorphism s
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http://dx.doi.org/10.1090/cbms/086
Conference Board of the Mathematical Science s
C B M S
Regional Conference Series in Mathematics
Number 8 6
Classification of Subfactors
and Their Endomorphism s
Sorin Pop a
Published for the
Conference Board of the Mathematical Sciences
by the
^^fc, American Mathematical Society
Providence, Rhode Island
$ &
with support from the
*%S5£*
National Science Foundation
%VDED^
Expository Lecture s
from the NSF-CBMS Regional Conferenc e
held at the University of Oregon, Eugene, Orego n
August 24-28 , 199 3
Research partially supported b y
National Science Foundation Grant DM S 890828 1
1991 Mathematics Subject Classification. Primar y 46L35 ;
Secondary 81E05 .
Library of Congress Cataloging-in-Publication Dat a
Sorin, Popa, 1953-
Classification of subfactors and their endomorphisms / Sorin Popa.
p. cm . — (Regional conference series in mathematics, ISSN 0160-7642; no. 86)
Includes bibliographical references (p. - ) .
ISBN 0-8218-0321-2
1. Von Neumann algebras. 2 . Endomorphisms (Group theory) I . Title. II . Series.
QA1.R33 no . 86
[QA326]
510 s—dc20
[512'.55] 95-1787 7
CIP
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10 9 8 7 6 5 4 3 2 1 0 0 99 98 97 96 95
Contents
Introduction i x
Chapter 1: Preliminaries 1
1.1. Index for inclusions of von Neumann algebras 1
1.2. Markov inclusions and Jones towers 9
1.3. The central sequence and ultrapower inclusions
associated to JSf C M 1 7
Chapter 2: Approximate innerness for subfactors 2 3
Chapter 3: Central freeness for subfactors 3 7
Chapter 4: More on central freeness: the type IIIi case 5 3
Chapter 5: The main classification result 6 7
Chapter 6: Applications 9 1
Appendix 10 1
A.l. A localization principle 10 1
A.2. A perturbation result 10 4
References 10 7
vii
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Introduction
The purpose of these lecture notes is to provide a more unified and to a certain
extent selfcontained presentation of the classification results for inclusions of von
Neumann factors of finite Jones index in ([Po2-5]). The classification is in terms
of the standar d invarian t GN,M o f the subfactor M C Ai. Thi s invariant i s
a lattice of inclusions of finite dimensional algebras associated with the Jone s
iterated basic construction for J\f C Ai. I t can be recovered from combinatoria l
data in a similar way inclusions of semisimple algebras can be recaptured fro m
their inclusion diagrams, with the graph of the subfactor, /V,.A/b providing most
of the information .
We will prove that if M C M. satisfies certain growth conditions, then GM,M
is a complete invariant for M C M (u p to the isomorphism class of M) an d that
in fact N C M i s isomorphic to (Afst C Mst) ® M, wher e J\fst C Mst i s the
model type Hi inclusion constructed from GM,M •
The growth conditions that w e have to assume are of two types. First , we
require IV, JW to be strongly amenable, i.e., to be ergodic and to satisfy a certain
F0mer-type condition. An d second, we require M C M t o be approximatel y
inner and centrally free. The condition on IV, Ai *s automatically satisfied when
^V,AI i s finite, i.e., M C M ha s finite depth. Th e approximate innerness and
central freeness are satisfied when Af, M ar e hyperfinite, provided the modular
group of M i s "independent" from GM,M > & situation that is automatically sat-
isfied if, for instance, J\f C M i s a finite depth inclusion of hyperfinite type IIIi
factors.
Thus, the main application of our general result shows that if an inclusion of
hyperfinte type IIIi factors has finite depth then it is isomorphic to its model type
Hi inclusion tensored by a hyperfinite type IIIi factor and it is thus completely
determined by GN,M •
This type IIIi cas e of the general theorem is what motivate d our work. I t
is the analysis of this case that le d us to the consideration of approximate in-
nerness and centra l freenes s fo r inclusions . Subfactor s o f finite dept h o f th e
hyperfinite typ e III i facto r 1Z appear a s ranges o f certain endomorphism s a
(or correspondences, i n the sense of Connes) wit h finite statistica l paramete r
(= (Ind(cr(7£ ) c 7£)) 1//2) i n the theory of superselection sector s of Doplicher ,
Haag and Roberts (as shown by Longo, [LI, 2], see also [FRS], [Fr]), and in the
recent work of Jones and Wassermann ([JW] , [W]) relating von Neumann alge-
bras with unitary conformal field theory. W e will end our notes by explaining
how the classification resul t for subfactors can in fact be used to show that, at
IX
