Operator Algebras and Applications: Volume 2 (London Mathematical Society Lecture Note Series)

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Author: David E. Evans | Masamichi Takesaki

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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England The books in the series listed below are available from booksellers, or, in case of difficulty, from Cambridge University Press.

4 17

27 34 36 39

40 42 43 45

46 49 50 51

54 57 58 59 60 62 65 66 68 69 74 76 77 78 79 80 81

82 83 84 85

86 87 88 89 90 91

92 93 94 95

Algebraic topology, J.F. ADAMS Differential germs and catastrophes, Th. BROCKER & L. LANDER Skew field constructions, P.M. COHN Representation theory of Lie groups, M.F. ATIYAH et a! Homological group theory, C.T.C. WALL (ed) Affine sets and affine groups, D.G. NORTHCOTT Introduction to Hp spaces, P.J. KOOSIS Topics in the theory of group presentations, D.L. JOHNSON Graphs, codes and designs, P.J. CAMERON & J.H. VAN LINT Recursion theory: its generalisations and applications, F.R. DRAKE & S.S. WAINER (eds) p-adic analysis: a short course on recent work, N. KOBLITZ Finite geometries and designs, P. CAMERON, J.W.P. HIRSCHFELD & D.R. HUGHES (eds) Commutator calculus and groups of homotopy classes, HJ. BAUES Synthetic differential geometry, A. KOCK Markov processes and related problems of analysis, E.B. DYNKIN Techniques of geometric topology, R.A. FENN Singularities of smooth functions and maps, J.A. MARTINET Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI Integrable systems, S.P. NOVIKOV et al Economics for mathematicians, J.W.S. CASSELS Several complex variables and complex manifolds I, M.J. FIELD Several complex variables and complex manifolds 11, MJ. FIELD Complex algebraic surfaces, A. BEAUVILLE Representation theory, I.M. GELFAND et al Symmetric designs: an algebraic approach, E.S. LANDER Spectral theory of linear differential operators and comparison algebras, H.O. CORDES Isolated singular points on complete intersections, EJ.N. LOOUENGA A primer on Riemann surfaces, A.F. BEARDON Probability, statistics and analysis, J.F.C. KINGMAN & G.E.H. REUTER (eds) Introduction to the representation theory of compact and locally compact groups, A. ROBERT

Skew fields, P.K. DRAXL Surveys in combinatorics, E.K. LLOYD (ed) Homogeneous structures on Riemannian manifolds, F. TRICERRI & L. VANHECKE Finite group algebras and their modules, P. LANDROCK

Solitons, P.G. DRAZIN Topological topics, I.M. JAMES (ed) Surveys in set theory, A.R.D. MATHIAS (ed) FPF ring theory, C. FAITH & S. PAGE An F-space sampler, NJ. KALTON, N.T. PECK & J.W. ROBERTS Polytopes and symmetry, S.A. ROBERTSON Classgroups of group rings, MJ. TAYLOR Representation of rings over skew fields, A.H. SCHOFIELD Aspects of topology, I.M. JAMES & E.H. KRONHEIMER (eds) Representations of general linear groups, G.D. JAMES Low-dimensional topology 1982, R.A. FENN (ed)

96 97 98 99 100 101 103

104 105 106 107 108 109 110 111

112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131

132 133 134 135 136

Diophantine equations over function fields, R.C. MASON Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN Localization in Noetherian rings, A.V. JATEGAONKAR Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time techniques for analysts and probabilists, L. EGGHE Groups and geometry, ROGER C. LYNDON Surveys in combinatorics 1985, I. ANDERSON (ed) Elliptic structures on 3-manifolds, C.B. THOMAS A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG Syzygies, E.G. EVANS & P. GRIFFITH Compactification of Siegel moduli schemes, C-L. CHAI Some topics in graph theory, H.P. YAP Diophantine Analysis, J. LOXTON & A. VAN DER POORTEN (eds) An introduction to surreal numbers, H. GONSHOR Analytical and geometric aspects of hyperbolic space, D.B.A.EPSTEIN(ed) Low-dimensional topology and Kleinian groups, D.B.A. EPSTEIN (ed) Lectures on the asymptotic theory of ideals, D. REES Lectures on Bochner-Riesz means, K.M. DAVIS & Y-C. CHANG An introduction to independence for analysts, H.G. DALES & W.H. WOODIN Representations of algebras, P.J. WEBB (ed) Homotopy theory, E. REES & J.D.S. JONES (eds) Skew linear groups, M. SHIRVANI & B. WEHRFRITZ Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Lectures on Fermat varieties, T. SHIODA Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds) Non-classical continuum mechanics, R.J. KNOPS & A.A. LACEY (eds) Surveys in combinatorics 1987, C. WHITEHEAD (ed) Lie groupoids and Lie algebroids in differential geometry, K. MACKENZIE Commutator theory for congruence modular varieties, R. FREESE & R. MCKENZIE Van der Corput's method for exponential sums, S.W. GRAHAM & G. KOLESNIK New directions in dynamical systems, T.J. BEDFORD & J.W. SWIFT (eds) Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU The subgroup structure of the finite classical groups, P.B. KLEIDMAN & M.W.LIEBECK Model theory and modules, M. PREST Algebraic, extremal & metric combinatorics, M-M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) Whitehead groups of finite groups, ROBERT OLIVER Linear algebraic monoids, MOHAN S. PUTCHA Number thoery and dynamical systems, M. DODSON & J. VICKERS (eds) Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds) Operator algebras and applications, 2, D. EVANS & M. TAKESAKI (eds)

London Mathematical Society Lecture Note Series. 136

Operator Algebras and Applications Volume II: Mathematical Physics and Subfactors Edited by DAVID E. EVANS University of Wales, Swansea

MASAMICHI TAKESAKI University of California, Los Angeles

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© Cambridge University Press 1988

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Preface

A symposium was organised by D.E. Evans at the Mathematics Institute, University of Warwick, between 1st October 1986 and 29th October 1987, with support from the Science and Engineering Research Council, on operator algebras

and applications and connections with topology and geometry (K-theory, index theory, foliations, differentiable structures, braids, links) with mathematical physics (statistical mechanics and quantum field theory) and topological dynamics.

As part of that programme, a UK-US Joint Seminar on Operator Algebras was held during 20-25 July 1987 at Warwick, with support from SERC and NSF

and organised by D.E. Evans and M. Takesaki. These two volumes contains papers, both research and expository articles, from members of that special week, together with some articles by D.B. Abraham, A.L. Carey, and A. Wassermann on work discussed earlier in the year.

We would like to take this opportunity to thank SERC and NSF for their support, and the participants, speakers and authors for their contributions.

D.E. Evans Department of Mathematics & Computer Science University College of Swansea Singleton Park Swansea SA2 8PP Wales, U.K.

M. Takesaki Department of Mathematics University of California at Los Angeles 405 Hilgard Avenue California 90024 U.S.A.

Contents Volume II: Mathematical Physics and Subfactors.

Some recent results for the planar Ising model D.B. Abraham

1

The heat semigroup, derivations and Reynolds' identity C.J.K. Batty, O. Bratteli, D.W. Robinson

23

C*-algebras in solid state physics: 2D electrons in uniform magnetic field J. Bellissard

49

Spin groups, infinite dimensional Clifford algebras and applications A.L. Carey

77

Subfactors and related topics V.F.R. Jones

103

Quantized groups, string algebras, and Galois theoryfor algebras A. Ocneanu

119

On amenability in type III factors S. Popa

173

An index for semigroups of *-endomorphisms of B(H) R.T. Powers

185

Coactions and Yang-Baxter equations for ergodic actions and subfactors A.J. Wassermann

203

Derived link invariants and subfactors H. Wenzl

237

Vii Volume I: Structure Theory; K-Theory, Geometry and Topology.

UK-US Joint Seminar on Operator Algebras Lecture list

K-theoryfor discrete groups P. Baum, A. Connes

1

Comparison theoryfor simple C*-algebras B. Blackadar

21

Interpolation for multipliers L.G. Brown

55

Elliptic invariants and operator algebras: toroidal examples R.G. Douglas

61

On multilinear double commutant theorems E.G. Effros, R. Exel

81

Loop spaces, cyclic homology and the Chern character E. Getzler, J.D.S. Jones, S.B. Petrack

95

The Weyl theorem and block decompositions R.V. Kadison

109

Secondary invariants for elliptic operators and operator algebras J. Kaminker

119

Inverse limits of C*-algebras and applications N.C. Phillips

127

Partitioning non-compact manifolds and the dual Toeplitz problem J. Roe

187

Cyclic cohomology of algebras of smooth functions on orbifolds A.J. Wassermann

229

UK- US Joint Seminar on Operator 91(gebras Lectures

H. Araki

Invariant Indefinite Metric for Group Representations.

W.B. Arveson

Connections for Semigroups of Endomorphisms of B(H).

C.J.K. Batty

State Spaces, Extensions and Decomposition.

P. Baum

Chern Character for Discrete Groups.

J.V. Bellissard

Using C*-algebras in Solid State Physics.

B. Blackadar

Comparison Theory in Simple C*-Algebras.

L.G. Brown

Interpolation for Multipliers.

R.A. Douglas

Elliptic Invariants and Operator Algebras.

R.L. Hudson

Quantum Stochastic Calculus and Dilation.

S. Hurder

Analytic Invariants for Foliation and their Applications.

J.D.S. Jones

Cyclic Cohomology, Loop Spaces and the Chern Character.

V.F.R. Jones

Subfactors and Related Topics.

J.T. Lewis

Bose-Einstein Condensation and Large Deviations.

R. Longo

Injective Subfactors Invariant under Compact Actions.

T.A. Loring

Embeddings into AF Algebras and Filtration of K-theory.

A. Ocneanu

On the Classification of Subfactors:

Strings and Galois Theory.

C. Phillips

Inverse Limits of C*-Algebras and Applications.

S. Popa

I. On a General Johnson-Parron Problem. II. Rigidity and Amenability in Type III Factors.

R.T. Powers

An Index for Continuous Semi-Groups of

J. Roe

Cutting Manifolds in Half and Cyclic Cohomology.

K. Schmidt

Automorphisms of Compact Groups.

V.G. Turaev

Yang Baxter Equations and Link Invariants.

A. Wassermann

Yang-Baxter Equation for Ergodic Actions and Subfactors.

H. Wenzl

On the Structure of Brauer's Centralizer Algebras.

J.D.M. Wright

Monotone Complete C*-algebras.

*-Endomorphisms of B(H).

SOME RECENT RESULTS FOR THE PLANAR ISING MODEL D.B. Abraham Department of Mathematics, University of Arizona, Tucson, AZ 85721 (on leave from Department of Theoretical Chemistry, South Parks Road, University of Oxford, Oxford)

Introduction

The planar Ising model has become one of the most important statistical mechanical systems for the study of phase transitions and critical phenomena.

Although there

are many rigorous results, such as correlation inequalities,

Peierls argument and the Yang-Lee circle theorem to name but three [reviewed by Griffiths, 1971], which are dimensionindependent in their validity and which lead to results of

considerable interest, only the planar model to date benefits from the added insights which stem ultimately from Onsager's tour de force [Onsager, 1944].

It is not the purpose of

this article to enter into a general review - for this the reader is referred elsewhere [Gallavotti, 1972] but rather to discuss two more mathematical aspects of the development of Onsager's solution.

The first item is the Yang-Baxter

system of equations for the planar Ising model in zero field with transfer in the (1,1) direction.

This work shows that

the Clifford-algebraic structure of the exact solution is a natural consequence of the star-triangle equations.

The

second item is a Fredholm system which turns out to be of crucial importance in understanding surface and interface problems, as well as the pair correlation function.

2

Abraham: Recent results for the planar Ising model Consider a planar lattice drawn on

A(N,M) = ((x,y): -Nsx 0

for

Jkl = 1

A(N,M)

We have

.

this is the ferromagnetic case.

;

It will turn out to be useful to introduce boundary fields H(i)

which supplement the constant field

H

.

They produce

a boundary energy EB({o}N,M) = -

E

H(i) a (i)

iEaA

The canonical configurational probability is PA({a}NAM) = Z

where

g

1

A

exp - g EA({a}N,M)

(1.3)

is the inverse temperature in units of the

Boltzmann constant and

normalises (1.3).

ZA

Consider the one-particle correlation function (B,N,M)

where

the measure (1.3) and been included.

B

is expectation with respect to

denotes that boundary fields have

First, let us consider the case

H = 0

.

At low enough temperatures, the Peierls argument, which is reviewed by Griffiths [1971] shows that lim

N,M--

(+,N,M)

> 0

(1.4)

3

Abraham: Recent results for the planar Ising model where

means that all spins on the boundary

+

+1

fixed to be

are

Since

.

(+,N,M)

lim

3A

but

1

-1

if

(2.34) k

which will be required in the spectral

decomposition of (1.9) if [0,PMI- = 0 are only non-zero if 0

Erl'M+

then matrix elements

or j4k EkM .

Such an

will be a polynomial of even degree in the Fermi operators

and the matrix elements can be obtained in principle at least, from the Wick theorem.

A typical example here is the

Abraham: Recent results for the planar Ising model

11

energy density.

The other case, when

is a great deal more

[0,PM]+ = 0

A

difficult, but has been solved [Abraham, 1978 a,b,c]. typical case here for

is the local magnetisation

0

For ease of notation we shall label (resp. W ESZM(+)) by

.

w ESZM(-)

Typically, we want

S (resp.a).

... G(an) aix Gt(-Sn+1)

where the vector on the left (resp. right) is in `Z,M

Such vectors are also eigenvectors of the

(res p.h,M +).

translation

defined by

T

T a xT-1

=

x -1

=

3

T a1 T

ax

J-1

M >_ j? 2

x aM

Thus the basic objects are

as explained in Abraham [1978a].

+ f1

is a linear form

Returning to (2.27), there is a linear

in the spinors.

dependence Ft(a)

=

M

E

8 e

i(S1a)

(3.2)

Ft(8) -1

and from (2.26) and (2.28) we have 1

{cos(6(S)-e(a))G(a) + i sin(6(S)-6(a))G

t

e

(3.3)

Thus the key matrix elements are

«_IG(an)

...

G(a1)I(D+>

from which the others follow by

using (3.3) and the vacuum properties of

J(D+>

.

In this

section we shall give some improvements over the original exposition which occured to the author in the intervening

12


time, mostly stimulated by the desire to understand various surface properties of the planar Ising model. The first area is the taking of the thermodynamic limit. We define n/2 FM((eia)n)

n

exp i E (aj+O(aj)) < D_IG(an) ...

= M

j=1 (3.4)

It has been shown that [Abraham 1978a]

2

E

M

+

FM ((z) n )

0(zl)

1

1+

I

z1/t-1

0(t)

Z1ESM

\

n

E (-1)J h(t,z.) FM(Al.(z)n)

=

(3.5)

2

where

h(t,z) = and {z

zM = -1}

(z)n = zl,

0(z) 0(t)

is given by (2.33).

H (z) :

1

zt- 1

(3.6)

The summation is over

We use the notation

.

and

... zn

tx(z)n

-1

=

z1..z J.-lzJ.+l..zn

IT

4

for

1

(3.10)

to which Wick's theorem may be applied using the linear the r.h.s. of (3.10) is a Pfaffian which

dependence (3.3);

is the square root of an antisymmetric determinant [Caianiello, 1953], to which the Hadamard bound may be applied giving sup n IFM(z)n1

n 1):

iw)2n+l)

and

2n+1 =

E

(-1) g(w

F(A.(eiw)2n+1)

1

(4.3)

where the last term is the usual Pfaffian with initial condition as in (3.16).

The reader should consult Abraham

Abraham: Recent results for the planar Ising model [1978b] for details of the function

and

g(w)

terms of the Wiener-Hopf factorisations of

17 in

f-(w1.w2)

0

.

The pair correlation function is given by Co

= lim M->°

1

I

0 n!Mn (w) EQ n(-)

M

n

2

IFM(e iw )

E

E

n

exp- E (IxIY(w.)+i w.) 1

Y J

J

M

n

(4.4)

First note that the series converges uniformly in (x,y) E

by the Weierstrass M-test;

2z 2

M

for any

thus the limit can be

taken term-by-term using the convergence results of section 3 to give

2

Tr

= E

J.f

1

n!(2Tr)n

0

d(w)

IFCo (elw) n n

I

n exp - E (IxIY(w.)+i w 1

J

Y

-7T

(4.5)

T < Tc

Notice that for Using the identity

the sum is only over even

(PfD)2 = det D

for antisymmetric

n

D

,

it

is clear that (4.5) is actually a Fredholm determinant, since the odd-n terms vanish (if odd, then

det D = 0

=

)

D = -DT

and

D

is

giving

d(l)

=

E

K 1n

(-1)n

(2Tr) n!

d(w) (4.6)

where

d(X)

n x n, n

J..f -7T

w1, ...wn

1

//I

(4.7)


18 with

w1...wn K

(4.8)

det K

w1...wn

where i h (wi)

1J

(4.9)

h (wi ) f_(wl,w2)

with

= exp -(xY(w) + iYw)/2

h (w)

(4.10)

A useful reference for the classical Fredholm theory is Since we have

Smithies [1965].

> (m*) 2 > 0

(4.11)

by the Griffiths inequalities [Griffiths, 1971 and references therein] and the definition of the low-temperature region, (4.6) gives

d(l) > 0

to simplify (4.7);

Thus we can use resolvent formulae

.

this is rather like the linked cluster

expansion in statistical physics.

Such an approach gets one

into inverse scattering, but that is another story. x = y = 0 getting

,

we can evaluate the l.h.s. of (4.6) trivially,

(m*)2

if_(wl,w2)

from (4.7) by obtaining the spectrum of

and evaluating the Fredholm determinant.

The work for

T > Tc

is slightly more complicated

because (4.5) is a sum over odd n jFx(eiw)2N+112

where

=

E

.

chain

But it turns out that (4.12)

det D

is antisymmetric, of even dimension.

D

simple graphical result [Kasteleyn, 1967]. coupling parameter series for

If

JX

This is a

Introducing the

as before allows us to rearrange the

small enough to give

< a(x,y) 0 ( 0 , 0 ) > ( X )

=

(h,(1 + X K2)h) d(X)

(4.13)

Abraham: Recent results for the planar Ising model followed by analytic continuation to The same idea with

x = y = 0

X = 1

since

19 d(1) 4 0

suffices to evaluate

and, incidentally, shows that only one subsequence actually occurs.

5. Conclusion

In these notes we have shown that the Jordan-Wigner transformation, which plays a central role in the algebraic development of the statistical mechanics of the planar Ising

model, is a natural consequence of the star-triangle relation and that the QISM is very useful in this respect since it provides motivation.

Then we outlined the matrix element

theory of the pair correlation function and its relationship to Fredholm theory.

The author thanks Professor D.E. Evans very much for the hospitality generously extended to him at the operator algebra symposium in Warwick.

20

Abraham: Recent results for the planar Ising model References

Abraham, D.B. (1976a). Oda Operators and Spinor algebras in lattice statistics: n-point tunctions for the rectangular Ising model. Commun. Math. Phys. 59, 17-34. Abraham, D.B. (1978b). Pair function for rectangular Ising ferromagnet. Commun. Math. Phys. 60, 181-191. Abraham, D.B. (1978c). n-point functions for the rectangular Ising ferromagnet. Commun. Math. Phys. 60, 205-213. Abraham, D.B. and Davies, B. (1988). The quantum inverse method and Bogoliubov-Valatin transformation. To be published. Abraham, D.B. and Martin-Lof, A. (1973). The transter matrix for a pure phase in the two-dimensional Ising model. Commun. Math. Phys. 32, 245-268. Abraham, D.B. and Reed, P. (1976). Interface profile of the Ising model in two dimensions. Commun. Math. Phys. 49, 35-46.

Aizenman, M. (1979). Instability of Phase coexistence and translational invariance in two dimensions. Phys. Rev. Letters 43, 407-409. Aizenman, M. (1980). Translational Invariance and instability of Phase coexistence in the two-dimensional Ising system. Commun. Math. Phys. 73, 83-94. Araki, H. and Evans D.E. (1983). A C*-algebra approach to phase transitions in the two dimensional Ising model. Commun. Math. Phys. 91, 489-503. Baxter, R.J. (1982). Exactly solved models in statistical mechanics. London: Academic Press. Caianiello, E.R. (1953). Combinatorics and renormalisation in quantum field theory. New York: Benjamin. Carey, A.L. and Evans, D.E. (1986). The Operator Algebras of the two dimensional Ising model. Contemporary Maths. Artin s Braid Group (to appear). Evans, D.E. and Lewis J.T. (1986). On a C*-algebra approach to phase transitions in the two dimensional Ising model II. Commun. Math. Phys. 102, 521-535. Gallavotti, G. (1972). The Ising model. Riv. Nuovo Cimento 2, 133.

Griffiths, R.B. (1971). Phase Transitions in Statistical Mechanics and Quantum Field Theory. ed. C. De Witt and R. Stora. pp 241-281. New York: Gordon and Breach.


21

Gutkin, E. (1986). A comment on Baxter condition for commutativity of transfer matrices. J. Statistical Pnys. 44, 193-202.

Higuchi, Y. (1979. On the absence of non-translationallyinvariant Gibbs states for the two-dimensional Ising model. Colloq. Math. Soc. Janos Bolyai 27, 517-533.

Kasteleyn, P.W. (1967). Graph theory and crystal physics, in Graph Theory and Theoretical Physics. ed. F. Harary. London: Academic Press. Kaufman, B. (1949). Crystal Statistics II: partition function evaluated by spinor analysis. Phys. Rev. /6, 1232-1243. Krein, M.G. (1958). Integral equations on a half-line with kernel depending upon the difference of the arguments. Usp. Mat. Nauk. (N.S.) 13, no.5(83), 3-120. English translation: A.M.S. translations, series 2, 22, 163 (1962). Lewis, J.T. and Sisson, P.N.M. (1975). A C*-algebra of the two-dimensional Ising model. Commun. Math. Phys. 44, 279-292. Lewis, J.T. and Winninck, M. (1979). The Ising model phase transition and the index of states of the Clifford algebra. Colloq. Math. Soc. Janos Bolyai 27, 671-679. Lieb, E.H. and Wu, F.Y. (1972). Two-dimensional Ferroelectric models in Phase Transitions and critical phenomena I, eds. C. Domb and M.S. Green. New York: Academic Press. Nachbin, L. (1970). Holomorphic Functions, domains of holomorphy and local properties. Math. Studies 1. Amsterdam: North Holland.

Onsager, L. 1944). Crystal Statistics I: a two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117-149.

Onsager, L. (1971). in Critical Phenomena in alloys, magnets and superconductors. eds. R.E. Mills, E. Ascher and R.I. Jaffee. New York: McGraw-Hill. Onsager, L. and Kaufman, B. (1949). Crystal Statistics III. Short range order on the binary Ising lattice. Phys. Rev. 76, 1244-1252. Reed, M. and Simon, B. (1971). Methods of Modern Mathematical Physics. I: Functional analysis. New York: Academic Press. Schultz, T.D., Mattis, D.C. and Lieb, E.H. (1964). Two-dimensional Ising model as a solvable problem of many fermions. Rev. Mod. Phys. 36, 856-871.

22


Smithies, F. (1965). Integral Equations. Cambriage Tracts in Mathematics and Mathematical Physics 49, Cambridge: Cambridge University Press.

Stephen, M.J. and Mittag, L. (1972). A new representation of the solution of the Ising model. J. Math. Phys. 13, 1944-1951. Takhtadzhan, L.A. and Fadeev, L.D. (1979). The quantum method of the inverse problem and the Heisenberg XYZ model. Uspekhi Mat. Nauk. 34 5, 13-63. English translation: Russian Math. Surveys 34 5, 11-68. :

Thacker, H.B. (1981). Exact integrability in quantum field theory and statistical systems. Rev. Mod. Phys. 53, 253-285. Widom, H. (1965). Toplitz matrices. M.A.A. Stud. Math. 3, 179. (ed. I.I. Hirschman).

THE HEAT SEMIGROUP, DERIVATIONS, AND REYNOLDS'S IDENTITY Charles J.K. Batty', Ola Bratteli2 and Derek W. Robinson3 Mathematics Research Centre, University of Warwick

1. Introduction Let A be a C"-algebra, and g a finite dimensional Lie algebra acting as a Lie algebra of closed *-derivations on A . We will consider the problem of when g exponentiates to a

representation of the simply connected covering Lie group G of g, as a group of *automorphisms of A, [JMI], [Rob 1], [BGJR1].

Let A. be the *-algebra of C,; elements for g , i.e. A. is the intersection of the domains of all monomials in the elements of g. A minimal requirement for g to exponentiate is that A. be dense in A , and we will assume this throughout this paper. Recall also that an operator S is called conservative if S and -S are dissipative, i.e. if 11(1 + A,S)(x)II IIxII for all x E D(S) and all ), E R, [BRI]. Let Sl, ..., Sd be a basis for g. It was proved in [BGJRI] in a general Banach space setting that g exponentiates to an isometric representation of G if and only if the following three conditions hold: (i)

The operators S1, ..., Sd are conservative.

(ii)

The closure 0 of the Laplacian A = -Y, 52 exists as a densely defined operator such

d k=1

that 2i generates a contraction semigroup St = exp (-tA) on A and

St AcA for t> 0 . (iii) There exists a constant C > 0 such that 116k Shc

1). The example is a projection, i.e. R2 = R , such that R is a conditional expectation in the sense

R(xyz) = xR(y)z for y E A, x,z a R(A), but R does not have norm 1, see Example 4.3.

To get further insight into Reynolds's identity, we finally show that if R is a positive normal operator satisfying Reynolds's identity on a von Neumann algebra M with a separating and cyclic vector such that the corresponding state to is R-invariant, then R decomposes as R = (1 - S)-'E where E is a normal conditional expectation (of norm 1) from M onto a von Neumann subalgebra N, and S is a a-weakly closed, a-weakly densely defined *derivation of N . This is Theorem 5.1, which is a straightforward generalization of GianCarlo Rota's corresponding theorem when M is abelian, [Rot 1]. Example 1.1.

(This approach was suggested by David E. Evans, and the counterexample was

provided by Man-Duen Choi). Let S be a derivation on a C*-algebra A , and assume that -82 (without closure) generate a semigroup of completely positive contractions. Then

(1 -

ES2)-1

= r dt ej eÒ82 is completely positive. Since (1 + s8)-1 (1 - £S)-' = (1 -

x282)-1

and (1 - eS)-' + (1 - eS)-' = 2(1 - 1282)-1 one could hope that this would imply that S = (1 + e5)-' , T = (1 are positive and that S is a generator by Theorem 4.1. However, there exists a C'-algebra A and bounded linear operators S and T on A such that ST and S + T are completely positive, ST = TS , but neither S nor T is positive. An example is A= M2 with eS)-r

26

Batty, Bratteli & Robinson: The heat semigroup

Tl

[a b d

and S=-T. Then S +T = 0, ST = TS and a -b ST

I-c

d1

.

Thus ST is an automorphism and hence completely positive. However, S and T are manifestly non-positive.

Note also that if K is any densely defined operator on a Banach space B , then

(0 Ol K 0 is a densely defined operator on A = B X B such that SZ = 0 . Thus it is not true in general that if - SZ is a generator then S is so. See [JMI, Example 8.51. 2.

Holomorphic semigroups and conservative operators

Theorem 2.1. Let g be a finite dimensional Lie algebra acting as a Lie algebra of closed d

conservative operators on a Banach space A . Let A = -F, 8k2 be an associated Laplacian. k=1

If A is closed and A generates a holomorphic semigroup of contractions on A , then g

exponentiates if and only if e tAAc:A for t> 0. Proof.

First note that, as in [Rob 1, Lemma 2.3], since Sk is conservative we have CIA(x)II < 11(1- c6k)(x)II + IIXII 11(1 + c6k)(l - eSk)(x)II + IIXII

= 11(1- e2

8k2)(x)II + IIXII

e2118k2(x)II + 211x11

for e > 0 and x E D(6 ). Thus 118k(x)I1 < el1Sk2(x)l1 +

IIXII

But the identity (1 - e2 Sg) = (1 + e6k)(1 - e6k) implies that

11(1 - e2 5k)(x)II ? Ilxll for d

x E D(6 ), thus

5k2

is dissipative and hence closable. As A = -Y_ 5k2 is closed and k=1

D(A) c D(8k2) it follows from H4 rmander's version of the closed graph theorem that there

exists a constant C > 0 such that II8k2(x)II < C(IIA(x)II + Ilxll)

for x e D(A), [Yos 1, Theorem II.6.2]. Combining this with the previous inequality we get IISk(x)II s eCIIA(x)II + (CC +

e) Ilxll


27

for x E D(A). But as a t° = S1 is a holomorphic semigroup, we also have an estimate of the form K

IIAS1xll
0 and Jim 11(£5(1 + £8)71)n (x)II = 0

n-

for each x e A . 8.

R = (1 + £8)-1 exists for all e > 0 and R(xy) = I (R(1 - R)" X) (R(1 - R)"y) n=0

for each pair x, y E A , where the latter series converges in norm. Furthermore, when the conditions 1-8 are fulfilled, the expansions

(£5(1 + ES)-1)n(x) = x - r dt e-` Q-1(t) e-1(X) and

(£S)" (1 + C5)-n--I(X) =

( dt C' Lno(t) e-8(x)

are valid, and II(ES)" (1 + £S)-i-111

this being the best possible general estimate.

0, (1 + E8)-1 exists and

4(p).

II(ES)n (1 + S)-n-pll

is uniformly bounded in n . (p = 0, 1, 2, ....)

For each e > 0, (1 +

6(p).

E8)-1 exists and

lim II(eS)n (1 + ES)-n-pll = 0

n-w-

(p=1,2,3,....) For each e > 0, (1 + ES)-1 exists and ES)-"-p(x)II

lim II(ES)n (1 +

=0

for each xE A. (p=0,1,2,....) Note in particular that condition 6(0) is not among the equivalent conditions; in fact if S unbounded generator then the spectrum of E8(1 + E8)-1 contains 1, so 11(F'8(l + ,)-1))nll ? 1 for all n

is

an

Proof.

1 => 2 and 1 => 3 are standard consequences of the Laplace transform formula dt e-` e(x),

(1 + ES)-1(x) =

see [BR1].

2 => 1: It is immediate from Hille-Yosida theory that S generates a one-parameter semigroup e'8 of contractions on A . To show that these are *-morphisms, one may for example apply the resolvent n-1 times to Reynolds identity to obtain (1 + t S)-n (xy)

n

(1 + L8)-n(x) (1 +

n

+

n

n S)-k (1+ n S) n-t+k(S(x))(1+ n

(1+

t2

s)-n(Y)

S)-n-l+k(S(Y)))

k=1

n

for x, y E D(S), and hence we get the estimate S-n(xY) - (1 + n

II(1 +

n

n

n2

IISx)II IIS(Y)II

(1 +

no)-n(Y)II

33

Batty, Bratteli & Robinson: The heat semigroup Letting n -+ °° , we deduce ets(xy) = e7'8(x) C'8(y),

see [BR1, Theorem 3.1.101. Thus a-is is a semigroup of *-morphisms.

3 => 2:

If R =

(1+eS)-t

is positive, it follows from Reynolds's identity

R(x*x) = R(x)* R(x) + R(((1-R)x)* ((1-R)x)) that the generalized Schwarz's inequality R(x)* R(x) 1

,

hence the spectrum of T is contained in the unit circle.

Repeating this argument, with 6 replaced by e6 for e > 0 , it follows that if z is contained in the spectrum of 6 , then I

1

-

I

1

1 1 + cz 1

for all e>0, i.e. Re z >- 0 . But the set of

_ 0 is the disc of radius l+z 1h around 1h , thus the spectrum of T is contained in this disc. In particular the spectrum


36

of T is contained in the unit disc, and intersects the unit circle only in 1. Now

(1-T) (1-zT)-1 RP = Y, Crn_r+') RP z" n=0

the power series having radius of convergence 1 and singular set (1). Furthermore N

III ('I"'-T"+') RPII = 11(1-r+') RPII n=O

uniformly bounded in n . It follows from [AOR, Theorem 4] that lim II(T"-Tn+1) RPII = 0 , which is 6(p+l). Note that if p = 0, this implication also follows n-+which is

from Katznelson-Tzafriri's stability theorem, [KT1, Theorem 1 with Remark], which states that if T is a linear operator on a Banach space such that II'1°II is uniformly bounded in n, then

lim IIT" - Tn+111 = 0 if (and only if) the spectrum of T intersects the unit circle in at most

n--)--

the point z = 1

.

6(p) => 7(p)

for p = 1, 2, 3 ...

4 => 7 (= 7(0)):

As in the proof of 4 => 6 (i.e. 4(0) => 6(1)) above one, uses Katznelson-

is trivial.

Tzafriri's stability theorem [KT1], [AORI] to argue that lim IIT"(1-T)II = 0 , n-*-

but as

Range (1-T) = D(6), it follows that

lim II'I°xII = 0

11-9=

for each x E D(6). Since D(6) is dense, it follows from 4 that this holds for each x e A , and that is 7. (In this argument we could also have used Arendt-Batty's stability theorem, [AB1, Theorem 2.4 and Remark 3.3], but the latter is formulated in the setting of oneparameter semigroups, and is more general than we need).

7 => 8:

Since e6(l+e6)-1 = 1 - R , it is clear from 7 that the remainder term in the iterated Reynolds's formula n-'

R(xy) _ E (R(1-R)kx) (R(1-R)ky) kO

+ R(((1-R)"x) ((1-R)"y)) tends to zero as n - oo . This proves 8. 8 => 3:

It is clear from the expansion 8 that R(xx*) >: 0 for all x e A , so R is positivity

preserving.

It follows from 7(p) that (1 R)"(x) tends to zero for each x r= Range (RP). As 7(p) => 3: in 7 => 8 it follows that

R(xx*) _

(R(I-R)kx) (R(1-R)kx)t >- 0 k=O

37


for x e Range (RP). Since Range (R) = D(S), which is dense, Range (RP) is also dense, so R is positivity preserving by closure. eS(1+eS)-1)n

This ends the proof of the equivalences in Theorem 4.1. The expansion of was obtained in the proof of 1 => 5, and similarly n

dt e7' n! ( at )n e-8(x)

Sn(1+8)--1(x)

dt ` dt

= r dt C'

)n

(e-`

a

n)

e`8 (x)

dt

.

Thus the best possible estimate is IISn(l+S)-n-tli

5 f dt ILn°(t)I

and this estimate is attained for A = C°(R), S =

dx

This ends the proof of Theorem 4.1, but we finally remark that condition

7,

lim II(s8(1+e6)-t)n(x)II = 0 , can be proved directly, without using Katznelson-Tzafriri's

n-+-

theorem, when S is the generator of an automorphism group, as follows: If A[-N, N] is the Arveson spectral subspace corresponding to a finite interval [-N, N], then the spectrum of the restriction of e8(1+85)-1 to A[-N, N] is a sector of the circle of radius 1/2 around '/z which does not contain 1. It follows that II(ES(I+ES)-1)n

n

I

A[-N, N]II = 0

But approximating a general element x with an element in some A[-N, N]

in norm, and

using that II(eS(1+sS)-')nll is uniformly bounded, it follows that

lim

n-w-

II(e8(1+e)-1)n(x)11

=0.

We next show that Reynolds's identity alone, as an algebraic identity, does not suffice for the implication 4 => 3 in Theorem 4.1.

Example 4.3. There exists a C'-algebra A and a *-linear bounded operator E on A such that E satisfies Reynolds's identity

E(xy) = E(x) E(y) + E((1-E)(x) (1-E)(y)) , and, moreover, 11(1-E)nll is uniformly bounded in n, but E is not positivity preserving.


38

Indeed, if E is any *-linear operator which is a conditional expectation in the sense

E2 = E and

E(xy) = xE(y), E(yx) = E(y)x

for x E E(A), y E A , then E satisfies Reynolds's identity. But a conditional expectation in this sense does not need to be positive. (This was pointed out to us by Jun Tomiyama, and he also provided the following example). Take any C*-algebra A with a sub-C`-algebra B such that there are more than one projection of norm one from A onto B . For example, if

A = L(H) and B is a maximal abelian subalgebra of L(H) without minimal projections, then there exists both a normal and a non-normal projection of norm one from A onto B, [Tom 1]. Let El, E2 be two such projections, and put

E = Et + 7l(El - E2) , where ? is a real number. As both Et and E2 have the same range one has El E2 = E2 and E2 El = E1, and hence E is a conditional expectation in the above sense. But if ? is chosen large (positive or negative) it is clear that E can be made not positivity-preserving. Moreover, as (1-E)" = (1-E), the sequence II(1-E)"II is uniformly bounded. We next show that even the resolvent of a derivation need not be positivity preserving:

Example 4.4. There exists a C`-algebra A and a densely defined *-derivation S on A such that the spectrum of S is empty, i.e. exists as a bounded operator for all e C , but S is not a generator.

Let A be the C`-algebra of continuous functions on [0, 1] vanishing at 0, and let defined on the set D(S) of f e A such that f'

S=

d

A A. The differential equation

Xf - f= g

where g(0) = 0 , with the initial condition f(0) = 0 , has the unique solution

f(x) = (? for each ?1. E C

.

S)-'(g)(x)

x

_ - ( dy ex(x-y)

g(y)

Thus (? - S)-1 exists, and we have the identity:

II( - S)-'II = sup eRe'i dy e Re(l)y 05x51

Re X

Red 1

if

Re 2,x0

if Re=0


39

We thus see that II(A, - S)-1

- 0 => a is of translations to the right given by

(e cs fl(x)

f(x-t) =

Note in particular that e 7 ' 6=0 fort >- 1

0 .

if

x>>-t

otherwise .

On the other hand

II(. - 8)-111 >

Re

if Re ? > 0 , so S is not a semigroup generator. This confirms that translation to the left is not well-defined in this example.

Applying the resolvent formula two times, we get

(X2 - S2)-t(g)(x) =

(), -

S)-1 (?

+ 8)-1(g)

x

=

1

f dy sinh (? (x-y))g(y)

for A, > 0 . This leads to the identity SZ)-'II =

Cosh(?,) - 1

for ? > 0, and hence - S2 is not a generator of a semigroup in this example, corroborating Corollary 3.2.

Finally, the identity

valid for A. * 0 , shows that the spectrum of (R - 8)-1 is {0} for all R E C . In particular this means that the spectrum of - E8(1 + = 1 - (1 + E8)-1 is { 11 fore > 0 , and the spectral radius formula implies that for any rl > 0 there is a Cn,E > 0 such that e8)-1

II(E8(1-E8)-1)nll

- 0, x E D(S).

Proof. The inequality w(R(x)* R(x)) 0 in this case. It may well be true that (1) is true for all positive t even for isometric representations of general Lie groups G , but this does not follow from the estimates in [BGJRI].

We consider the two cases of small positive t and all positive t separately. Proposition 6.1.

Let S be a holomorphic semigroup on a Banach space B with generator

A, D be a core for A with S t B c D for all t> 0 , and H : D-411 be linear and

44


closable. The following conditions are equivalent. (1)

There is a constant Cl such that IIHStII

(2)

C t

for

0 < t:5 1

.

There is a constant C2 such that IlHxll

(3)

0 , and H : D - B be linear and closable. The following conditions are equivalent: Proposition 6.2.

(1)

There is a constant C1 such that ¶AS f

i

C

or a t

46 (2)

Batty, Bratteli & Robinson: The heat sernigroup There is a constant C2 such that IIHXII (2).

For any e>0,

i

IIH(1+e2A)-Ill = 11 dt e-` HSehII

0. As before we define by mean of the projection method the following effective hamiltonian :

Hen(B,z) = IIHII + IIH(1-II)

1

z1- (1-rl)H(1-11)

(1-II)HFI

(20)

This is a norm analytic function of z in a complex neighbourhood of the band, and z belongs to the part of the spectrum of H in this band if and only if z belongs to the spectrum of H(B,z), again an implicit equation for z. At last we introduce the orthonormal basis B B obtained from the W, 's by diagonalising the matrix of inner products. Then every element W of tfie subspace generated by'f1 can be identified with the sequence (f(a);aE r) of its coordinates which is square summable. Thanks to this description one gets [107]: Theorem 3: The band hamiltonian Ht(B,z) given by (20) can be expanded as :

Hen(B,z) _

E (z;a') W(a')

with

(W(a')f)(a) = eirzeBm(ua') f(a + a')

(21)

a'er

where the W(a)'s obey the Weyl commutation relations. Moreover, the coefficients E.t(z;a) of the expansion satisfy :

E,,(z;a) = $ (a) + 0 (IBI)

(22)

where the $ (a)'s are the Fourier coefficients of the energy band function E(k). If at and a2 generate r, one has :

W( mlal+m2a2) = e411'"2 Um1V'"2

with

U V = e2hca. V U

(23)

0

From (22) the energy band function E(k) gives rise to the effective hamiltonian after performing the Peierls substitution up to an error of order B and the projection method give an exact effective band hamiltonian.

55

Bellisard: C*-algebras in solid state physics

III- The Rotation Algebra III-1) TonologvIn the previous section we have seen that every operator of interest can actually be expanded in term of trigonometric polynomials in two unitaries U and V such that :

UV= e21 VU

(1)

The parameter a is the ratio of the magnetic flux through the unit cell of the Bravais lattice, and the flux quantum We (see §II eq(11)). In the strong field expansion however, we have seen that a must be replaced by 1/a. The C*algebra generated by two such unitaries is called the rotation algebra and will be denoted by Aa . It has been introduced and studied by M.Rieffel [82] in 1977. However, in the early seventies A.Grossman [44] showed that the Von Neumann algebra generated by them is type II when a is irrational. This result was a source of difficulties in understanding the spectral behaviour of Bloch electrons in a magnetic field. Actually it is now known that : Theorem 4 :

1) Aa= rla+t ;. and A., are isomorphic if and only if (x = ± a' (mod. 1) 2) If a is irrational, A. is simple. 3) Let W 1 and W2 two qxq unitary matrix such that :

W1W2= e"4gW2W1 W2q=W2q=1 (2) then APq is isomorphic to the set of matrix valued functions F(k) over R2 such that : (Wtm2W2 1°t) F(k) (Wtr2W2ì)* = F(k + m)

m = (Mt,m2)E Z2

(3) 0

Remark : The periodicity in I/a for a strong field was known by the physicists from a long time: Landau noticed it in 1930 [63] but doubted if it where possible to observe it. He was not aware of the experiments of De Haas and Van Alven [68] at about the same time exhibiting such periodicity which remained mysterious quite a long time. It was not before the work of L.Onsager in 1952 [70] that a

semi-classical theory was provided for the phenomena which proved to be one of the basic experimental method in the measurement of the Fermi surface.

Corollary : If a is irrational, every non zero representation of A. is faithful.

0

A trace on Aa is defined as the unique state r such that : T (Umt V1O2) = 0

Pr

if

m = (mi,m2) * 0

(4)

For every a the trace on A. is faithful. 0

Since a is proportional to the magnetic field, it is liable to vary continuously. The theorem 4 shows that A. does not vary continuously. However, as shown by G. Elliott the family (Aa; (XE R) is a continuous field of C*Algebras [31,108]. A simple way to construct this field, consists in using the Universal Rotation Algebra A defined as the C*Algebra generated by three unitaries U,V,and X, such that: [U, 711 =[V1 X1 =0 UV=7XVU (5) Then we define by pa the canonical *homomorphism from A onto A. such that :

Pa (U) = U

Pa (V) =V

pa (A) = e2ixa

1

(6)

Elliott's result [371 is then equivalent to : Th

:

1) Let A be an element of the Universal Rotation Algebra. Then the maps a->Ilpa(A)II

and a->T{pa(A)) are continuous.

2) Let H be self adjoint in A. Then, the gap boundaries of the spectrum of pa(H) are continuous functions of a. 0


56

111-2) Differential structure-

As remarked by A.Connes [28], one can see as a non commutative 2D toms by identifying U and V with the coordinate functions. This allowed to define two derivations as follows both on

AandA. :

a1U = Nit U

a1V=0

a2V = 2ia V

a2u = o

(7)

These derivations commute and are actually the infinitesimal generators of a two parameter group of automorphism representing an action of the 2-toms defined by : (8)

11x.Y( Um1Vm2) = e2n(m1x+m2Y) Um1Vm2

Ck(A.) will denote the intersection of the domains of every powers of order at most k of these derivations. Thus A belongs to Ck(Aa) means that the map (x,y)-> ilx (A) is Ck in the norm topology. One also introduces the Sobolev space 9!$ as the Hilbert space obtained by completing Ck(Aa) (for k>-s) under the norm :

II A II$ = (2(I(-0/4rt2+l)'nAI2))1f2

where

A = a12+a22

(9)

Using the Fourier expansion in terms of U and V, it is possible to define this norm for any non negative real value of s. This family of space possesses many of the properties of the usual Sobolev spaces. 111-3) The Chem class of a Droiection-

Let now P be a projection of A.. If P belongs to H, the Chem class of P is defined by analogy with the 2-toms as :

Ch(P) = (1/2in) ti (P[a1P,a2P])

(10)

Then one can check that Ch assumes the following properties [6, 29] :

i) Invariance : let P and Q be two equivalent projections in 1f namely such that SS* = P and S*S = Q for some S in C'(V., then Ch(P) = Ch(Q;. ii) Additivily : let P and Q be two projections in xt such that PQ = QP = 0, then : Ch(P®Q) = Ch(P) + Ch(Q)

The invariance property implies in particular that Ch extends to all projections of Aa for any projection is equivalent to a C1 projection [29]. Moreover it implies the omotony invariance namely if s->P(s) is a norm continuous family of projection in AJ., Ch(P(s)) is constant for two projections close enough are equivalent [75]. The main result concerning the Chern class is the following : Th

r m 6: If P is a projection of Aa, its Chern class is an integer. 0

This result was proved in this particular case by A. Cones in 1980 [28], but later results of Connes allow to extend it in a much bigger generality in order to be used for the Quantum Hall Effect [18]. A sketch of the proof will be given at the end of this paper.

111-4) A Ito like derivation "How do gaps boundaries of the spectrum of H vary whith the magnetic field ? This question motivates us to introduce in A the derivation with respect to a. More precisely let A be polynomial in U,V and X. One can expand A as follows:

a(m;a) W(m) p(A) = a met

with

W(m1,m2) =

We define the operation a by the following formula :

eixam,mZ

Um' V

(11)

57

Bellisard: C*-algebras in solid state physics p(aA) _ Y,

aa(m;a)

as

W(m)

(12)

CI(A) will denote the completion of the set of polynomial under the norm : IIAIIdt = IIaAII+IIa1AII+Ila2All

(13)

This operation satisfies the following rules [107]:

Theorem 7:

1) If A and B belong to C'(A) :

a(AB) = aA B + AaB + i/4tt (a1Aa2B - a2Aa,B)

(14)

2) If A belongs to CI(A) and if it is invertible, its inverse belongs to CI(A) and

a(A-I) =

-A-t (aA + i/4n(a,A A7Ia2A - a2A A-Ia,A)) A-'

(15)

3) Streda's formula : if PE C1(A) is such that for a in the interval (a,b), pa(P) is a projection, then one gets in (a,b) :

a 2(pa(P)) = Ch(pa(P)) as

(16)

0

Corollary : The C*algebra A has no non trivial projection. 0

The proof of 1) and 2) in theorem 7 is a simple matter of calculation, and to prove 3) it is enough to remark that a projection satisfies P2 = P, that r (AB) = 'r (BA), since 'r is a trace and that

? r(p (A))/aa ='t(pa(aA)).

`tfhe proof of the corollary starts with the Streda formula. If P belongs to CI(A), by theorem 6 the right hand side of (16) is a fixed integer for all a. Since the left hand side is the derivative of a periodic function of a, we must get Ch(pa(P)) = 0 . We then integrate both members of (16) from zero to one and we get a formula which extends to any projection of A. Now, from Pimsner and Voiculescu [77], we know that 'r(p (P)) = m+n a, and from the Streda formula it follows that n = 0. Since P is a projection and't(p0(P)l is continuous in a, we get in = 0 or 1. The trace being faithfull (prop. 1), it follows that pa(P) is either 0 for all a's or 1 for all a's. Remark : The Streda formula presented here is the mathematical version of a formula relating the derivative of the density of states with respect to the magnetic field to the Hall conductivity [see this relation in 18]. In the original work by Streda [89] another contribution was added to take into account the effect of the disorder in the crystal.

58


IV- Hofstadter like Spectra IV-1) The snectn>rn of the Harper onerator-

In the section II we saw that the effective hamiltonian describing the quantum motion of a 2D Bloch electron in a uniform magnetic field can be expanded in a non commutative Fourier expansion in the operators U and V in the Universal Rotation Algebra. However in practice the exact expression may be quite complicate and if we only want to get qualitative properties, simplifications are required. The simplest approximation was proposed by Harper [49] in 1955 in dealing with square lattices, whereas Claro and Wannier [25] studied the corresponding model for triangular lattices (eventhough they claimed having studied the hexagonal one) and R.Rammal investigated recently the hexagonal lattice [81].'Ihe square lattice is characterized by a complete symmetry of the hamiltonian under a n/2

rotation in space, which in turn means that it must be invariant under the automorphism R of defined by [(U) = V and (3(V) = U*. If we retain in the Fourier expansion only the lowest order terms we end up with the Harper model :

H=U+U*+V+V*

(1)

Let us notice however that the physical meaning of this approximation is different at high and low magnetic field : in the former case it consists in approximating the potential itself by its first Fourier coefficients, whereas in the former case, it is the energy band function in momentum space which is approximated. Thus the strong field-weak field duality has also something to do with the position momentum duality by mean of Fourier transform.

W

(i) yit.i. (t)-'lie spectmm of the Harper7(ami(tonian H = U+U*+V+V', where U V = e2iaa VU as a function of the parameura. (ii)-7rre same for the 7(ani(tonian H(s) = U + V+ o(eizav _ e imV *)U + h.c. with a = 0.1 (Taken from st:f.

[26ii])

59

Bellisard: C*-algebras in solid state physics This model has been the focus of attention ever since. Among the main contributions let us notice the work of M.Ya.Azbel' [12] in 1964 who guessed a rule for building up the spectrum. Independently in 1965, W.G.Chambers [24] using a semi classical approach proposed by Onsager [70], Adams II [2], Blount [21] and A.B.Pippard[78], rediscovered the rule together with the practical algorithm to compute numerically the spectrum. As in the work of Azbel' whose justification

is still today quite obscure, he realized that if a = 1/(al+l/(a2+...)) is decomposed into continuous fraction expansion, each original band is divided into at subbands, each of which being divided into a2 subbands and so on. Nevertheless he "doubted whether this result is of great practical interest" an interesting comment in view of the future developments of the subject. D.Langbein [42, 64] gave a rather detailed calculation of the subband structure of (1) whithout getting far enough however to see the global structure in term of the magnetic field. D.G.Hofstadter [53] made it in 1976 and deserves the credit of having found a fascinating fractal structure (see fig.li) by a time where it was no longer shameful to consider these monsters in Physics. Actually the precise rule in getting the subbands in the Hofstadter spectrum is a little bit different from the Azbel' and Chambers one, as was explained by Hofstadter : the middle bands have a different pattern but the rule may also be described in term of a modified continuous fraction expansion. Let us also indicate that if one breaks the square isotropy, one gets the so called "Almost Mathieu" operator [14,86]: H(µ) = U + U* + µ (V+V *)

-4

0

-2

0

(2)

2

L

1

a- (upper fig.).Cntica! umperature versus magnetic fa i for a square network, made of superconlucting a4minrutn.'1St arrows indicate the magnetic field values corresponding to integral number of f1UX quama a= $ o per unit of of the network; one sees a parabolic background corresponding to the bu& properties of the superconductor (see [100]). (lower fig.)-A magnified view of the first period of the critical line in reduced units with a view of the seccondary dips at a .1/4, 1/3, 2/5, 1/2, 3/5, 2/3 and 3/4.4he parabolic background has be en removed Dots repres en t the

theoretical values ca(ulaud for rational p/q (q 5 30). Taken from Rff. [73].


60

It has been originally introduced by S.Aubry [7, 8] in connection with the study of 1D organic conductors and their representation by mean of a quasi periodic potential. He then introduced the "Aubry duality" using an idea of Derrida and Sarma, which leads to the existence of a metal-insulator transition for the 1D representation of this hamiltonian. In algebraic terns Aubry's duality mean that

H(µ) and (3(H(µ)) = µH(1/µ) have the same spectrum (an elementary statement in our framework since 0 is a *automorphism). More precisely if E is the energy, the substitution µ-+14t, E->E/µ does

not change the spectrum. It does not change the density of states either and through the Herbert-Jones-Thouless formula [52, 93] permits to show that for p.1 the Lyapounov exponent is positive, leading (with some care [86] to localization [55, 74]. IV-2) The Grenoble experiments -

It turns out that the groundstate energy of the Harper hamiltonian and the corresponding models for other lattice symmetries, are describing the metal-superconductor transition curve for an array of superconductors in the magnetic field-temperature plane. The connection goes back to a work by P.G.de Gennes [40] and Alexander [3] in using Landau the case of a

regular array, square or hexagonal lattice, Pannetier, Chaussy and Rammal [72] showed that the transition temperature for the superconductor in the array, submitted to a uniform magnetic field B, is given by a rather simple function of the groundstate energy E(a) of the Harper hamiltonian, where a is still defined as in §II eq.1 1. They realized experimentally such an array with the hexagonal symmetry and were able to fit

perfectly the numerically computed curve: in particular they found not only a flux quantization at integer values of a but they were able to observe the quantization at the rational values as it goes from the Hofstadter spectrum (see fig.2). Since then the experiment has also been performed on a square lattice [73], on a quasi periodic and the Penrose lattice [13], and the Serpinsky gasket [43]. Later on they realized that the electric conductivity for an square array of dirty metals was related through the theory of weak localization to the Green function of the Harper hamiltonian [32, 33, 34]. It gives a spectacular experimental evidence that indeed weak localization results [19], even though hardly rigorous, are quite accurate in describing the electronic properties of weakly disordered cristals. More recently they went even farther by relating the magnetization of a superconductor in an array, at the transition, to the derivative of the groundstate energy of the previous hamiltonian [39, 99, 101] by using the Landau theory for superconductors of type II due to Abrikosov [1]. The

numerical calculation of this derivative is quite easy to perform and again the experimental measurement is in spectacular agreement with the theoretical formula (see fig.3).

Actually the theoretical formula for the derivative aE/aa (see theorem 10 below) was firstly derived by M.Wilkinson [103] in a very original paper which deserves more attention than it got up to now. Using a delicate semiclassical analysis leading to corrections due to the presence of a "Berry's

0

vN 10 v

(b)

-3

0

0.5

1

1.5

2

0.5

1

/ma 1.5

Tia.3- (a)-E.Werimental derivative of the magnetisation dM/dT versus the reduced flux a _ Q/+O atT -3A5Kin a square naurorkpf suptrconducting indium. (6)-'Ite numerical derivative dTc/dH of, the theoretical mean field transition temperature. dT /dH is vrovortional to aE/aot. Taken from Ref. 1391.

2


61

phase"[20, 87], he produced the exact value of the derivative of the gap boundaries with respect to the magnetic field. Rammal interpreted the result in term of the local density of state at the top of the band, near the rational values of the flux and remarked that it could be measured in term of the magnetization. For this reason we will call it the Wilkinson-Rammal formula In the present section we are going to derive this formula for a Ck (k >2) element of i by a completely different method than the one used by M.Wilkinson. Indeed a major defect of the Wilkinson calculation comes from the fact that it works only for polynomials of partial degree one in either U or V. It could probably be extended with some effort to polynomial of higher degree, but there is little hope that it could work for other examples. We will use instead our Ito derivation and a purely algebraic framework to derive it. However, prior to this calculation, we need a good knowledge of the weak magnetic field limit which can be performed via a semi classical analysis. To achieve this goal we have used ideas developped recently by P.Briet, J.M.Combes, P.Duclos [22, 28], and by B.Hellfer, J.Sjostrand [50, 51]. IV-3) Semi-classical results-

Let 9k, a) be a continuous function of the variables k = (kt,k2)e R2 and an (-E,E) for some >0. We assume that it satisfies the following properties:

(i) His periodic with respect to k of period 2n in each component of k. (ii) If 9(=EmE z2 h(m,a) eikAm is its Fourier expansion (where kAm = ktm2-k2mt), then either :

119(IIa1= sup lm z2 Ia'h(m,(x)/aa4(1+ImI)k < m

for some k > 2

(3a)

for some r > 0

(3b)

or the h(m,a)'s are holomorphic in a in a strip of width r and III 9(III, = supi«lijmE z2 Ih(m,a)I erlmI < oo

(iii) For each a in (-c,E) the function k-> 9((k,a) has a unique regular minimum in each cell of period. Without loss of generality one can assume that this minimum is located at k=0 for a=O and that 9((0,0) = 0. Correspondingly we define the quantized of Has the following element of .%:

H((x) =Emez2 h(m,a) W(m)

W(m) = et101"tm2 Umt V-m2

(4)

The groundstate energy E(a) is defined as the infimum of the spectrum of H((x) in A. . Our first result concerns the asymptotic behaviour of E(a) as a->0 namely : DrDrem 8: Let 9(satisfy (i),(ii),(iii) and let H(a) be given by (4). Then the groundstate energy of H(a) is given by : 2

E(a) = 27t lal where S = Min(3, k).

det(2

kk

,

(0,0) + a

DH

as

(0,0) + 0 (Ial)

(5)

0

The proof of this result is actually an immediate consequence of the following result :

Theorem 9: Let 9(satisfy (i),(ii),(iii) and let H((x) be given by (4). Then there is E,, > 0 and ec 5 E depending only on 9(such that if an (-ec,Q the spectrum of H(a) below Ec is contained in the union of the intervals In =[Er((x) - 6((x), En((x) + 6(a)] where if 9(satisfies (3a) :

E.(a) = (2n+1)21t lal dettrz(1/2 D291(0,0)) + a a9(/aa(0,0) + 0 (lalbn)

(6)

0< 6(a) 2. Here, LP(H) is the Schatten ideal of bounded operators Con H such that ICIP is trace class [85]. With this construction we can prove the following theorem [18, 107):

The=m 1

:

The Chem character of any projection P in 9f1(A) is an integer. It is given by the

Connes formula :

Ch(P) = Ind(Fp)

P - almost surely

where Fp is the restriction of {PFP)+, to the space PH.

(17) 0

To give a hint in the proof of this theorem let us introduce some technical tools. A graded commutator is defined as follows on the set of bounded operators on H:

TT

[T,T'l = TT' -

(18)

and a graded trace Trs is given by :

Trs (T) = 1/2Tr ( GF[F,T])

(19)

One also defines a character by the following trilinear map on A(B) : t (A0, A1,A2) = 1/2in t (AO {a1A1 a2A2 -a2A1 a1A2 })

(20)

in particular [29] : (i) it is cyclic :

(ii) it is closed :

t

t (An, Al

) = t (A1,A ,A

t (A0,A1*

3) + t (Ao,A1,A2A3) - t (A3Ao,A1,A2) = 0

70


Moreover, as in [29] we get the following Comas formula (again the magnetic field is implicit here) :

It (Ao, A1,A2) = Ja dP((o) Trs (A0[F,A1][F,A2])

(21)

One remarks that since [F[A] belongs to LP(H) for p > 2, the right hand side of (5) is well defined. If now P is a projection in C (91(B)) one sees that :

Ch(P) = '[ (P,P,P)

(22)

An important remark lies in the following lemma [107] :

T

Lemma 2: The formula (22) extends to every projection in the Sobolev space '4(91). 0

The main ingredient in the proof of the integrality of the Chem character is the remark proved in [29,

app.l,p.86 ] that : Trs (P[F,P][F,P]) = Ind(FP) = Ker (Fe) - Ker (Fp*) = n(?.,w)

(23)

where FP is the restriction of (PFP)+_ to the space PH. Since F commutes to P up to a compact operator, and since F2 = 1 it follow s that F is invertible in PH up to compact operators, and therefore it is a Fredholm operator. In particular Ind(FP) is an integer! In fact this integer may depend

upon the disorder. However thanks to the homogeneity condition (eq.4) one has (including the explicit dependence with respect to (o) : Ind((P0) FP

0)

)+) = Ind((P.

U(a)FU(a)*P.

P1. U(a)FU(a)* P1'W = P1,0) F PI.. + K

)+J

(24)

(25)

(0

where K is some compact operator. This is because the gradient of X/IXI converges to zero at infinity. It implies : n (A,w) = n (? ,Taw) E Z for all a in R2 (26)

Since the probability measure dP on i2 is ergodic invariant, it follows that n(7t,(o) is almost surely a constant and therefore we have proved that the Hall conductivity is an integer whenever 4 is finite. It is even possible to compute this integer. Let H(s) be the operator obtained from the initial model (eq.3) by replacing the potential V by sV (0_<s via

(.,.) = where P:W-'W is a fixed complex conjugation. The algebraic Clifford algebra Y0(W)

over W is the algebra with identity I generated by the elements of W where the

Carey: Spin groups and Clifford algebras

79

multiplication satisfies

w1w2 + w2w1 = (wl,w2)I; wl,w2 E W

Now 60(W) may be completed in an appropriate topology to form a *

C -algebra t'(W) but we will not need this fact explicitly here. Associated with

every splitting W = W+9W_ into Hilbert subspaces W+ isotropic with respect to ( ,

) there is a representation of UA(W). We parametrise such splittings by self adjoint

operators Q : W-+W with Q2 =I where W+ are the +1 eigenspaces of Q (so Q = Q+ - Q_ in terms of its spectral projections Q+ onto W+). This (Fock) representation FQ is constructed as follows. Introduce the alternating tensor algebra

A(W+) over W+:

where AkW+ = kth exterior power. Define for each xEW+,C(x):A(W+) -+ A(W+) by C(x)v = xAv, vEA(W+)

Equipping A(W+) with the natural pre-Hilbert space

.

structure we complete it (and use the same symbol for the completion) and define

FQ(w) = C(w+)+ C(Pw_) where w = w + + w_EW+9W_ and the * denotes the Hilbert space adjoint. Any bounded linear transformation G on W satisfying GT def PG*P = G-1 defines an automorphism of 4'(W) via w -+ Gw,wEW .

The set of all such G's forms the complex orthogonal group. We let Ores(W) = {G IG orthogonal, GQ-QG is Hilbert-Schmidt}

If GEOres(W) we write G =

a 2 x 2 matrix of operators defined by

80


the splitting W = W+®W_ and note that it is not difficult to prove that the off diagonal operators in G are Hilbert-Schmidt while the diagonal ones are Fredholm of index zero.

The subgroup of Ores(W) consisting of G satisfying G* = G-1 , or equivalently PGP = G is well-studied and goes back to Shale and Stinespring [35].

It is well known that for such G there exists a unitary operator I'Q(G) on A(W+) such that

rQ(G)FQ(w)rQ(G)-1 = FQ(Gw), wEW and that QG-GQ being Hilbert-Schmidt is the appropriate necessary and sufficient condition for the existence of rQ(G), [35].

We are interested in part in the generalisation of this result to Ores(W) There are numerous reasons for this interest and we list a few here. (i)

Segal and Wilson used a subgroup of Ores(W) in their study of the KdV equation [32]. The work of Date et al [11] on the Landau-Lifshitz equation (which we mention briefly in the applications section) suggests

that the methods of Segal and Wilson can be extended to the

Landau-Lifshitz equation using Ores(W) (ii)

'

It turns out that exactly solvable models in two dimensional quantum field theory are closely connected with the representations of Ores(W)

and its sub-groups. For example: The massless Thirring model [9], the Federbush model [28], [29], the Luttinger model [5], the continuum limit of the Ising model [17] and the monodromy fields of Palmer [16], [17]. In

particular these last two applications require results which utilise infinite dimensional analogues of the SMJ analysis and hence are not covered by

the established lore on 'Bogoliubov transformations'. (iii)

Related to (i) and (ii) is the representation theory of loop groups, vertex

81


operators and string theory ([8], [32], [34] and references therein). It is

possible that some new ideas will emerge from studying these in the

context of 0res(W)' 1.2 Structure of 0res(W):

Now 0res(W) has a natural topology arising from using either the norm or strong topology on the 'diagonal' components and the Hilbert-Schmidt topology on the 'off-diagonal' components in the representation G = [A(G) B(G)1 determined by the

CG DG

splitting W = W+®W- . ker D(G) (mod 2) is a continuous

In this topology the map

homomorphism from 0res(W) to

identity of 0res(W)

.

whose kernel is the connected component of the

We let SOres(W) denote the latter subgroup.

At first sight it would seem that a subgroup of relevance here would be that

generated by those G for which there existed a geV(W) with gwg 1 = Gw . This is the group introduced for finite dimensional W in SMJ [31,1] and called the Clifford

group there. In the infinite dimensional case this requires either G+1 or G-1 to be trace class (cf. Araki [1]) and this group is thus too small to be of much use. It turns out to be of more interest to consider a slightly larger group SOQ(W) = {GESOres(W) IA(G)®D(G) - I is trace class} .

This latter group could be loosely described as the group of Q-inner automorphisms.

It should become clearer later why this is the appropriate object to study. More motivation comes from some of the applications [16-24].

A basic fact about SOQ(W) is the existence of the decomposition SOQ(W) = SOS Q(W)BQ

where BQ = {GESOQ(W) IG =

*

*']I and

SOIRQ(W) = {GESOQ(W) IPGP = G}

82


The group BQ is the analogue of a Borel subgroup of O(n,C) and in fact one can view the Fock representation as defining a line bundle over SOQ(W)/BQ. I will not develop this Borel-Weil analogy here (however see [34]).

1.3 The Clifford Group

When W is finite dimensional SMJ [31,1] introduced the group Y(W) of

invertible elements of e(W) with

gwg 1 = G on W . When W is infinite dimensional we can similarly

define 10(W) as the group of invertible elements of t'0(W) with

gwg' for some complex orthogonal G

= G w, wEW

.

There is a homomorphism T from po(W) to the complex orthogonal group given by choosing T(g) in the following way for g c Yo(W):

gwgl = T(g)w if g is even gwg' = -T(g)w if g is odd (note that elements of i'o(W) are even or odd depending on whether they are in the

1 or -1 eigenspace of the involution on i'0(W) defined by the map w-4 -w, wEW). Now, each wEW is odd and from the formula

wxw2 wlw-x,xCW it is clear that w induces the reflection in the hyperplane orthogonal to w . As every element of po(W) may be written as a product of reflections [3] there is a naturally

defined involution r on po(W) given by


83

g = wl..wki wk...wl = gT so that if gEY0(W), grgEc* and the map g - gTg = nr(g)

is a homomorphism. If W is finite dimensional its kernel is called Pin,(W) and there is an exact sequence I-+7L2m+Pin,(W) ' OC(W) -+I

which restricts on the connected component of the identity SO,(W) to give I -+ a 2-+ Spin,(W)

T SOC(W) -+ I

It turns out that these facts have infinite dimensional analogues with SO,(W)

replaced by SOQ(W) and Spin,(W) replaced by a group SpinQ(W) whose definition is not immediately obvious, but whose construction is one of the main objectives ofthe preprint of Palmer and myself.

The construction of SpinQ(W) does not, on its own, solve the problem of providing the missing analytic details in SMJ [31]. They consider examples of

complex orthogonals G which are not in Ores(W) and hence for which PQ(G) simply does not exist as an operator. However, analogies with the vertex operator construction [6], [32], [34], suggests that one should be able to approximate the SMJ

"operators" by genuine operators FQ(G') with G' in Ores(W) in such a way as to explain what rQ(G) really means. This has in fact been done for some of the examples considered by the Kyoto school (for example, the massless Thirring model

[9]). The idea of introducing Ores(W) is simply that it is a bigger group than those groups of Bogoliubov transformations studied previously and so provides more 'room'

in which to look for approximations to the SMJ "operators".

Now let W be infinite dimensional and note that the map T defined above


84

has the property that for gEleo(W), T(g) is in the group O0(W) = {G IG complex orthogonal and G-I is finite rank}

If we introduce the involution r and homomorphism nr on Yo(W) as in the finite dimensional case and then look in the kernel of nr for those gE,10(W) with T(g)ESO0(W) (=Oo(W)nSOres(W)) we obtain a subgroup denoted Spino(W)cG0(W)

,

i.e.Spino(W) = {gEY0(W) IT(g)ESO0(W) , nr(g) = 1}

.

There is then an exact sequence I -4 z2-' Spino(W)

T

SOO(W)

This is as far as one can go with algebraic methods.

I can now describe SpinQ(W) , as loosely speaking, the closure of Spino(W) in an appropriate topology. To explain how this works in more detail I need some technicalities.

1.4 Preliminaries on Fock states Let StQ = 1 ® 0 ® 0 ... denote the "vacuum" state in A(W+) . Define for

gEyo(W) Q = <S1Q,FQ(g)fQ>

In general we are interested (following SMJ [31]) in the "correlations"

Q = <StQ,FQ(gl)...FQ(gn)QQ> and these may be expressed in terms of the Q i=1,...,n (see section 2) so that it is of interest to give formulae for the latter. In fact these formulae (some of which are given below) are one of the main reasons for the route we have chosen to SpinQ(W) . If one were not interested in calculating correlations one could follow the more direct


85

argument in [34]. However we have not found a direct route from [34] to the formulae below.

We introduce a'Wick-ordering' map 0Q from the algebraic exterior algebra

A(W) to (i)

'0(W) as the unique map satisfying

0Q1=1 (1=1®0®...)

(ii) 0Q(AAB) = 0Q(A)0Q(B) if AEA(W+) or BEA(W_)

.

(cf.[12]).

Now for GESO0(W) it is a simple algebraic construction to find

with gwg 1 = Gw and hence to define rQ(G) = FQ(g)

.

(This is contained in [31,I]

and discussed further in Palmer [13].) To see how this works note firstly that for

GESO0(W) we may as well assume W is finite dimensional (since G = I except on a dimensional subspace). Now if {e k Ink=1 is a basis of W and ekk is a dual basis under the pairing defined by (.,.) (so that for example if IlekII = 1, ek = Pek is a

suitable choice), we introduce for each P-skew adjoint operator R on W (i.e. RT =

-R) the element E.Re.Ae.

,

of A2(W) which we also write as R . These arise

naturally from GESO0(W) by setting RQ(G) =

and we have

(G-1)(Q_G+Q+)-I

the

LEMMA 1: ([31], [13]). For W finite dimensional and ge '(W) then (1.1)

g = Q OQ(exp

(RQT(g))

Here the exponential is calculated in the exterior algebra. Note that given G = T(g)

(1.1) defines g provided we have a formula for Q in terms of G

.

Note also

that when (Q_G+Q+) is not invertible this formula fails. There is an analogue [13] however, which takes care of ker(Q_G+Q+) . We will not use it here. Now we are ready to state


86

LEMMA 2:

Q = nr(g)det(Q_T(g)+Q+)

(1.2)

Thus for nr(g) = 1 lemma 1 and lemma 2 determine g in terms of G = T(g) up to sign. The latter ambiguity is inevitable since we are dealing here with a double cover. I believe that (1.2) is related to the Pfaffian formulae in [34] and if so this would provide the link between [34] and the preprint of Palmer and myself.

Now (1.1) and (1.2) hold when W is infinite dimensional for gEYo(W). One

simply works inside an appropriate finite dimensional P and Q invariant subspace of

W. Thus for GESO0(W) we may use lemmas 1 and 2 to construct gEY0(W) and hence define

FQ(G) = FQ(g) Note that since P:W-iW extends to define an involution on '0(W) written

g-'g, gE '(W) we can define g = gr and it follows that FQ(g)* = FQ(g

so that

we have a *-representation of Yo(W) . 1.5 Constructing the infinite dimensional spin group

In closing up Spino(W) in an appropriate topology the only real difficulty is to control which sheet a sequence is converging on. The topology on Spino(W) is nearly

fixed by the covering map T, for SOo(W)cSOQ(W) comes equipped with a natural topology whichfor the decomposition G

-

(C(G D(G)]

'

amounts to imposing the trace norm topology on the diagonal components and the

Hilbert-Schmidt topology on the off-diagonal components.

To control which sheet we are on we use the *-representation FQ . Thus we

introduce a metric p on Spino(W) given by:


87

P(g1,g2) = P(T(gl),T(g2)) + IIFQ(gl)YQ - FQ(g2)fQII

where " is the metric on SO0(W) . Then SpinQ(W) is just the closure of Spino(W) in this topology. The advantage of this definition is that in order to prove that it works one needs to construct simultaneously, an action of SpinQ(W) on the Fock space A(W+) by densely defined, invertible, but unbounded operators. It is this last fact which makes the introduction of some analysis necessary. A second problem

is that to analyse p we need to consider Q , for sequences {gn}n_1 even when the sequence Q converges to zero. Now we have skirted the possibility

that Q = 0 or equivalently that ker(Q_T(g)+Q+) 10 . In fact of course the set of GESOres(W) with ker(Q_G+Q+) = 0 (i.e. Q # 0) is just an open subset and we need a technique for handling G outside this subset. I do not want to go into how this problem is solved here except to say that it makes the constru tion of SpinQ(W) by this method rather complicated in its details.

The upshot of this then is the existence of a group SpinQ(W) and an exact sequence

where the covering map T extends that defined previously. Now a corollary of this is

the existence of a representation of SpinQ(W) on A(W+)

.

If we let 0 denote the

closure of the orbit of 1Q under the action of Spino(W) and let .0 denote the dense

linear subspace FQ(i'0(W))0 of A(W+) then there is a strongly continuous

representation rQ of SpinQ(W) on .0 (by operators which are unbounded in general) such that

FQ(g)FQ(w)rQ(g)-l = FQ(T(g)w) where the equality holds only on .0. Now we come finally to our main result. To obtain SOres(W) from SOQ(W)


88

we need to include the group of all complex orthogonals on W which have the form

I00 (G r)0-1J relative to the decomposition W=W+®W_ . Here G0T denotes the l

o

transpose of G0:W+ -' W+relative to (.,.) . This latter group we denote GL(W+) and form the semidirect product

SpinQ(W)xa GL(W+)

where xa denotes the fact that GL(W+) acts by conjugation on SpinQ(W) (one needs to check that this action is well defined). This group acts on .0 as follows. Define r(Go) = 1 ® Go ® Go Go 0 ... and hence an action of Go on

Extend this to all of 91 by writing

r(Go)rQ(g)1Q = rQ(GogG01)r(Go)1Q . Then the representation of SOres(W), rQxr , defined by

rQxr(g,G0) = rQ(g))r(G0) factored by its kernel gives a group which we denote SpinQ(W)

.

One then has the

exact sequence (1.3)

I -4 c.* -4 SpinQ(W)

T SOres(W) -' I

Clearly SpinQ(W) acts by operators on 91 which implement the automorphisms of sg(W) defined by the corresponding elements of SOres(W)

To get Ores(W) we

need only give the action of the reflections defined by w E 'o(W) and this is easily done.

From the viewpoint of the Kyoto School the significance of the exact sequence

(1.3) is indicated in T. Miwa "An introduction to the theory of r-functions" Springer Lecture Notes in Physics Vol.242 pp.96-142.

Now (1.3) is pretty well the best one can do in the direction of producing


89

infinite dimensional analogues of the finite dimensional results in [31]. To go further

and use them in a direct way to make rigorous the applications in [31] requires further analysis.

§2 APPLICATIONS

2.1 Preliminaries An account of the underlying general strategy of SMJ may be found in "Monodromy, solitons and infinite dimensional Lie algebras" by M. Jimbo and T.

Miwa in Vertex operators in Mathematics and Physics (ed. J. Lepowsky, S.

Mandelstam, I.M. Singer, Springer Berlin 1984). Their idea in a nutshell is that particular spherical functions (or limits thereof) for the representation of Ores(W) provide solutions of non-linear differential equations including the Painleve equations (in the case of the scaling limit of the Ising model correlations), the KdV and K.P.

heirarchy and the Landau-Lifshitz equation. To describe this work fully is beyond the scope of this lecturer. Instead I wish to indicate how the discussion in the preceding section is relevant to making sense of the "vertex" operators used in [31] and [11].

2.2 Ising model, continuum limit To save space I will assume that the notion of taking a continuum limit of a lattice theory is understood. In heuristic terms the Ising model case is explained in [31]. A more precise mathematical discussion may be found in [14], [15]. In any case it

is generally agreed that in the continuum limit the Ising model becomes a free fermion

theory, describable in terms of a Fock representation of the Clifford algebra. The non-triviality of the Ising model arises because one is not interested in the fermions or Clifford algebra elements as such but in the 'spins' (which may be thought of as defining elements of the complex orthogonal group) or more precisely the continuum

limit of the spin operators. The latter may be described as 'operators representing' elements of the complex orthogonal group which do not lie in Ores(W)' Clearly, making sense of what is meant by operator representing' is where the difficulties arise.

90


Since the quantum field theory constructed from the continuum limit of the spin

operators has a non-trivial S-matrix [23] it is not surprising that significant mathematical problems need to be solved.

The continuum limit of the Ising model is constructed on the Hilbert space W

= L2(IR,c2). The complex conjugation P takes the form

0 Al

P=

IA 0 "

where (Af)"(p) = fA(p) with the Fourier transform f of f being defined via "

f (x) = 1 f f (p)elpxdp We can now form the Clifford algebra i6(W) and consider the representation

given by taking up Q to be multiplication by [gy(p) 0 1

where E(p) =

0

E(p) J

1 p>0 .

-l p0 p_ 4

finite von Neumann algebras and M

N c M

there is a pair

tr

where

restricted to <M,eN>

is a

eN N)

.

III

If

with

N

and

are factors then

M

factor and then

N c M

<M,eN> = {M,eN}"

is the orthogonal projection onto If

with

are

is a faithful normal normalized trace

one considers the von Neumann algebra

,

L2(M,tr)

,

.

The "basic construction" of the theory is as follows.

on

n ? 3

,

.

CM:N]tr(eN) = 1

on

L2(N,tr)

(tr

CM:N] < -

iff

.

Ocneanu has made great progress on classifying subfactors of the hyperfinite II1 talk.

factor with given index, which he will explain in his

It would appear that the classification is complete for index < 4

Wenzl has constructed many examples of subfactors in index > 4 additional property talk.

N'

n M = L 1

.

with the

He will no doubt explain them in his

Pimsner and Popa have shown that for property

set of subfactors up to inner conjugacy is countable!

T

II1

factors the

For factors like

the group von Neumann algebras of free groups the existence problem for index values of subfactors, even < 4

,

is wide open.

In this talk I shall largely avoid internal questions of the theory and talk about relations with other parts of mathematics and physics.

The exception will be the first topic which I feel to be of

fundamental importance.

104

Jones: Subfactors and related groups Commuting squares

I

A commuting square will be a quadruple

BO = B1

of finite von Neumann algebras together with a

U

faithful normal trace

AO 2 Al

tr

on

with the

B1

U

(equivalent) properties. a)

0 from b)

is the trace preserving conditional expectation

to

B1

L (B ,tr) space

c)

(EB0

EB (A1) = AO

and

BO).

L (A ,tr)

L2(Bl,tr)

are commuting subspaces of the Hilbert

with intersection

EAO(x Y) = EAO(x)EAO(y)

if

x e B0

L2(A0,tr) and

.

y e Al

.

(I owe this last

condition to de la Harpe).

It is an easy exercise to see that if

A,M

and

N

are finite

von Neumann algebras with traces, then

ANM91EARM9N U

U

Afll®1=AR1NN Popa's notion of orthogonal pairs of algebras is

is a commuting square.

exactly that of commuting squares with

A0 = t 1

squares has evolved along with subfactors.

.

The notion of commuting

It appears explicitly in

Wenzl's thesis and in Pimsner and Popa's "Entropy and index for subfactors" paper.

One reason for the importance of this notion for subfactors is

BO- B1 the following.

If

U

U

is a commuting square, then consider the

A0 c Al

basic construction for the pair BO E B1 U

U

A0 c Al

B2 =

U

A2 = {A2,eB 0

The existence of a faithful normal trace on

B2

extending

tr

which makes

B1 ` B2 U

U

Al c A2

is a commuting square is, under suitable irreducibility assumptions, the same as the Markov property for that trace, i.e. tr(xeB1) = r tr(x)

for

105

Jones: Subfactors and related groups and some

x E B1

T

1 R +

It is well established that the Markov

.

condition will continue if one iterates the construction so we have, automatically from the first square a whole tower BOcB1 U

..

2 B2 S

U

BiS U

U

AO = Al 2 A2 E

Ai E

..

of commuting squares and a privileged faithful normal trace on Under the same irreducibility assumptions the unions

all

i

UAi

give

.

N c M

factors

Ill

when the

GNS

for

Bi

and

UBi

construction is applied.

B. C M U Moreover

is a commuting square for all

U

i

which

A. c N N c M

allows one to approximate the basic construction for Ai = Bi

and in particular to calculate

,

to identify

N' n M

in this process.

[M:N1

Wenzl

by those for

It would be important

.

has made important contrib-

utions to this.

A suggestive reason for the interest of commuting squares comes from the connection between subfactors and braids.

For some while

it has been known that, if one considers the projections

e i

for a subfactor defining

N c M

Mi+1 = <Mi,ei_1> , M1 = N , M2 = M ,

where

gi = tei - (1-ei)

,

2 + t + t-1 = CM:N) ,

of the union of the braid groups if

aiaj = ajai of variables

1i-j1

ei -> gi

>_ 2>)

in the tower

BW (=

by sending

is unitary and one may show that

in

alg(l,eN)

yields representations

so that

v Q v' -v' Q v

subfactors

can be naturally identified g = 2eN-1

(up to sign)

is actually the so that g Mg-1 =

Thus the commuting square coming from the braid group is

the same as the one used in statistical mechanics.

If

dim V > 2

the

permutation matrix will again do the job so this lends even more weight to replacing the braid group structure by commuting squares in general. In fact it seems likely that subfactors give, more or less canonically, statistical mechanical models of an exactly solvable kind. One should look at the relative comnutants in the tower M2 C ...

.

N c M c Ml C

Such models have been worked out in some cases by Pasquier.

They generalize the Andrews-Baxter-Forrester models.

One begins with a

Jones: Subfactors and related topics

109

Coxeter graph and decrees that the states of the atoms are the vertices of the Coxeter graph.

The model then only allows states with the property

that neighbouring atoms are assigned neighbouring vertices on the Dynkin diagram.

w(a,b

The weights

I

are the vertices of

(a,b,x,y

x,y)

neighbouring atoms - this is not a vertex model) are related to eigenvectors of the incidence matrix of the Coxeter graph - a condition forced by integrability).

The same structure occurs in Ocneanu's analysis of subfactors. III

The Yang Baxter equation

In the preceding discussion of vertex models there is a consistency condition.

Considering different ways to reorder to the following likely condition on

R

V 51 V 9 V

one is led

:

R12(e'e')R13(e',e")R23(e",e) = R23(e",e)R13(a',e")R12(e,e')

(YBE)

given a solution of

Moreover the direction can be reversed: ing only on

one can construct

0-e'

t(e)

and

(YBE)

Thus one is led to look for solutions of the YBE. similarity with the braid group relations.

depend-

YBE

implies (*). Note the

The simplest solution of real

interest is that corresponding to the "ice-type" or six vertex model solved by Lieb.

One has

dim V = 2

The

.

matrix is then a

R

4x4 matrix,

in fact up to a scalar the following 1

0

0

0

0

b

c

0

0

c

b

0

0

0

0

1

In field theory the

YBE

factorizability of the

b = sinh(e)/sinh(a-e) c = sinh(a)/sinh(A-e) (0

as below)

has a simple geometric interpretation as S-matrix.

The idea is that an

2-particle interactions.

action can be decomposed as a composition of Then if there are

3 particles

1,2,3

operator for an interaction between 0

,

and i

there are 2 ways to decompose the

to the picture.

One reads off the

S12(0)S13(e'+8)S23(")

and

n-particle inter-

Sij(e) j

is the scattering

with relative "velocity"

3 particle interaction according

YBE

= S23(e')S13(e'+e)S12(e)

110


IV

Knot theory The easiest way to visualize knots and links in three

dimensions is by way of (generic) plane projections.

If one records

crossing data one gets a more than adequate picture of a link,

e.g.

Since there are only double points as singularities of the projection, the situation locally resembles that of a one may consider vertex models with no and ask the question: invariant,

i.e.

ZZ 02Z

lattice so

e-dependence on the projection

is it possible that the partition function is an

depends only on the link up to isotopy in

3 space?

(The assymetrizing of the transfer matrix approach shows how to deal with the different components of a link differently - just choose a different vector space for each component.) Such an invariant would be of no interest since we have not taken into account the crossing data and all that could remain would be the number of components.

can be given a sign

+ =

However if the link is oriented, crossings

< ,

to choose a different set of weights w_

for negative.

- =

w

,

and it is natural

at crossings,

w+

for positive,

Then it is possible that non-trivial invariants could

be obtained from partition functions.

Without some background this sounds

like absurdly wishful thinking but right from the braid - von Neumann algebra discovery of new knot polynomials it has been known that there is a connection with statistical mechanics and Kauffman actually defined using a model on knot projections (not obviously a vertex model).

In

VL


111

fact it is entirely conceivable that the collection of invariants obtained in the simple minded way outlined above is faithful on oriented There are certainly enough to determine

links!

V

obtained by Lipson and corresponds to the following that

t

is given by

R

cosh a

0

0

sink a

-sinh a -cosh x

0

0

-cosh x -sinh a

0

0

R-matrix (we assume

from now on)

0

sinh a

The first was

.

0

cosh a

As shown by Lickorish and Lipson, the link invariant so obtained is the moment generating function for the random variable on the sublinks of the link given by linking number with the rest of the link. Further invariants obtained by Lipson are specializations of VL

corresponding to the element of

inner product on

V

given by a Euclidean

End(V Q V)

.

But to return to the story, one must find conditions on the two solutions

w

for the partition function

ZL

to be an invariant.

Three dimensional isotopy is translated into link diagrams by the Reidemeister moves, which come in three types according to the number of crossings involved.

type I

type II

type IIII To show that

ZL

is an invariant it suffices to show that it does not

change under moves I, II, III. R

Let us examine what this implies for the

matrix.

Type I

By some ideas emphasized by Kauffman, one may ignore type I.

Type II

In fact there are two situations to consider

and (A)

(and their mirror images)

112


The thing is that the diagrams for have less states than those for

and and

(A)

(B).

be that a sum over internal variables

a

c

n

11

So the condition must cancels unless

b a = b

c = d

,

.

This is fairly obviously just a way of

d

saying that the

R-matrix, call it

W+

R-matrix provided one identifies

IIB

condition coming from

End(V Q V) = End(V) Q End(V)

= 1

as an endomorphism of IIA

w_

V Q V

in

one finds that the In fact if we write

is different.

there are transpose operations

t1 =

t2 = id R(transpose)and the condition for invariance

and

Rt1(R-1)t2

is

is the inverse of the

If this is done for

the appropriate manner.

(transpose) R id

R

R,

I have not yet made much sense of this condition

.

but it must be natural. Type III

All the various possible orientations etc., can be reduced to

precisely the

YBE

(with no

0-dependence).

So we get the result that to every invertible matrix R E End(V I V)

satisfying

Rtl (R-1 ) t2 = 1

and the

YBE

there is an

oriented link invariant. If one limits oneself to the theory above it is rather difficult even to obtain can try to use the

VL

a

Make sure your projection is projection R+(s1,s2) ZL =

the values

C°°

has 6 edges

(

of

YBE

E

states

but, taking a cue from the physics, one

variable in the

given by

YBE.

This can be done as follows.

and choose one point per edge of the Then suppose we have a solution

w+(a,blx,y)(e1,s2,e)and define

w (a,bIx,y)(el,02,e) where

it

for a given crossing

crossings e1

and

e2

are the changes in angle between the points on

the left incoming and right outgoing, and right ingoing and left outgoing curves respectively.

And

a

is the change in angle between the left out-

going and right outgoing curves.

Once again we can write down the conditions for this invariant.

ZL

to be a link

One now has to consider dependence on the choice of the points.

This turns out to be a simple property involving

R(e1+a,e2,e).

Once

is independent of the points it is an invariant of planar isotopy and

ZL

113

Jones: Subfactors and related topics one may consider invariance under Reidemeister moves as before. getting into detail let me say simply that corresponding precisely to the

VL

Without

is obtained from a solution

matrix for the ice-type model.

R

It is also intriguing that one can see the relevance of matrices in this picture as it is possible to find choices

t-

for

(R,t)

which one may choose as many points as one likes on the knot and use the R-matrix

t-matrix inbetween points that do not have a crossing and the between points that do.

The equation

Rt1(e)t2(e')R-1

= t2(e')tl(e)

is

then implied by the second Reidemeister move

R tl Rt2 R 1 Quantum groups

V

Thus far we have begged the important question of how one is The payoff is good, as we have seen, to

to come up with solutions of YBE.

every solution we are likely to get an exactly solvable vertex model a subfactor, a knot invariant and a factorizable be a completely integrable

S-matrix.

(In fact there will

dimensional quantum field theory as well.)

1 + 1

We have hinted at the existence of interesting solutions.

Quantum groups

are a machine which was developed in order to construct a wide variety of solutions of

The simplest case came from a study of the following

YBE.

commutation relations (all this theory was pioneered by Fadeev and the Russian school, culminating in Drinfeld's formalism of quantum groups. Woronwicz was led to simlar objects for different reasons): CH,E] = 2E st(2,h):

CH,F] = -2F

[E,F] = (sinh (hH/2))/(h/2)

If non-zero

h -> 0

we recover st2

in its usual presentation.

we must work on the level of the enveloping algebra.

h

point is that these relations admit a tensor product operation: H,E,F

on

VQW

by

and

V

W

satisfying st(2,h)

H=H&1+1fit H E

E 1k e xp (

F=F&

) + e xp ( +

11

H) 6l E

AF

,

one may define

For The

given

H,E,F

on

114


then the new

H,E,F

satisfy

s9.(2,h)

exponential gives another solution

as an element of

.

Changing the sign in the

H',E',F'

conjugating

End(V El W)

formally one defines the associative algebra

R-matrix comes in

and the

H,E,F

to

H',E',F'

More

.

as the (suitably

Uh(St2)

completed to take care of the power series) algebra with presentation sk(2,h)

The tensor product operation is then the existence of a co-

.

product

o

Uh(st2) -> Uh(st2) W Uh(st2)

:

will be an element of where

Uh(st2) & Uh(st2)

a e Aut(Uh(st2) 0 Uh(st2))

an exact formula for YBE

and the universal

matrix

R

satisfying RA(x)R1 = Qoo(x)

is defined by

a(x6ly) = yflx

.

In fact

is very useful as it will give a solution of

R

for every representation of

finite dimensional rep. of st2

Uh(St2)

.

Jimbo has shown that every

can be deformed to one for

Uh(st2)

so we will get solutions of YBE for every positive integer.

I have

checked that the YBE solutions are of the right kind to give knot invariants so one has polynomials for every irrep of be determined by

st2

They will

.

of cables of a knot but this should not be construed

V

as a negative fact since the object is to understand the polynomials and

not to create more. Turaev has worked out the explicit formulae for the so(2n+l) sp(2n) so(2n)

series in their identity representations.

specializations of the Kauffman polynomial. specializations of the

s to

One obtains

in general gives

2-variable Alexander polynomial

PL

.

For the record, Drinfeld gives the following formula for the

R-matrix in Uh (s t2) & Uh (s t2) E

R =

ekh/2 n

k=O

:

(eT)

r=l (e

{exp 4EHQH + k(H611-19H)]}Ek Q F

-1)

For subfactors, the interest in quantum groups is quite direct. R E End(V 9 V)

Given by

R1,R2,R3 ....

action.

Then VI

one considers the subalgebra of

fl°° End(V) generated

which is contained in the commutant of the quantum group

R2,R3 ...

will generate a subalgebra.

Field theory One of the most important aspects of statistical mechanical

models is that there should exist a non trivial quantum field theory as the lattice spacing tends to zero. the

If it exists it can be defined in terms of

n-point or correlation functions which will be defined for a finite

lattice spacing as

115

Jones: Subfactors and related topics Ef(Z1)...f(Zn)exp(-sE(a) = a Z

where

f

is the variable of interest and

are lattice points.

Z1 ... Zn

If the lattice spacing tends to zero keeping is hopefully a sense in which the limit of

Z1,..,Zn

fixed in

1R2 there

exists.

Provided these limit functions satisfy certain conditions (Osterwalder Schrader axioms) one is then guaranteed the existence of a field theory (operator valued distributions on a Hilbert space etc.) for which the n

point function,

are the boundary values of an analytically continued

.

Thus the Euclidean

field theory is that naturally comes out of statistical mechanics.

The

most interesting points at which to let the lattice spacing go to zero are the critical points for which long range phenomena occur.

The homo-

geneity of the system suggests that the field theory will have some kind of invariance under local conformal transformations. This possibility is even more strongly suggested by the knot theoretic approach where the scaling limit could be approached is a

IL'

totally haphazard fashion

which would entirely forget any privileged directions. In any case a local conformal transformation (away from zero, will have the form

z ->

E

an zn

and one expects the Lie algebra of

these symmetries to act on the field theory.

Since the action may be

projective, one obtains an action of the Virasoro algebra which has presentation on generators

c, {Ln,n e 7Z} 3

CLn,Lm] _ (n-m)Ln n + c CC,Lm] = 0

V M

n, -m

.

In an explicit theory theory one may calculate "stress energy tensor"

T(z) =

E

Ln zn

.

Ln

from the so called

The element

c

is the so-

nclL

called central charge and in an irreducible representation it will be a scalar.

If the theory is physical the

Ln's

will be subject to some

116


unitary condition on a Hilbert space.

then

Ln = L_n ,

and

Qiu and Schenker have shown that this imposes restrictions on the

Friedan, scalar

It appears to be

c

in an irreducible representation.

c = 1 -

6

Namely

for some integer m >_ 3

.

c 0

There is a general feeling

m(m+l)

that these discrete values of

c

correspond in some way to the 4cos21r/n

values for the index for subfactors.

The feeling is reinforced by many

observations, perhaps the most convincing of which is that the algebra of ei's for

CM:N] = 4cos2 ,r/n

does occur as transfer matrices in statistical

mechanical systems for which I am told, the corresponding central charge is 1

-

6

It is also true that the original proof of restrictions on

n(n-1)

index values is morally the same as Friedan Qiu and Schenker's proof - they

consider a sesquilinear form on finite dimensional spaces of increasing dimension defined by the Hilbert space inner product restricted to the span

of

L_1,L_2, .. L_k

applied to an eigenvector of

L0

.

positivity of the inner product gives the restrictions.

The constraint of In the subfactor

proof one considers the inner product given by the trace on the sequence

of vector spaces of increasing dimension given by

alg(l,el, ... ek)

Positivity of this inner product gives the restrictions. It seems to be important to understand this connection.

Since

the field theory is in fact determined by the statistical mechanical model, one is naturally led to consider actions of the Virasoro algebra on the algebra generated by the

ei's

.

The "right" action would be the one which

gave the right action on the field theory in the scaling limit though

at

this stage one would settle for the right central charge. An important first step has been taken by Connes and Evans

who used as a guiding principle the fact that for algebra is actually the even

CAR

,

the

ei

or Clifford algebra where explicit

formulae are known for the field theory. ei's

CM:N] = 2

By solving certain equations in

they were able to generalize these formulae to obtain actions of

Virasoro (and more) for arbitrary

[M:N]

central charge is always the same as for

.

But it turns out that the CM:N] = 2

.

In a us.eful paper for mathematicians, Tsuchiya and Kanie con-

centrate on a specific model (E-model on

SU(2)

with Wess-Zumino-Witten

term) treated by Knizhnik and Zamolodchikov and express its mathematical content in purely mathematical terms.

(Representations of the affine sR,2

117

Jones: Subfactors and related topics They explicitly calculate the

algebra and Virasoro algebra).

functions

equation with regular singular points on o = {(Z1..Zn)

I

Zi = Zi

n

point

and show that they satisfy a differential

for some

monodromy representation of

to

# D

i

the singularities being on

,

Then they calculate the

.

and find precisely the braid

,rl(to - 0)

group representations that come from 4cos2 ,r/k 3k

central charge is of the form

.

subfactors.

But the

Unfortunately there do not seem to

k+2

be any field theory models realizing all the

1

values without

6

-

m(m+l)

going through the scaling limit.

It is important to note, in fact, that even the existence of the

1

values of

6

-

requires a non-trivial trick due to Goddard

c

m(m+l)

affine Lie algebras

and Olive:

are acted on by the Virasoro algebra

V

and the Virasoro operators can be written down explicitly in the enveloping algebra of the affine Lie algebra (the'Segal Sugawara" form) (If

(Ta Tn) = c CabTm+n + k m am,-n sa,b

L= const n

E

:

m

Ln

satisfies Virasoro with some

of

V ,

V

:TaTa: =TaTam0

(

Given the appropriate subalgebra h

simply subtracting the Virasoro operators for

h

from those for

gives a new representation in which the central charge is the difference

of the central charges.

No such operation is known on subfactors which

will "subtract" indices but note that the first proof of the existence of subfactors of index

4cos2 ,r/n

simple associative algebras.

used a pair of finite dimensional semiThe analogy remains.

Finally we would like to draw the audience's attention to the

Associated with Gell-Mann

"operator algebras" of quantum field theory.

one has the equal time commutation relations CA(x0,x), B(x0,Y)J = z Dn(x-y) On(xO,Y) n

where the

On(x)

derivatives. and

that

On

.

are local fields and

Dn

are

s functions and their

The "algebra" is defined by the particular

choices of

Wilson proposed a generalization where one supposes in fact

On

118

Jones: Subfactors and related topics A(x)B(y) = E Cn(x-y) On(x) is a space-time coordinate and

where

x

x-y

Thus the

.

Cn(x-y)

are only powers of

Cn(x-y) are structure constants for an "algebra" and one

idea was to use associativity constraints to help in the calculation of n-point functions.

the

Polyakov applied this in the case of conformal

field theory.

It seems important to try to understand the many examples of "operator algebras" in terms of our operator algebras = C

and von

Neumann algebras.

The following references may be useful in understanding the

text. von Neumann algebras V. Jones. "Index for subfactors".

Invent. Math 72 (1983) 1-25.

H. Wenzl. "Representations of Hecke algebras and subfactors".

Thesis,

University of Pennsylvania (1985). M. Pimsner and S. Popa.

"Entropy and Index for subfactors".

Ann. Sci. Ec.

Norm. Sup. 19 (1986) 57-106. Knot theory D. Rolfsen.

"Knots and Links".

L. Kauffman.

Publish or Perish, Berkeley, 1976.

"State models and the Jones polynomial" to appear in topology.

V. Jones.

"Hecke algebra representations of braid groups and link poly-

nomials".

To appear, Annals of Math.

Physics

J. Lepowsky, S. Mandelstam and I.M. Singer. Mathematics and Physics". R. Baxter.

"Vertex operators in

MSRI publications, Springer Verlag (1984).

"Exactly solved models in Statistical mechanics".

Academic

Press, London 1982.

A. Tsuchiya and Y. Kanie. ]P1

"Vertex operators on conformal field theory on

and monodromy representations of braid groups".

formal field theory and solvable Lattice models",

To appear in "ConAdvanced Studies in

pure mathematics, Konokuniya.

Other V. Drinfeld.

"Quantum groups".

To appear, proc. ICM 86.

QUANTIZED GROUPS., STRING ALGEBRAS AND GALOIS THEORY FOR ALGEBRAS

Adrian Ocneanu Department of Mathematics, The Pennsylvania State University, 215 McAllister Building, University Park, Penn. 16802, U.S.A.

ABSTRACT

We introduce a Galois type invariant for the position of a subalgebra inside an algebra, called a paragroup, which has a group-like structure. Paragroups are the natural quantization of (finite) groups.

The quantum

groups of Drinfeld and Fadeev, as well as quotients of a group by a non-normal subgroup which appear in gauge theory have a paragroup structure. In paragroups the underlying set of a group is replaced by a graph, the group elements are substituted by strings on the graph and a geometrical connection stands for the composition law.

The harmonic analysis is similar

to the computation of the partition function in the Andrews-Baxter-Forrester models in quantum statistical mechanics (e.g. harmonic analysis for the paragroup corresponding to the group

22

is done in the Ising model.)

The

analogue of the Pontryagin van Kampen duality for abelian groups holds in this

context, and we can alternatively use as invariant the coupling system, which is similar to the duality coupling between an abelian group and its dual.

We show that for subfactors of finite Jones index, finite depth and scalar centralizer of the Murray-von Neumann factor

R

the coupling

system (or alternatively the paragroup) is a complete conjugacy invariant. In index less than 4 these conditions are always satisfied and the conjugacy classes of subfactors are rigid: there is one for each Coxeter -- Dynkin diagram

An

each of the

and E6

D2n, and there are two anticonjugate but nonconjugate for and

E8

diagrams.

Thus underlying the rigidity of the Jones

index there is a crystal--like rigidity of the position of subfactors of

R.

120

Ocneanu: Quantized groups

INTRODUCTION

THE PROBLEM

A subalgebra B2

Al

of an algebra

B1

and a subalgebra

A2

of an algebra

are conjugate (or have the same position) if there is an isomorphism

0: B1 - B2

with

0(A1) = A2.

In what follows we study the possible

positions of a subalgebra into an algebra, i.e. the conjugacy classification of subalgebras.

This problem is an analogue of the classical Galois theory

for fields, although in our case we want to impose no a priori restrictions similar to the normality and separability of field extensions.

It is

convenient to study the problem in its technically simplest form; we shall consider a subalgebra centralizer in

B

finitely generated.

A

of a complex algebra

and finite index in

B, i.e.

which has scalar

B

B

as a module over

A

is

Since we are interested in the position of the

subalgebra, the isomorphism class of the subalgebra in itself should be ideally irrelevant; moreover the algebra should be rich enough to have subalgebras in interesting positions.

The algebra used in what follows is the Murray-von Neumann factor

or the hyperfinite

R,

Ill factor in von Neumann's classification, also called

121

Ocneanu: Quantized groups in the sequel the elementary von Neumann algebra.

The algebra

R,

which is the weak closure of the Clifford algebra of the real separable Hilbert space, is a factor (i.e. it has scalar center) which has very many symmetries (e.g. any locally compact group can act outerly on

R.)

A strong

uniqueness theorem of Connes implies that any closed subalgebra of is a factor, i.e. has trivial center, is isomorphic either to Thus any finite index subfactor

itself.

all the information in the inclusion of

canonical trace

tr

between

1

0

N C R

and

R

MatnC

is isomorphic to

or to

R

R, and

comes from the relative position

and not from the structure of

R

in

N

of

N

which

R

N.

The algebra

R

has a

with the trace of idempotents varying continuously is a continuous geometry, or a type

(R

I11 factor in the

sense of von Neumann,) and is the weak closure of an ascending union of finite dimensional subalgebras (i.e. dimensional.)

R

is hyperfinite, or approximately finite

The classification theorem of Murray and von Neumann states

that any hyperfinite II1 factor is isomorphic to

R; in our context this

guarantees that the closure of all finite dimensional constructions done below will bring us back to

If

A C B

A with index

with [B

:

R.

B

elementary, then

A] = dim AB E [1,+-]

B

is a projective left module over

computed by means of the trace.

The index was introduced (with a different but equivalent definition)

122


by V. Jones, who proved a remarkable rigidity theorem: if [B [B

:

A] E (4 cos2(7r/n); n = 3,4,...).

A] < 4, then

:

We shall show in the sequel that

underlying the rigidity of the index there is rigidity for the conjugacy class of the subalgebra A: for any classes of subfactors of

R

there are at most

n a 3

with index

4 cos2(nr/n).

4

conjugacy

The construction will

also explain and give a short proof of the rigidity of the Jones index.

THE GALOIS FUNCTOR

We introduce a conjugacy invariant for a finite index subfactor factor

of a

N

M, consisting of a pair of graphs, an involutive bijection of

their vertices, and a connection, which is a cohomological invariant defined on small loops in the graphs called cells.

Consider the algebraic tensor products where

times)

'even

NXN

and odd vertices

subbimodules of and

MYN

M®n, n = 1,2,... with

NNN

M ON

..

ON

M

(n

N - N

subbimodules of

men,

'odd the equivalence classes of irreducible

if the restriction

the identity bimodule

N

be a bipartite graph having even vertices

Let

the equivalence classes of irreducible

n = 0,1,...

M - N

MOO = N.

M®n = M

NYN

k

contains

edges between the classes of NXN

is distinguished and marked

k

times. *,.

The class of

The graph

6

is

123

Ocneanu: Quantized groups connected, and the finiteness of the index guarantees that the graph locally finite.

The graph

is called the induction - restriction graph of

°a

Construct analogously the restriction - induction graph

N C M.

is

ej

3f

of

N C M,

with even and respectively odd vertices the equivalence classes of irreducible M - M

and respectively

of

are defined as before, and the vertex

M

bimodule

N - M

subbimodules of

MMM, is distinguished.

vertices classes of irreducible

Men, n = 1,2,....

The edges

a,,, the class of the identity

We have in fact four graphs, having as N - N, N - M

,

M - N

and

M - M

bimodules

respectively, and a typical picture could be the following.

N - N

M - N

bimodules

bimodules

t

N - M

M - M

We compare now two ways of inducing an

N - N

bimodules

bimodules

bimodule to

an

M - M

bimodule: inducing first on the right hand side or inducing first on the left hand side.

Although the end result is the same, the decompositions into

irreducible components at each stage yield, when compared, essential new information in the form of a scalar for each cell consisting of one edge in each of the four graphs. The corresponding map is called a connection.

124


'

As an example, in the case when M action of a finite abelian group cardinality of

G

C

is the crossed product of

by an outer

N

the graphs depend only on the

G

while the connection gives the pairing

G X G -' C.

Because of the existence in our context of the contragredient of a bimodule,

these graphs are two by two isomorphic, and the picture can be condensed to the two graphs

&

and

N.

We have to record the involutive bijection

r,

called the contragredient map, on the vertices of , U H, which maps the class of a bimodule onto the class of the contragredient bimodule (r

is marked with

dotted lines below.)

Choose now an irreducible bimodule from each equivalence class and a bimodule intertwiner for each edge of the graphs, as well as antiintertwiners corresponding to each pair of contragredient vertices; this is called a representation of

(ti,

If, r).

A cell

quadruple of (oriented) edges of

Rd U N

c = (al,a2,a3,a4)

with the range

becomes now a yi

of

ai

equal to

Ocneanu: Quantized groups the contragredient of the source

There is an

for which

I

125

xi+1

of

xi E ee0en; let

(indices mod 4).

ai+1

NXN

be the representative of it.

The composition of the intertwiners and antiintertwiners corresponding to the successive edges and pairs of vertices

ai+3, (yi+3,xi)

N - N

ai, (yi,xi+1)' ai+1, (yi+1'xi+2)'

viewed as (anti)intertwiners of

intertwiner of the irreducible bimodule

N - N

NXN, and hence a scalar

its equivalence class

tj U 3f;

W(c)

If,

invariant of

the Galois invariant of

We call

N C M.

The radius from

r1

(fd,

3f,

[M

[W])

is a conjugacy

is called the depth of

of the graph

be infinite even if the index

r,

N]

W,

[W]

modulo the choice of the representation of (Fd,

r)

N C M.

N C M, and can

is finite; however when

[M

:

N] < 4

the depth is finite.

Define a function with values in representing

Qt+, by

x, where

µ, called the Haar measure, on the vertices of t U 3f u(x) = dim(PX) P,Q E (M, N).

if

pX

Q

is a bimodule

It is not hard to see that if

the incidence matrix (or the laplacian) of

,

bimodules is an

(with suitable normalizations.) This way we obtain a complex valued map called a connection, on the cells of

..

U If, then

a

e µ = J9.µ, with

is

126


= [M

ev

N]

and

1/2,

and µ(*,) = µ(*H) = 1. In the finite depth case, when

are finite, by the Perron - Frobenius theory µ

3f

eigenvector with positive entries of a P =

normalized at

is the only

*H, and

and

But by an elementary theorem of Kronecker, if a matrix with

II a II.

natural entries has norm less than 2, then the norm must be n = 3,4,....

This yields the rigdity of the index

[M

:

N] _

2 cos(rr/n), 2,

since the

same method shows that infinite graphs have Perron eigenvalues larger than or equal to

2.

THE INVARIANT

It is possible to axiomatize the Galois invariant of subfactors with finite depth.

Any quadruple

(ej, W, r,

[W])

N C M

satisfying the axioms (and

in particular any Galois invariant) is called a coupling system.

There are

initialization axioms, describing the graphs and the connection around then local and global axioms for the connection. when

2

opposite vertices are fixed the connection

matrix indexed with the two halves of the cell. that for each choice of the normalization, unitary. P4 U H.

A cell has

Let

(Q1, 2)

2

edges, and

yields a complex

The main local axiom states

fixed vertices, this matrix is, up to a

The connection

be an

W

4

*, and

n-string on

W

P4

gives a transport for strings in and let

vl,v2

be paths on

tj

127

Ocneanu: Quantized groups with even length transport

k, source

Tvl v2((f1,f2))

*

of

combination of pairs of paths sources

xi

and respectively

computed as follows.

and ranges (C1,f2)

along

and respectively

The coefficient

The

x2.

is a linear

(v1,v2)

n, common range and

having length

(n1,,j2)

x2.

xl

of

c

(711,7)2)

is

Consider the diagram

i

V2

vl

i

I

I

I

I

I

I

in which the space inside the contour is filled with cells of the coupling system

(f,,

lf, T, [WI); such a filling with cells having matching walls is

called a surface.

The contribution of each surface to R W(cell1)

.

celll

cell2

ranges over the cells in the right half.

ranges over the cells in the left half of the surface and

on

if

The coefficient

c

of

is obtained by summing up these products over all possible surfaces.

If the length of

(f1,t2)

is the product

([ W(cell2)

where

(711,172)

c

vl

and

v2

is odd, a similarly defined transport moves

into pairs of paths on the dual graph is defined analogously.

N.

The transport of strings

The main global axiom states that the


12 8 transport

of any string is

T. 1

only on the range y = r(x).

0

unless

v1 = v2, and

T,,,,,

depends

'v 2

x

v, and will be denoted in the sequel by T. with

of

Thus the the transport of strings based at

transport, i.e.

f,

and M

are

geometrically

is

*

a parallel

Remark the formal

.-flat.

similarity between the computation of the coefficients of the transport and the computation of the partition function in statistical mechanics on the one hand, and between the parallel transport and string multiplication and the action and respectively Witten product in string theory on the other hand.

THE MODEL

Starting from a finite depth coupling system

(ej,

3f,

r,

[WJ)

we

construct a model of finite index subfactor of the elementary algebra

R,

called the string model.

For

n = 0,1,2,...

as follows.

An

which have source common length linear basis

n

construct the string algebra

n - string p

on

is a pair

(g,77)

An

of the graph of paths on

r,, common range (which is called the range of (which is called the length of

p.)

n-strings and product defined by

(11,'72)

The algebra

i

Pj

p)

A.

and has

129

Ocneanu: Quantized groups where is

is the Kronecker symbol.

8

(71,C).

The algebra

An

into

Z

(t,i))

(called an FD-algebra in the sequel.)

Cm- algebra

by mapping an

An+i

of a string

(t,71)"`

is a finite dimensional semisimple algebra over

C, or a finite dimensional

We map An

The adjoint

over all edges

of

{,

n-string

into the sum

(t,n)

having source the range of

tr

denotes the composition of paths by juxtaposition.

(t,+7); here

This map is an algebra

homomorphism which is injective (here and in the sequel we assume that more than one edge) and thus yields an algebra

6, define the measure p

For the graph

N,

of

ej,

Ae, i. =

has

A_ = U An.

on the vertices as the unique

eigenvector with nonnegative entries, normalized by incidence matrix

rd

p(*

)

= 1,

of the

Define now a positive trace

#.µ.

tr: A. -4 C, which is a linear functional given by

tr(Q'7)) =,-n.(x) 8(t,+7) where

is a string of length

(t,77)

eigenvector of

An

4 An+1

0..,

and thus gives a trace

II.112

11.11-bounded

by

x.

Since p

is an

this trace is compatible with the inclusions

in addition to the uniform norm L2-norm

with range

n

11.11

on A .

tr

On the algebra A define

inherited from the

11xII2 = (tr(x*x))1/2.

Ca-algebras

Complete now A with respect to

11.1j2-convergence to obtain a von Neumann algebra

of completion would yield

C'([O,1])

An, the

from

A

(this type

the algebra of polynomials on


13 0

[0,1].) Construct analogously the n-string algebras the string algebra

B

as the completion of U Bn.

called the string algebras of the graphs

and

on the graph

Bn

The algebras #f,

A

If, and

and

B,

are isomorphic according

to the theorem of Murray and von Neumann to the algebra

R; in particular they

retain no information about the graphs.

We now embed the string algebra embedding for each

n

An

into

into the string algebra

A

Bn+i;

B

by

this yields a model of subfactor.

We cannot, in general, use graph maps such as the embeddings of

An

into

An+1; a more general class of maps, the cell maps, is needed (quite surprizingly any filtered map between locally finite dimensional algebras

U Cn

and U Dn

is a cell map.)

where

(p(p) = E c(p,a) o

p

and

A linear map

rp: An - Bn+1

are strings in

o

and respectively B.

An

and the coefficients

c(p,a) E C.

We use the connection

coefficients

as follows.

Let

c(p,1J)

p = (p+,p_)

has the form

and

W

to define the

o = (o+,a_), and

construct a diagram

0'_

where the vertical lines are edges of cell;

Pd U ?f

label the cells from left to right as

The coefficient

c(p,a)

is then the sum

such that each square is a

e l l. ..

,

cn, cn,

....ci.

131

Ocneanu: Quantized groups £

W(c1) W(c2) ... W(cn) W(cn)

for all possible choices of vertical edges, where conjugate.

W(ci)

... W(c2)

denotes the complex

This is an instance of parallel transport.

W

The local unitarity of the connection that the map

(p

is a

yields then the fact

*-homomorphism compatible with the inclusions

An C An+1' Bn C Bn+11 which preserves the trace and thus gives a homomorphism (p: A -I B

of the string algebras.

The inclusion

string model of subfactor on the coupling system

(p(A) C B is called the (Pd,

-r,

If,

[W]).

The main

results in the sequel show that for subfactors with finite index, scalar centralizer and finite depth of the hyperfinite

factor

II1

R, the Galois

invariant and the string model are functors inverse to each other, and thus the coupling systems classify subfactors up to conjugacy.

The parallel transport given by the connection several morphisms of the string algebras W

:

map

An

9 Bn+1

Bn -4 An+1.

An

W

and B.

is used to construct The homomorphism

can be extended, with essentially the same definition, to The square

r = (D'(p

:

An

4 An+2' Bn -4 Bn+2

is called the

canonical shift, and is a generalization of comultiplication in Hopf algebras. In a coupling system arising as a Galois invariant, the shift to Morita equivalence.

is connected

r

The information in the coupling system can be

recovered from the graph

Pd

together with the canonical shift

1'

on its

132


string algebras A.

and the pair

can be axiomatized.

(ed, r)

With the

parallel transport we also construct the mirrorings, which are involutive

antihomomorphisms yn :

For instance on

A2n

n-string (f1,C2)

of the string of

7)1

and

transport

into

Then

Tx((t1,C2))

path inverse.

the mirroring yn

('71,12)

n2.

A2n H A2n, B2n 4 a B2n, A2n+1 H B2n+i' is an antiautomorphism which maps the

yn((f1,g2)) = £ c (711,772), where the coefficient

is computed as follows. c

of

is the coefficient of ((1,f2)

from

s

to

Let

x

c

be the common range in the parallel

(72",171")

x, where

"

denotes the

The mirrorings are generalizations of the coinvolution on Hopf

algebras, and in the Galois invariant they are coming from the contragredient map on bimodules.

THE RANGE THEOREM

The following theorem shows that any coupling system with finite depth appears as the Galois invariant of a subfactor.

THEOREM

Let

(,,

If, r, [W])

The string model built on (6, 3f, r,

[W]), where

W

be a coupling system with finite depth.

(e', M, T,

[W])

has Galois invariant isomorphic to

is the connection complex conjugate to

W.

133


The proof of this result begins by constructing open string bimodules as follows.

x

Let

an even vertex of the graph

be

open n-string is a pair range such that

has source

f

linear space with basis of

An(x)

into

of paths on

(g,n)

*

and

open

(*,x)

n

A(x)

algebra A

on

U An(x)

of 6.

has source

n-strings.

'7)

having common

n

An(x)

Let

x.

(*,x)

A

be a

An embedding and finally the

An(x)

are defined the same way as for the string

The n-string algebra (

n Z 0.

of length

Pd

An+l(x), an inner product on each

completion

and

ej

An

acts on

An(x)

by

(f0'70) = 8 (n,E0) (f ,n0)

This action is compatible with the inclusions and the trace and inner product for A(x)

An

which gives

action of

An

n-string on transport

p

An(x), and give an action of the string algebra

and

the structure of a left

A(x) An(x)

on

on the right defined as follows. a E S(n).

and let

6

from

on the right.

A(x) becomes an

*

to

This gives

A - A

A-module.

x

on

There is an Let

Now use the connection

A

W

p

be an

to (parallel)

and with the result multiply the open string

An(x)

bimodule.

a right

We have

a

An-module structure and thus A(*) = A

as

A - A bimodules.

The main part of the proof of the theorem consists in showing that the bimodules constructed analogously for each vertex of

6 U 34

are irreducible

and that their intertwiners correspond to the edges of the graphs.

134


The irreducibility of the open string bimodules, which is an asymptotic property, is reduced using the mirrorings

yn

to an ergodic property of the

random walk associated to the graphs coming from the Perron - Frobenius property.

THE CLASSIFICATION THEOREM

The string model algebras are by construction approximately finite dimensional.

The following result shows that for approximately finite

dimensional algebras this is the most general type of inclusion, in the finite depth case.

THEOREM

Let

N

be a subfactor with finite index, scalar centralizer and

finite depth of the elementary factor conjugate to the string model ('a,

If, r,

[W))

Then the subfactor

M.

is

constructed on the conjugate

Astr C Bstr

of the Galois invariant

N C M

(j,

If, r,

[WJ)

of

N C M.

The proof starts with two constructions associated to the inclusion

N C M.

The upward construction is

the downward construction is a subfactor the endomorphisms of

N

as a

P-module.

N C M C End(NM)

P C N C M

such that

M ON M

and

M = End(NN),

These constructions, called the

135

Ocneanu: Quantized groups basic constructions, were first systematically used by V. Jones.

They can be

iterated to give a sequence ... C M-2 C M_1=N C MO=M C M1 C M2 C ... The sequence

(M0)n>_11 called the tower, is canonical. In the sequence

(Mn)n60, called the tunnel, the choice of any finite subtunnel is unique only up to an inner automorphism of

(Mk)n6k.l

can be used to insure that the

and

(M_n' n M)n=1

span

N

and

By the first part of the proof, these centralizers are in

fact a string model naturally built into the subfactor

N C M.

The freedom

of choice for a finite number of steps of the tunnel is an inner automorphism of

N;

if instead we could use an inner automorphism of

M

then the

spanning could be obtained by a classical hyperfiniteness argument.

The

latter case can be reduced to the former via a decoupling theorem, which states roughly that it is possible to choose the tunnel so that breaks into a tensor product

MO

almost

M_n ® (M_n' n MO); this is analogue to an

asymptotic abelianness argument in quantum field theory.

The decoupling

theorem is proved using the ergodic theory of the associated random walks on graphs, together with the study of central sequences in ultraproducts of the algebras.

The main ingredients of the proof are the following.

free ultrafilter on

THEOREM

Let

N, and W denotes the

A C B C C

Here

w-ultraproduct algebra on

be finite von Neumann algebras, with

approximately finite dimensional. Then

B

w

is a

M.

137


(A' fI B")' n& = A V (B' n C)".

This generalizes to paragroups the fact that outer automorphisms of the elementary algebra are outer on central sequences.

THEOREM

Let

N C M

be a subfactor with finite index, finite depth and

scalar commutant of the elementary factor.

Consider the tower of extensions

V Mk, and

MO = N C M1 = M C M2 C M3 C ... C

let

A = M' n m-.

k

Let Let a)

index b)

index c)

EM

denote the conditional expectation onto the subalgebra where

_ 1 11 µ 112

The subfactor

M ® A

2

is the

M.

12-norm of the Haar measure

of M has scalar relative commutant and

t. The subfactor

M' n N"

of

M' n M"

has scalar relative commutant and

t. Let

M

and

el, e2,...

be a subfactor and standard projections making

the following tower standard

M C M ® A C ML = <M ® A, el> C

<M ® A, el' e2> C

..

Then the following tower is standard

M' n N" C d)

µ.

Let

M' n w C <M' n MW, el> C

<M' n mw. e1, e2> C

x e M' n M". Then

E

M' n to

(x) = E- (x) = lim E (M_k)u (x) M"

k4-

138


uniformly in

for

Ilxll 5 1, where

... C M_2 C M-1 C MO C M1

is

any standard tunnel.

Example.

An

For the subfactors

the index is

[M

:

N C M with graphs the Coxeter-Dynkin diagrams while their central sequences have

N] = 4 cos2nn+l

index [M'

:

M' n N"]

= (n+l)(4

n mw The algebra

M C M ® A

Remark that in general if x = (xn)n e M' n M"

is called a core for the subfactor

N C M

N C M.

is as above, for a sequence

the termwise conditional expectation EW(x) _ (EN(xn))n

is not centralizing for

M, which would be the case for cross products or

fixed point algebras of group actions. even part of M,

sin2n+1)-1

E-(x) = (EM(xn))n

an inner perturbation by a unitary in

Nevertheless, although the core is not

is in

M' n N.

The core is unique up to

M.A, which does not change E- (x).

SUBFACTORS OF INDEX LESS THAN 4

After showing that the Galois invariant is faithful, we are left with the task of finding the possible coupling systems.

While a complete

description for all the values of the index is out of reach

(e.g. all finite

139

Ocneanu: Quantized groups groups are in the range of the invariant,)

it is possible to get a feeling

of the general situation from the classification of subfactors with index less than

4, that is up to the first accumulation point of the index values.

In this range all subfactors have automatically scalar centralizer and finite depth.

The fact that the norm of the incidence matrix is less than

forces each graph to be a Coxeter - Dynkin diagram

An, Dn

or

2

E6, E7, E8.

Conditions stemming from the biunitarity of the connection show that in this range the graphs

tj

and

If

are isomorphic.

Due to the simple structure of

these graphs, it is not hard to find all the connections which satisfy the local axioms; up to isomorphism there is one for each diagram there are two on each diagram

D2n

the global axioms eliminate one connection for each

for each diagram

E7.

D2n

or

D2n+1'

differing by the contragredient map and

there are two conjugate to each other for each diagram

connections on

An

Dn

Finally

E6, E7, E8.

and the pair of

Thus there is one subfactor for each diagram

An, one

and a pair of opposite conjugate but nonconjugate

subfactors for each diagram

E6

and

E8.

PARAGROUPS

The coupling system has been defined as a Galois invariant containing as much information as possible, by using an analogue of the coupling

140


between an abelian group and its dual.

This approach is convenient for the

exhaustion part of the classification properties: it is easier to find all groups in a given class by working with both the groups and their representations.

For other purposes it may be useful to have the analogue of

the group itself; in our context this is called a paragroup (para = standing for.)

A local description of the paragroup associated to an inclusion as before is the following.

rm

Let

ej

N C M

be the induction - restriction graph and

the restriction to the even vertices of

A macrocell consists of eight edges of

t

of the contragredient map

r.

as follows

with endpoints alternatively equal and respectively contragredient to each other (contragredient vertices are connected by a dotted line in the figure.) To each such macrocell

c

we associate a scalar

called the macrocell connection. connection

W

Wm(c), with the map

One way to construct

of the coupling system

(ii,

computations for the parallel transport.

from the

is analogous to the

If, r, W)

We fill

Wm

W

c

with cells


141

where the inner edges belong to the graph

W

on the cells and sum over all ways of filling define

Wm

directly, in a way analogous to

W

and then take the product of It is also possible to

c.

W.

[Wm])

(e,, Tm

The triple

forms a paragroup, and can be axiomatized in a manner similar to that of the coupling system (3f, Tm, [Wm])

system

(c,,

There is an analogous dual paragroup

(c,, H, T, [W]).

built on the restriction - induction graph [W])

If, T,

3f, and the coupling

plays the role of the Pontryagin - van Kampen duality

coupling between these paragroups.

The process of finding

N

and

W

from

(e,, Tm, [Wm]) is the analogue of the representation theory for (finite) groups.

It is also possible to give a global description of a paragroup as follows.

If

is an n-string on the graph

p

then the parallel transport n-strings on

Pd

with source

only need to know

Wm.

Tx x.

The pair

maps

p

Fd

and

x

is an even vertex of

into a linear combination of

Remark that in the construction of (P,,

(Tx))

where

x E 'even

Tx

we

is the global

form of a paragroup, can be completely axiomatized and contains all the

142

Ocneanu: Quantized groups (g, Tm

information needed to recover the local form

[W'J)

As an example, consider the way in which a group appears as a paragroup. M

Let G.

be the cross product of

by an outer action of the finite group

N

Then the induction - restriction graph

and its even vertices are indexed by a single edge labeled

G

has a single odd vertex *

S

at the unit

with

g between the vertex

g

contragredient map Tm connects the vertices

and

and

g

SG.

g E G.

for each g-1.

For

There is

G = Z3

The

the

picture is the following.

A macrocell

c

is labeled as follows.

g

h h-1

1-i I

I

k-1

1

Then

Wm(c)

is

1

if

multiplication table of p = ((1,g),(l,g))

ghkl = 1 G.

k

and

0

otherwise, and thus yields the

The parallel transport of a 2-string

to the vertex

h

is

the string

Th(p)

=

((h,hg),(h,hg)).

The picture for a finite quantum group is analogous, with the parallel

143

Ocneanu: Quantized groups transport yielding the comultiplication.

QUANTIZED DYNAMICAL SYSTEMS

In classical mechanics, a dynamical system consists of a group on a measure space algebra

(X,µ).

Instead of the space

group acts by an automorphic action setting, the algebra

L°'(X,p)

a: G -+ Aut A.

of

automorphic action Neumann algebra

B(H).

X, on which the

In a quantum mechanical

is replaced by the noncommutative algebra

of bounded operators on a Hilbert space A

one can work with the

of bounded measurable functions on

A = L°'(X,p)

*-subalgebra

X

H,

The quantized dynamical system consists of an

a: G -+ Aut A

of the group

on the noncommutative von

G

A.

be accompanied by a corresponding quantization of the group noncommutative algebra

an algebra

A

should

has more symmetry than its automorphism group

by a group action

then there is an action

X

G, since the

Consider, for instance, a crossed product algebra B

B(H)

or by a weakly closed

There is growing evidence that the quantization of the space

can detect.

acting

G

a: G -- Aut B.

a-: G" -> Aut A, where

If the group G"

A = B >4 G G

is abelian,

is the dual of G

Pontryagin - van Kampen sense), with aQ(E xgg) = E a(g)xgg

of

for g 6 G.

(in the

144


a E G action

xg a B. and the algebra a-.

If the group

representation

G

o: G --> MatnC

B

consists of the fixed points of the dual

is compact nonabelian,

there corresponds a homomorphism

a;: A -+ A 0 MatnC: E xgg -> E xgg 0 o(g). G-

of

G, with

G"

then to an irreducible

This is an action of the dual

viewed as a quantum group [Dr] (Hopf algebra,

ring group [Ka], Kac algebra [Ta,ES]), and its fixed point set (y a A; a;(y) = y 0 1, a e Irr G)

A - A 0 MatnC algebra

A.

of

MatnC.

a6

a, then

n

E

ij

B.

Homomorphisms

should be viewed as quantized symmetries of the

The homomorphism ao

conjugate of projection

such as

is again the algebra

is generalized invertible: if

(a;- 0 idn) o a contains the identity

eij 0 eij

E MatnC 0 MatnC, where

eij

aidA

is the

under the

are the matrix units

In certain situations, such as the quantum Yang-Baxter equations

in statistical mechanics, one has to consider similarly defined actions of quantum groups which are neither groups nor group duals [Dr].

Harmonic

analysis concepts such as the Haar measure, the Fourier transform, the Pontryagin duality theorem and the Takesaki duality for crossed products extend naturally to the quantum group setting.

What is then the suitable quantization of the group concept which should accompany the quantization of the space?

Ideally, like the finite abelian

groups, the category should be closed under quotients by subobjects and

145

Ocneanu: Quantized groups under duality, with the bidual of an object isomorphic to the object.

Quantum groups are closed under duality, but only quotients by normal quantum Separable field extensions can be viewed

subgroups are again quantum groups.

as indexed by a quotient extension and

H

G/H, where the group G

corresponds to the normal

This category is closed under

is a subgroup of G.

quotients.

A natural answer is to let the quantization of spaces to algebras determine the quantization of groups via Galois theory.

The quantized groups

should appear as those (quantized) symmetries of an algebra which leave the subalgebra invariant.

If a group

A, the position of the algebra A gives back the group

G.

G

and

A

inside the crossed product

B

with as

8(A) = Al

Al

B

ABA =

as an

®

in

(in which case

B1), then the groups

A direct way to see this in the case when

a factor, i.e. has scalar center, is the following. of

B = A X G

is another such crossed product and

0: B -> Bi

has the same position in

are isomorphic.

G1

acts freely (i.e. outerly) on an algebra

If B1 = Al X G1

there is an algebra isomorphism we say that

G

A

is

The unique decomposition

A - A - bimodule into irreducible subbimodules is

AAgA, and the tensor product

AAgA ®A AAhA

gEG

A - A - bimodule to

AAghA.

Thus

G

is isomorphic as an

is obtained as the group of equivalence

classes of irreducible subbimodules of

ABA with composition given by the

146


tensor product over

A.

Similarly, the group-like object which we introduced

appears as the position invariant for a subalgebra inside an algebra.

In fact

it is convenient to work with both the group - like object and its dual, which form a coupling system.

ampliation

A ® Matn

Homomorphisms of a von Neumann algebra

A

into an

(i.e. quantized symmetries, as described above)

correspond to bimodules

AXA.

Thus the invariants introduced in terms of

bimodules can alternatively be given in terms of quantized symmetries, and the paragroup and coupling system are a way to organize these quantized symmetries in a group - like form.

In these notes we work under the finite index assumption, and obtain the quantized analogues of finite groups.

The fact that even in this context the

theory is expressed naturally in geometric terms suggests that an extension of these methods and results to a differentiable context could have very interesting consequences in quantum physics and geometry.

147

Ocneanu: Quantized groups APPENDIX A: AXIOMS FOR COUPLING SYSTEMS AND PARAGROUPS

NOTATION

a)

Let let

s(e)

r(e) E 00)

and

unoriented graph on

ei(1), with

A path path

f

Po

f

of length

of

t

is

_ (el,...,en)

JfJ

In the sequel

0

is a vertex

e H e"

= n, and let

x,

is an unoriented graph. with

r(f) = r(en) with

s(ei+l) = r(el).

and length

s(f) = r(f) = x.

JfJ

The = n.

The inverse

and the composition of the paths

(en,en"1,.... el and

,

(e1,e2,...,en)

s(f) = s(el), range

' = (fl,...Ifm)

= (e1,...,en,fi,...,fm). with

For an edge

as an oriented graph endowed with an involution

is an n-uple of edges

A path

and edges 01).

Let

with

Path'

r(E) = s(f)

is the path

denote the set of paths

PathXn) = (f E Path(n); s(f) = x)

PathXny = { a PathXn); r(t) = y).

The cardinal of

and

PathXny

is denoted by

p(x,y;n).

b)

MEASURE GRAPHS

A standard measure graph

&

E

denote its source and range. We shall consider at

s(e") = r(e).

has source

f

6(0)

be a graph, with vertices

,

is a couple

is a graph which is connected, locally finite and unoriented, with

the number of edges adjacent to a vertex uniformly bounded.


14 8

The graph

has a distinguished vertex

S

* = *,

adjacent to only one

edge and has no cycles of odd length; its vertices 00) classes,

'even

and

The measure µ

''odd

are divided into two

according to the parity of their distance to

*.

,

is a map

P(0)

)

1R+

with

a

Let

j (*) = 1.

the Laplacian operator (having matrix the incidence matrix) of

Pd.

denote The

measure must satisfy the following axiom.

Harmonicity

(H)

The measure µ

oµ=flµ

is an eigenvector for the Laplacian

with .6 EHt+.

A measure with this property will be called a Haar measure on the

graph ej.

REMARKS

We conjecture that for infinite graphs, the following axiom must be also satisfied

(PE)

Path ergodicity lim

E I

-k p(*,x;k) y(y)

-

.8-k-1

e where

e e P,(1)

and

p(*,y;k+l) la(x)

,

= 0

x = s(e), y = r(e).

For finite graphs the condition (H) determines the measure p completely, P = 11 n 11

theory.

and (PE) is implied by (H) via the Perron - Frobenius

For infinite graphs the range of the eigenvalue

6

for harmonic

149

Ocneanu: Quantized groups measures is

[

+ - [, and it is an open question whether (PE) implies

11 A 11,

that P _ 11a ll .

In general if

is the supremum of the number of edges with the same

n

source, then the operator norm

sup n

with union

11 a 11

satisfies

for an ascending family

'n

Fd

n

and

n

n1/2 $ 11 a

of subgraphs of

t,.

For finite graphs, Hoffman and Shearer have shown that the accumulation points of the range of the eigenvalue rn = «1/2 + a-1/2

where

xn+1 - 1 - x -

...

xn-1

-

with and

a

with initial point

*

(rn

I

n = 1,2,...) U [rcr' +°°['

the positive root of the polynomial

rcr = (p1/2 + rp-1/2

(r1 = 2, r2 = 2.019, rcr = 2.058.)

The measure µ

are

fi

with

The lowest eigenvalue

the golden ratio.

(p

fi

> 2

= 2.007.

is

determines a stationary random walk on the graph

6,

and transition probabilities proportional to

µ. The

condition (H) is known in statistical mechanics as detailed balance, and means that the transition probabilities for the reverse random walk are asymptotically proportional to in

steps to a vertex

k

x

µ; alternatively, the probability of arriving

on a path

(

does not depend on

condition (PE) is equivalent to the fact that if

Pk(e)

that the random walk uses the unoriented edge

from the step

then

c)

11

Pk+1 - Pk 11

-)

0.

LOCAL COUPLING SYSTEMS

e

¢. The

is the probability k

to

k+1,


15 0

A local coupling system is a quadruple = (,,µ) lt63 V .

and

Let

are disjoint standard measure graphs, with

34 = (3f,µ)

be the vertex in each graph adjacent to *.

**

The contragredient map preserves the measure,

(CI)

Initialization

(CP)

Parity

(s,3f,r,W).

r

is an involution of

P(C) U W(C)

which

µ o r = µ, and satisfies the following conditions.

even

even) = r(odd)

The connection

W

=

even Wodd

is a map which associates to any cell

(al,a2,a3,a4) consisting of four oriented edges s(ai+1) = r(r(ai)) W(al,a2,a3,a4) E C,

(WI)

i = 1,...,4;

a5 = al,

al,a2,a3,a4

a number, the ener

having the following properties.

Inversion symmetry W(a4",a3",a2",aI-)

= W(al,a2,a3,a4) where

denotes the edge inverse.

of

& U If

with

151

Ocneanu: Quantized groups (WR)

Rotation symmetry W(a2'a3,a4'a1) =

where -

(WU)

W(a1,a2,a3,a4)-

denotes complex conjugation.

Bi-unitarity

For any vertices

x,y E P(U) U ,(O)

the matrix

where the indices are pairs of edges

(U1 j)

i = (a1,a2), j = (a3,a4), with edges

al,a2'a3'a4 E 01) (j W(1), s(ai) = T(r(ai-1)) = xi, a0 = a4, having fixed ends the connection

x1 = x,

is unitary,

i = 1,...,4,

x3 = y, and U is a renormalization of

W p(x1)-1/4µ(x2)1/4µ(x3)-1/4µ(x4)1/4 W(a1'a2'a3'a4).

Ui,j =

The empty matrix is considered unitary.

d)

CELL CALCULUS

We extend the map

W

from cells to a map defined on contours as follows.

A contour is a quadruple (1,2,3,t4) Ifil

= Iti+2I' i = 1,2

A surface

s

and

with

t1'(2't3'f4 E Path(fd U H) having

s(ti+1) = r(r(f1)), i = 1,...,4

where

f5 = f1.

is a family of cells

(c(i,j)) = ((c(i,j)l,c(i,j)2,c(i,j)3,c(i,j)4)) i

j = 1,...,n,

having matching walls c(i,j)2'

c(i+l,j)4 = c(i,j+l)1 = c(i,j)3-.

The boundary

3s

of the surface

s

is the contour

(fl'(2''3'N)

where


15 2

ti = c(n,l)1 a c(n-1,1)1 u

...

c c(1,1)1

f2 = c(n,m)2 o c(n,m-1)2 0

...

. c(n,l)2

f3 = c(1,m)3 o c(2,m)3 c

...

o c(n,m)3

(4 = c(1,1)4 o c(1,2)4 0

...

o c(l,m)4.

For the surface

s, the energy W(s) =

Finally, for a contour

IT

i,,j

W(s)

is the integral product

W(c(i,j)) E C.

c, the energy

W(c)

is the sum

W(c) =1 W(s) s

over all surfaces

s

If the contour

with boundary

c

c.

is degenerate, say It1l = 0, then we let

W(c) = S(f2,f4"'); similarly, if If2) = 0, then W(c) = 8(t1 f3-). It is convenient to extend

W

to arbitrary quadruples of paths, by letting

If (f1'f2'f04)

W(f1'f2'f3'f4) = 0

is not a contour.

REMARK There is a striking formal similarity between the computation of the energy of a contour and the expression of the partition function theory, where

Z =

E

exp(-S),

and the action

S

Z in string

is given by

surfaces

S = f

surface

metric

e)

g

I(X,g),

with

of the surface.

GLOBAL COUPLING SYSTEMS

I

a form in the coordinates

X

and the induced

153

Ocneanu: Quantized groups The local coupling system

is a global coupling system if the

(4,3f,r,W)

following two global axioms are satisfied.

(PT)

The parallel transport axiom.

W(k1,f2,f3,f4) = 8(t1,f3-) S(f2,f4-). for any contour with s(ti) = r(fi)

(GC)

i=1,2,3,4.

E

The global contragredient axiom.

x E ,(G) U W(G)

For any W(c) # 0

such that

there is a contour

s(f1) = s(t3) = w,

c = (f1,f2,t,,t4),

s(t2) = x,

with

s(f4) = r(x).

This axiom takes care of the compatibility between the contragredient map and the cells.

REMARKS

The connection

W

defines, for paths

f, n

with

It)

= (ni, a transport

Tt"q: Xx,y , Xz,t where

x = r(s(t)), y = r(s(,7)), z = r(r(f)), t = r(r('rj))

space of formal linear combinations of paths T

p

with

and

Xa b

is the

s(p) = a, r(p) = b, by

(P) = E c(P,Y) Y Y

with

c(p,Y) =

E E.

It can be shown that the parallel transport


15 4

axiom is equivalent to the fact that for any paths we have

unless

= 0

T

in which case

= n

the transport from

The axioms

*

and

(PT)

f

is a parallel transport.

are independent of the local axioms

(GT)

and are independent of each other. (PT)

s($) = s(n) =

depends only on r(t),

Te

77

i.e.

with

f,n

It can be shown that for the axiom

needs to be checked only on contours with edges

i.e. with length twice the distance from

*

monotonous,

ti

to the midpoint; a finite graph

has only finitely many such contours.

f)

COUPLING SYSTEMS

Let

be a local coupling system.

(,,3f,r,W)

connection

W

A perturbation

consists of a complex unitary matrix

to each pair of adjacent vertices

x,y E 00) U

(u(a,b))a b

(0), where

u

of the

associated

a,b E Path")

with u(a'",b-) = u(a,b)-

for any

a,b.

The perturbed connection is

W#,

where

W#(bl,b2,b3,b4) _ u(bl,al) T(5-9-55-97 u(b3,a3) u b4,a4

E

W(al,a2,a3,a4)

a1,a2,a3,a4 Here

al,a2,a3,a4

are edges with

The local coupling systems equivalent.

s(ai) = s(bi), r(ai) = r(bi),

(6,3f,r,W)

and

(6,3f,r,W#)

i = 1,2,3,4.

are called

The global properties (PT) and (GC) are preserved under

155

Ocneanu: Quantized groups equivalence.

A coupling system coupling system (e,,f,r,[W]) W*

of

W

µ' o 6= u,

is the equivalence class of a global

(e,,3f,r,[W])

(ej,)f,r,W)

The coupling systems

under perturbation.

and

are isomorphic if there is a perturbation

0: 6 -- ej' If -4 #'

and a graph isomorphism

r' -0=9-7- and

,

W'

a A= W#.

with


15 6 APPENDIX B.

THEOREM

THE COUPLING SYSTEMS FOR INDEX LESS THAN 4

The coupling systems

(y,3f,r,[WJ) of dimension less than

are

4

described below. The corresponding paragroups have graph (ej.Teven'[WmacrocellJ)' where

is the restriction of the contragredient map

7-even

vertices of connection

ti, and the macrocell connection

Wmacrocell

r

to the even

is computed from the

W.

The coupled graphs

The vertices of the graphs respectively

dimension less than other.

with

6

and and

1

4, the dual graphs

3f

are labeled

1,...,n

and

the distinguished vertices.

1'

P6

and

if

In

are isomorphic to each

A vertex and its contragredient are joined by a dotted line.

Remark that in the graphs

Bn

for

n = 4k

are contragredient to each other while for

n = 4k+2

selfcontragredient.

1

3

n-1

3'

n-i'

2

1/

A., n=2k

na2 1'

the endpoints they are

n-1

and

n

157

Ocneanu: Quantized groups 1

3

n-2

n

3'

n-2'

n'

An, n=2k+1

na3

1

'

n-3

3

n-1

i i

-2

®n, n=4k n 1 4

n-2'

it

3'

n-3'

n-1'

n'

1

3

n-3

n-1

n

n2

®n, n=4k+2 n

n

6

n-3'

1'

3'

1

3

6

3'

6'

E6

it

n-1'

n'

158

Ocneanu: Quantized groups 3

1

8

5

7

4

2

i

E8

i6'

/2\ /4Nv 7\. 3'

1'

5'

8'

There is one coupling system for each of the graphs

and

An

ID2n, and

there are two nonisomorphic coupling systems for each of the graphs E6

and

E8.

The dimension

The dimension of each coupling system (and thus the index of the corresponding subfactor) is

4 cos2 m,

where

m

is the Coxeter number of

the diagrams. An: ®n: E6: E8:

The Haar weight

Let

a = m

with

The Haar weight µ

m

the Coxeter number of the graph.

on the vertices labeled

is given by the following expressions.

1,2,...;1',2',

..

159

Ocneanu: Quantized groups An

µ(k) = µ(k') = s(k)/s(1),

k = 1,...,n.

0n

µ(k) =µ(k') = s(k)/s(l),

k = 1,...,n-2,

µ(n-1) = µ((n-l)') = µ(n) = µ(n') = 1/2.

En µ(k) = µ(k') = s(k)/s(1),

k = 1,...,n-3,

µ(n-2) = µ((n-2)') = s(n-2)/s(2), µ(n-1) = µ((n-l)') _ (s(n-2)s(2))/(s(3)s(l)), µ(n) = µ(n') = s(n-2)/s(3).

The connection

We specify a cell by the vertices edges (el,e2,e3,e4), where We let

j = j'

and jr = j

ki E (1 ,..

for

which are the sources of its

(ki,k2,k3,k4) ,n,l',

.n').

j E (1,...,n).

The cells are given by the following expression, common to all graphs

An' mn' E. Let

m

be the Coxeter number, and

E

_ - i exp[

i

2 m

µ(k3) 1/2

The curvature of the cell On the cell

c = (kl,k2,k3,k4)

c = (kl,k2,k3,k4)

the connecion

W(c) = E R(c) 8(kl.k) + E

is

R(c) = (p(kl) µ(k2) µ(k4)

W

takes then the value

R(c)-1

8(k2,k4).

The above formulae give two connections corresponding to the two choices

160 of

Ocneanu: Quantized groups VI-7-1.

For the graphs

There is a

fin' the two connections are equivalent.

connection, equivalent to these, with For the graphs

mn

W(c) E R

for all cells

c.

the two connections are nonequivalent but they are

isomorphic using a symmetry

n-1 H n

in the graph

On the macrocells,

&.

these connections are equivalent, and there is an equivalent macrocell connection with real values.

For the graphs

En, the two connections are nonisomorphic and correspond

to nonconjugate subfactors; changing from a subfactor Nop C Mop

N C M

to the opposite

switches between these connections.

a

The subfactors with diagram An

of the hyperfinite

isomorphic to the subfactors generated by the projections

II1 factor en

R

are

introduced by

Vaughan Jones, and are refered to henceforth as the Jones factors.

For the integer values

1, 2

and

3

in the above dimension range the

paragroups and coupling systems are also given by the following familiar constructions.

In dimension

1,

2

and

3

the paragroups with diagrams

respectively

04

In dimension

3, the paragroup with graph

S3/S2

correspond to the groups

of symmetric grouups.

Z1, !2 A5

and

A2, A3

and respectively

Z3.

corresponds to the quotient

The coupling systems are given by the duality

coupling between thes paragroups and their duals.

For the hyperfinite

II1

161

Ocneanu: Quantized groups factor

the classification theorem yields the following results.

R

COROLLARY

All subfactors of index R 2 C R

and to

R C R >4 Z2

2

of

R

are normal and are conjugate to

for a free action of

Z2

on

R.

0 The normality is Goldman's theorem and the rest is Connes' classification of actions of

Z2.

COROLLARY

All normal subfactors of index to

R C R M !3

3

or for a free action of

All nonnormal subfactors of index to

of

R >4 S2 C R >4 S3

on

23 3

are conjugate to

R

of

R 3 C R

R.

R

are conjugate to

and to the Jones subfactor of index

RS3 C RS2,

3.

a

The first part of the corollary is Sutherland's description of normal subfactors as cross products together with Connes' classification of the actions of

Z3.

The second part of the corollary is new.

REMARKS

In each-graph the distinguished vertices

and

*

is a vertex with the

16 2


smallest entry of the Perron eigenvector

In dimension less than

4

µ.

the maximum number of nonisomorphic coupling

systems (and thus of nonisomorphic subfactors of same index hyperfinite Ill

< 4

subfactor, cf. the classification theorem) is

of the in

4

n

dimension

4 cost 30

REMARKS ON THE PROOF

The fact that the graphs must be Coxeter - Dynkin A, follows from Kronecker's theorem on matrices of norm

ID,

E

diagrams

< 2.

The position of the star at a vertex with minimum Perron eigenvector entry is a corollary of the axioms.

The contragredient map is determined by the restrictions on the number of paths joining vertices imposed by the biunitarity of the connection.

The exhaustion of the equivalence classes of connections proceeds as follows.

Along each leg of the diagrams there is enough freedom given by the

equivalence to make all choices equivalent; this takes care of the case In the cases

IDn

and

E11

An.

there remains a choice at the triple point of the

diagram which yields nonequivalent connections.

Although there are no subfactors with diagrams

ID2k+l

or

E7,

there are

163

Ocneanu: Quantized groups connections on each of these, as well as on

ID4k

and respectively

with the endpoints of the short legs selfcontragredient and

ID4k+2

respectively contragredient to each other. The connections are given by the same formulae as the ones in the theorem, but fail the global axioms. graphs ID 2k

ID2k+1

and

E7

The

are eliminated by the parallel transport axiom, while

with the wrong contragredient map satisfy the parallel transport axiom

but fail the global contragredient axiom.

These connections can be used to

construct string model subfactors with correct index.

When the parallel

transport axiom is not satisfied, the induction - restriction graphs might not be the starting ones.

If the parallel transport axiom Is satified but

the global contragredient is not, the induction - restriction graphs will be the initial ones, but the contragredient map will be different from the map started with.

164


BIBLIOGRAPHICAL NOTES

There are several lines of research brought together by the approach described in the sequel.

The two main directions are the study of subfactors A

and the study of actions of groups (or group - like objects) on algebras.

[Go,1960] stating that a

first result common to both was Goldman's theorem subfactor of index

2

is the fixed point algebra of a twisted action of

Together with the classification by Connes [Co2,1977] of the actions of

!2.

8n,

this result shows that up to conjugacy there is only one subfactor of index

2

of the hyperfinite II1 factor.

In a breakthrough paper less than

4

of the index are

Vaughan Jones showed that the values

[Jo3,1982]

He did the first systematic

4 cos2 (7r/n).

study of the basic construction, which had appeared in the thesis of Skau [Sk,1977]

N C M.

and iterated it to obtain the tower of extensions associated to

The projections given by conditional expectations satisfied the Jones

relations

ei eitl ei = r ei

ei ej = ej ei

and

for

1i - j1

: 2, which

gave representations of the braid group yielding new polynomial invariants for knots and links [Jo5,1985]. These relations also appear in statistical mechanics as the Temperley - Lieb relations [TL,1971]. conversely, the subalgebra generated by generated by such projections

e1, e2,

e0, e1, e2,

examples of subfactors with index

for

...

4 cos2 (rr/n)

graphs the Coxeter - Dynkin diagrams

An_1.)

...

Jones showed that inside the algebra

r = (4 cos2(+r/n))-1 gives

(the Jones factors, having

The graph of the tower of

relative commutants, which was introduced by Jones as a conjugacy invariant of

165

Ocneanu: Quantized groups subfactors, was shown to have floors stabilizing eventually Dynkin

A, D

or

E

diagram which repeats itself

to a Coxeter -

(since the vertices of the

floors match); this raised the question of describing this graph up to the stabilizing level and proving an analogous result for arbitrary finite index. Techniques in this direction were developed independently by Wenzl [We2,1985] and us [0c3,1985].

The answer is provided by the introduction of the

principal graph in [0c3,1985] which was later shown to coincide with the induction - restriction graph described in the present paper; the diagrams of the tower of relative commutants are then path algebras on the principal graph.

A path algebra of a graph was subsequently also introduced by Sunder

[Su,1986].

The thesis of Wenzl contains general methods for finding the index

and the relative commutant of subfactors given in a nonstandard form, with which he obtained the values

sin2(k)/ sin2((k+l) r)

for the index of the

subfactors coming from the Hecke algebras. He later computed the graph associated to these subfactors, also described independently by us.

A.

Wassermann [Wa,1987] has introduced a different method of computing the tower of relative commutants for quotients of groups.

Using McKay's results on

discrete subgroups of SU(2), Jones constructed examples of subfactors with graphs extended Coxeter - Dynkin diagrams

A", D", E"

and index

topics related to subfactors are discussed in the monography

Various

4.

[GHJ].

These papers raised several questions about subfactors of index < 4 of the hyperfinite Ill factor: whether there exist up to conjugacy only finitely many

subfactors with the same index, whether the subfactor of index N2 ((p

= Golden Ratio)

correspond to

is unique, whether the only subfactors of index

23 and

S3/S2

3

and whether there are any nontrivial sequences


16 6

in the subfactor which are centralizing the factor, answered in this paper.

For a subfactor with finite index

N C M Pimsner and Popa

[PP1,1985]

constructed an orthogonal basis of the module NM, gave a general definition of the index and defined the entropy.

They proved the gluing lemma for

iterated basic constructions [PP2,1985] also obtained independently by us [0c3,1985].

Several of these techniques and results were carried over to the

infinite index case.

The standard projections, the Jones relations and the

module basis have been extended to subfactors of infinite index, (in the discrete - compact case) by Herman and us [HO] using the operator Sorin Popa has studied infinite index subfactors

valued weights of Haagerup.

which are amenable or have Kazhdan's property

T

[Po,1986].

The extension

of the index to subfactors of arbitrary type was done by Kosaki [Ko,1986]. Phan Loi developed the type III theory further and showed among others that subfactors of finite index III

,

with a' = Xp/q

t

where

of a factor of type p,q E IN

and

p q s c

III,

must be of type

[Lo,1987].

A different direction of research, which was our point of departure, was the study of outer actions of groups on operator algebras. classified the actions of

!n

and

Z

Alain Connes

on hyperfinite factors [Col] [Co2]

The former result was extended to actions of finite groups on the hyperfinite

II1 factor by V. Jones [Jol] while the latter was generalized

to actions of discrete amenable groups on factors of type

II, and more

generally to actions of discrete amenable groups with trivial modular invariant on arbitrary factors by us [Ocl]. In unpublished work, we

167


classified prime actions of compact groups on type II factors, using the Takesaki duality for actions.

The dual of a compact nonabelian group can be

viewed as a quantum group (Hopf algebra, Kac algebra) which forced us to develop the ergodic theory for actions of these group - like objects.

Our

point of departure in 1985 was the remark that the golden ratio subfactor

N C M

is the fixed point algebra of a homomorphism

satisfies, up to an inner perturbation, the relation (id, o)

v: M --> Mat2 0 M which o -41 = a ® id, and thus

behaves like a group (or rather like a quantum group or Kac algebra.)

Another remark was that the tower of extensions for a crossed product consists of the successive crossed products by the group and its dual.

This proved to

be the case in general, and the group like objects called paragroups were announced in

[Oc2,1985]

and presented at the AMS meeting in January 1986,

together with a short proof of the classification of finite group actions using subfactors.

This method of classification was extended to subfactors of

finite index and finite depth and presented at meetings in July 1986. The canonical shift unified such objects as the group multiplication, the comultiplication in Hopf algebras (for the case of crossed products) and the canonical endomorphism previously introduced by Doplicher and Roberts [DR] (for the case of embeddings corresponding to representations of compact groups.) The harmonic analysis on quotients of groups by nonnormal subgroups displayed a local structure, with all the information contained in numbers associated to cells.

Such an approach could be obtained for general

subfactors, with the graphs and cells coming from the induction and restriction of bimodules.

The methods systematized the previously developed

theory and explained the rigidity of the index and the apparition of the

16 8 diagrams

Ocneanu: Quantized groups A - D - E

in the tower of relative commutants proved by V. Jones.

The local approach was detailed enough to allow the description of all the subfactors of index less than 4, with the surprizing conclusion that the diagrams

Dodd

and E7

do not appear as induction - restriction graphs of

subfactors (since they are not geometrically flat.)

These results were first

obtained by computer modelling, and the computations involved in them proved to be very similar to those in the extensions by Pasquier of the Andrews Baxter - Forrester models in statistical mechanics [Pa,1986].

169

Ocneanu: Quantized groups REFERENCES

[Br]

[Col]

0. Bratelli:

Inductive limits of finite dimensional

C*-algebras.

Trans. A.M.S. 171 (1972), 195-234.

A. Connes: Une classification des facteurs de type III. Ann.Sci.Ec.Norm.Sup. 6 (1973) 133-252.

[Co2]

A. Connes: type III.

[CO3]

A. Connes:

Periodic automorphisms of the hyperfinite factor of Acta Sci. Math. 39 (1977), 39-66.

Outer conjugacy classes of automorphisms of factors.

Ann. Sci. Ec. Norm. Sup. 8 (1975), 383 - 419.

[DR]

S. Doplicher and J. Roberts: Endomorphisms of C*-algebras, cross products and duality for compact groups. Preprint.

[Go]

M. Goldman: On subfactors of factors of type II 1. J.

[GHJ]

Mich. Math.

(1960), 167-172.

F.Goodman, P. de la Harpe and V.F.R. Jones: Coxeter Dynkin diagrams and towers of algebras I - III.

Preprint.

[HO]

R.Herman and A. Ocneanu: in preparation.

[Ho]

A. Hoffman: On limit points of spectral radii of nonnegative symmetric integral matrices. Springer Lecture Notes 303 (1972).

[Jol]

V.F.R. Jones: Actions of finite groups on the hyperfinite III factor. Mem. A.M.S. 237 (1980).

17 0 [Jo2]

Ocneanu: Quantized groups V.F.R. Jones: Sur la conjugaison des sous-facteurs des facteur de type II 1. C.R.Ac.Sci.Paris t. 286 (1977), 597-598.

[Jo3]

V.F.R. Jones: Index for subfactors. Invent. Math., 71 (1983), 1-25.

[Jo4]

V.F.R. Jones: Index for subrings of rings. Contemp. Math. 43 (1985), 181-190.

[Jo5]

V.F.R. Jones: A polynomial invariant for knots via von Neumann algebras. Bull. A.M.S. 12 (1985), 103-112.

[Ko]

H. Kosaki: Extension of Jones' theory on index to arbitrary factors. J. Funct. Analysis, 66 (1986), 123-140.

[Lo]

P. Loi: On the theory of index and type III factors. To appear in Comptes Rendus Ac.Sci.Paris.

[MN]

F. Murray and J. von Neumann: On rings of operators

III - IV.

Ann. Math. 37 (1936), 116-229; Ann. Math. 44 (1943), 716-808.

[Ocl]

A. Ocneanu: Actions of discrete amenable groups on von Neumann algebras. Springer Lecture Notes 1138 (1980).

[0c2]

A. Ocneanu: Subalgebras are canonically fixed point algebras. A.M.S. Abstracts 6 (1986) 822-99-165.

[0c3]

A. Ocneanu: A Galois theory for von Neumann algebras. Notes of a UCLA lecture. Nov. 1985.

[0c4]

A. Ocneanu: Path algebras. Notes from a series of lectures,


171

A.M.S. meeting in Santa Cruz, 1986.

[Pa]

V. Pasquier: Operator content of the A-D--E lattice models. CEN - Saclay preprint.

[PPI]

M. Pimsner and S. Popa: Entropy and index for subfactors. Ann. Sci. Ec. Norm. Sup., t. 19 (1986), 57-106.

[PP2]

M. Pimsner and S. Popa: Iterating the basic construction. INCREST preprint, 1986.

[Po]

S. Popa: Correspondences. INCRFST preprint, 1986.

[Sk]

C. Skau:

Finite subalgebras of a von Neumann algebra. J. Funct.

Anal. 25 (1977), 211-235.

[Su]

V. Sunder: A model for AF algebras and a representation of the Jones projections. Preprint 1986.

[TL]

H.N.V. Temperly and E. Lieb: Relations between the percolation and colouring problems ....

[Sh]

J. Shearer: On the distribution of the maximum eigenvalue of graphs.

[Wa]

Proc. Roy. Soc. London A 322 (1971).

I.B.M. preprint 1987.

A. Wassermann: Ergodic actions of compact groups on operator algebras I - III. U.C.Berkeley preprints.

[Wel]

H. Wenzl: On sequences of projections. Preprint, Univ. of Pennsylvania, 1984.

172 [We2]

Ocneanu: Quantized groups H. Wenzl: Representations of the Hecke algebra and subfactors. Thesis, Univ. of Pennsylvania, 1985.

On Amenability in Type III Factors Sorin Popa

Introduction. In [8] we gave a proof of Connes' fundamental theorem that injectivity implies hyperfiniteness for finite von Neumann algebras. It consisted in first proving a certain Folner type condition for injective algebras, and then in combining this Folner condition with a local Rohlin lemma to get a local approximation property. Using a maximality argument, the latter gives the proof. We show here that by using a slightly stronger local Rohlin lemma, one can derive a similar proof of injectivity implies hyperfiniteness from Connes' Felner type condition ([1],[2]). Although we will use the same ideas as in [8], by reordering some of the steps in [8] the proof will become here more conceptual and shorter. In fact the proof in [8] can also be shortened using this reordering of the arguments. 1.

1.1

Notation. Throughout this paper, M will denote a finite von

Neumann algebra with a fixed normal finite faithful trace t (t(1) = 1), 11x112 = t(x*x)1/2, xEM will be the Hilbert norm given by t, and L2(M,t),

the completion of M in this norm. We let 'U(M) denote the unitaries in M.

1.2

Injectivity. The algebra M is called injective if there exists an

M-invariant state 4yo on B(L2(M,t)), i.e., Wo satisfies Wo(xT)=W0(Tx) for

all xEM, TE B(L2(M,t)). Such a state is called a hypertrace on M and

may be regarded as the operator algebra analogue of an invariant mean on an amenable group (see [23).

174 1.3

Popa: On amenability in type Hi factors

Hyperfiniteness. The algebra M is called hyperfinite or

approximately finite dimensional, if there exists an increasing sequence

of finte dimensional subalgebras Mn for which (UMn)-=M (the bar indicating weak closure - this of course implicitly means that the predual of M is separable). 1.4.

Murray-von Neumann global approximation property.

It

was proved in [6] that for von Neumann algebras M with separable predual, the hyperfinitenss (as defined in 1.2) is equivalent to the following approximation property: given any finite subset F=Cvl,...,vn)

of unitary elements in M and E>0, there exists a finite dimensional *-sualgebra M1CM and elements vi',...,vm'E M1 so that Ilvi-vi' II 2< E for

all i. This property is in fact independent of the separability of M, and

when speaking about hyperfiniteness of an arbitrary finite von Neumann algebra M, we will mean that each countable decomposable summand of M has this approximation property.

was pointed out, implicitly in [3] and explicitly in [8], that in fact for M to have the property 1.4, it is sufficient that any reduced algebra of M satisfies 1.5.

The

local approximation property.

It

the following: given any finite set of unitary elements v1,...,vn and E>0,

there exists a projection f and a matrix subalgebra Mo having f as unity,so that II vifvi*-f 11 2 < C II f 11 2, and so that M0 contains elements V1',.. ,vn' with II fvif - vi' II < E II f 11 2, for all i. Indeed, by a simple maximality argument this local approximation property implies the global

approximation property of Murray and von Neumann.

Thus to prove that an injective finite algebra is hyperfinite (in the sense of 1.4) we only need to prove the local approximation. Moreover, since the injectivity of M is clearly inherited by its reduced algebras, we only need to prove that if M is injective and v1,...,vnE M, c>0, then there exists a projection fEM and a matrix algebra M0 with unity f, which contains some elements vl',...,vn'E Mo for which l l fvif - vi' 1 1 2 s c 1I f 1I 2 and Il vifvi* - f112 < E II f 1I 2 for all i. A digression on the Connes-Feldman-Weiss theorem. Let (X,I i) be a nonatomic probability space and G a discrete countable

1.6.

Popa: On amenability in type III factors

175

group acting properly outer and measure preserving on X. Let A = L°°(X,µ), M=AXI G. The Connes-Feldman-Weiss theorem [3] states that given any finite set F9 G and any E>0, there exists a matrix

*-subalgebra M1CM with matrix unit (eij)i j so that the diagonal A1=(eii)i" is in A, the normalizer of Al in M1 is included in the normalizer of A in M, and there exist vl',...,vn' in the normalizer of A, with II vi - vi' 11 < C for all i.

The key part in the proof of this theorem consists in proving a suitable local approximation property similar to I.S. In its simplest form the proof of the local approximation goes as follows: Let (ug)gE GCM be the unitaries implementing the action of G on A. Let F'CG be a finite set and E> O.By what we call the commutative local Rohlin lemma, there is a nonzero projection eE A so that eg(e)=0 if gE F'-(1), or equivalently, euge = 'C(ug)e. In particular, let F'=K-lFK where lE F, KCG are finite sets. If we denote eg.g = ug.eug-1, then gE K gives a matrix unit for a matrix subalgebra Mo with support

f = I ugeug 1 gE K

and by the local Rohlin lemma for each hE F we have

fuhf = I

C(ug, hg)eg'g E Mo.

g,g'E K

Now by the. Folner condition of G, if F is fixed, then K may be chosen so that card(FK\K) < C card K, which in turn means that Iluhfuh 1-f1122<EIIf1122, hEF.

This shows that (uh) can be approximated under an almost invariant projection by a matrix algebra. The reader should keep in mind this argument because it is an approximate version of it that we will prove in the next section. 2.

Proof of the local approximation property. In this section we

shall, for simplicity, assume that M is a separable type IIl factor. 2.1

The noncommutative local Rohlin lemma. The local Rohlin

lemma we need here is slightly stronger than the one in [7], [8]. Its proof is a simple consequence of results in [8] as will be shown in

176

Popa: On amenability in type IIl factors

Section 3.

2.1.1. Lemma. Let F'CM be a finite set and E'> 0. There exists a finite

set P' of mutually orthogonal projections e' in M so that t(le')?1-c', and so that for each e'E J3; 7-yE F' Il e'yei'- t(y)e' 1122

e' ll e' 1122,

0, and let e'E J3)'. Then we have

IyE F' II e'x j*vxie'- t(x j*vxi)e' 11 22 < C' II e' 1122. In particular, since 1E F, we have: II e'x j*xle' - Si je' 1122

O be chosen so that C" o. With the notations in 2.1.2 we get from the definition of the Hilbert-Schmidt norm and from the above observation:

II

eE 9 vE F

I1v(2 xie®xi*)v* - I xie®xi* II 2

HS

=

= 7-vEF IIv(1ixi(7_e)®xi*)v* - lixi(le)®xi*IIHS 2 ilv(,ixi(&xi*)v* - hixi®xi*IIHS 5 7-vEF

2

< (C/8)2 II Jixi®xi* IIHS2

5 2 (C/8)2 II 1xi (Me)®xi* IIHS2

= 2 (C/8)2 LE ? fill xie®xi*ll HS 2.

It follows that there exists an eE P so that llv(Zxie®xi*)v* - Ixie®xi*IIHS < C/4 Illxie®xi*IIHS,

for all vE F.

If ui are the partial isometries approximating xie as in 2.2, then we get llxie®xi* - ui®xi*IIHS = llxie - uil12 5 C"II a 112 = C" Ilui®xi*IIHS,

and thus (*)

ll v(Iiui(Dxi*)v* - 1iui®xi*IIHS < E/4 II Ziui®xi* IIHS+3m II xie-ui II 2 s (C/4 + 3m1/2 E") II Xiui®xi* II r for each n. To see this,

let the group S of all the permutations over IN act on R properly outer and ergodically (e.g., as Bernoulli shifts), and let KnCS be an increasing sequence of finite subgroups with UKn = S,,. If Rn=RKn are

the algebras fixed by Kn, then Rn will do. For BCM a von Neumann

subalgebra we let EB denote the unique normal trace preserving conditional expectation onto B. Then by 1.2 in [7] we have II ERn(x) 'C (x) 11 2 *0 for each xE M. Let n be such that Gy II ERn(y) - 'C(y)1122 < (E/4)2.

Since Rn'f1R=(C, by [7] there exists a finite dimensional abelian subalgebra AOCRn so that IyII EAO'f1R(ER(y)) - 'C(y) 11122 < (E/4)2.

Moreover, by [7] there exists a finite refinement Al of AO in R so that if y'=y-ER(y), then Iy11EAl'f1M(y')1122 < (E/2)2.

We thus get Iy II

EA1'f1M(y)--C(y)l1122

< 27- II EA1'f1M(y')1122 + + 27. II EA1'f1M(ER(y)) - -C(y)1 1122 < E2,

Thus, if e1,...en are the minimal projections of Al then we are done. QED

3.3. General finite von Neumann algebras. In order to make the proof of injectivity implies hyperfiniteness in S2 work for arbitrary finite von Neumann algebras, we only need a suitable more general Rohlin lemma:

Popa: On amenability in type Ul factors

3.3.1

181

General noncommutatve local Rohlin lemma. Let M be an

arbitrary finite von Neumann algebra, and let N be a von Neumann subalgebra of M with N'f1MCN, and let E be the canonical conditional expectation of N onto Z(N). Let also be a normal finite trace on M. 'C

Given any finite set F'SM and E>O, there exists a partition of unity e1,...,en with projections in N so that lyIIlieiyei - E(y)1122
0 we can extend at to all of 98(15) by One checks that at is an Eo-semigroup of 98(15).

Intuitively one feels that the index of this Eo-semigroup should be one.

The semigroup St is generated by an operator with deficiency

indices (0,1).

So our problem is how to associate the number one

with this E0-semigroup.

We begin by noting that for this example there exists a strongly continuous one parameter semigroup of intertwining isometries (i.e., there is a strongly continuous semigroup of isometries U(t) so that U(t)A = at(A)u(t) for all A e 8(15) and t

-1 0).

The intertwining isometries U(t) act on the n-particle

spaces by the relations

193

Powers: An index for semigroups t...... xn- t) for x.

(x

t

i

= 1,..,n

-

= 0

otherwise.

More specifically we define

Let -d be the generator of U(t).

df = lim (f t-!0+

- U(t)f)/t

where the domain D(d) of d is the set of all f e $ so that the limit exists in the sense of norm convergence.

Let d* be the

Note that since the U(t) are isometric

hermitian adjoint of d.

d is skew-hermitian so -d* is an extension of d (i.e. -d* D d).

Then it follow from the theory of hermitian operators (see e.g. Dunford and Schwartz (3) Chapter XII section 4) that each

f e D(d*) can be uniquely expressed in the form f = f0+ f++ f- where There are no solutions to the equation

d*f + = ±f+ and foe D(d). d*f

= -f

since this would imply for g e D(d) that

d (f,U(t)g) = -(f ,dU(t)g) = -(d*f ,U(t)g) = (f ,U(t)g). _ dt

-

-

-

Hence, (f ,U(t)g) = et(f ,g) for g e D(d) and this contradicts the Hence, each f e D(d*) can be

fact that the u(t) are isometric.

uniquely expressed in the form f = f0+ f+ with foe D(d) and f+ e W d with d

*

f += f +.

Note that the space of such vectors f + is precisely

the space 2 of vectors f e Fj so that U(t)*f+= e-tf+ for t > 0. We define a bilinear form
on D(d*) as follows

= 3'a(f,d*g) + 33(d*f,g). A straight forward computation show that if f = f0+ f + and g = 90 + g+ with fo,g0e D(d) and f+,g+e n then = (f+,g+).

194

Powers: An index for semigroups

Hence, the bilinear form
is positive on D(d*).

Let s be the generator of at.

Specifically we define

6(A) = limt40+ (at(A) - A)/t where the domain D(6) is the set of all A E B(F5) so that the above limit exists in the sense of a--weak convergence.

One easily checks

that D(6) is a v-weakly dense *-subalgebra of 19(F5) and 6 is a closed

*-derivation of 13(6) into +b(Fjj). First we will show that if A E 1)(6) then AD(d) C D(d) and dAf = -6(A)f + Adf.

To this end suppose A E D(6) and f e D(d).

Then we have

t-1(I - U(t))Af = t-1(I

- at(A)U(t))f

= -t-'(O( t(A) - A)U(t)f + t-'A( I - U(t))f. --) - 6 (A) f + Adf

as t --) O+.

Hence, Af e D(d) and dAf = -6(A)f + Adf.

Now suppose f e D(d*),

A e D(6) and g e D(6) then we have

(Af,dg) = (f,A*dg) = (f,dA*g) + (f,6(A*)g) _ C(Ad*f + 6(A)f),g). Hence, we have for A E 1)(6) that AD(d*) C D(d

and for f e D(d*)

d*Af = Ad*f + B(A)f.

(*)

Using (*) we will show that the mapping A -) A gives us a *-representation of D(6) on D(d*) with respect to the bilinear form we have

To this end suppose f,g a D(d*) and A e 13(6).

Then

195


= 33(d*f,Ag) + 3'a(f,d*Ag) = 33(A*d*f,g) + 3'a(f,Ad*g) + 3'k(f,5(A)g)

= 33(d*A*f,g) + 3a(A*f,d*g) _ If f e D(d*) and = 0 then f e D(d).

Hence, the mapping

A - A gives a *-representation of 13(5) on the quotient space D(d*) mod D(d) with inner product

We will call this

representation of D(b) on this quotient space na. The question arises, what is this representation? consider the vector F1= 02 e x e Y5

1

To see this

(the one particle space).

A

straight forward computation shows that = 0 for all

It follows

f e Do= {f a L2(O,oo) with df/dx a L2(O,oo) and f(O) = 0).

that _ (Fo,p(a(f1)...... a(fn)*)F0) for

all polynomials p in the a(f) and a(f)* with f e Do.

Hence, the

state associated with F1 for the new representation na is the Fock state.

A little work shows that (see [5] for details) shows that

F1 is a cyclic vector for na.

Hence,

ncc is unitarily equivalent to

the Fock representation.

Now see why the index of this representation is one. the index as the multiplicity of na.

We define

Since n« is irreducible na has

multiplicity one.

Now suppose we repeat the above construction only this time

we replace 9< = L2(O,oo) by K = L2(0,00) ® ... ® L2(O,oo) (n times). Then 9< is the space of n-component functions on [0,00).

If we repeat

the above construction and form the inner product

and construct

the representation na we now find there are n functions one particle space) given by < F

( i

)

, p ( a ( f 1 ) ...... a ( f m

) F ( J ) )

F(i)e S5

1

(the

5i 02 e x e fit and we have = 6ij (F0 p(a(f1),....,a(fm)*)F0).

We

196


see then that na is the direct sum of n copies of the Fock representation.

Then we say this Eo-semigroup is of index n.

For

future reference we will call these examples the CAR flows of rank n. Given any Eo-semigroup at of 93(r5) with an intertwining semigroup

of isometries U(t) so U(t)A = at(A)U(t) one can define the inner product

and construct the representation na.

that na is insensitive to inner perturbations.

We should note

If at is an

Eo-semigroup of 93(5=5) with generator S we can define a new generator S1 given by 61(A) = S(A) + [h,A] where h e 93(!-j)

is skew hermitian.

Then S1 is a generator of an E0-semigroup dt of 93(.

).

If -d is the

generator of U(t) then we define d1= d - h then we find -d1 is the generator of a semigroup W(t) which intertwines 'fit so that W(t)A =

0t(A)W(t) for all

A

e 93(555).

The bilinear form

one constructs

from d1 is equal to the bilinear form one constructs from d. the representation na is unchanged by inner perturbations.

Hence,

The

representation na is then something which classifies at up to "outer conjugacy".

We define the index of at to be the multiplicity of na. can show the following (see [5] for details).

Suppose at and Bt are

Eo-semigroups of 93(f3) with intertwining semigroups Ui(t) for

Suppose at and at are of indices i1 and i2.

One

i

= 1,2.

Let Yt = at® Bt be the

tensor product of at and dt where Y(A ® B) = at(A)

t(B).

Then

U(t) = U1(t) ® U2(t) is an intertwining semigroup for Yt and one can show that the index

i

of Yt satisfies

i

6 i1+ i2.

(I expect the

equality sign holds.) The index theory

I

have outlined is based on two assumption.

One is the existence of an intertwining semigroup U(t) of isometries.

197

Powers: An index for semigroups The second assumption is the implicit assumption that the intertwining semigroup U(t) is unique.

In my own work I considered

this second assumption to be an obvious fact.

It was not until

been working on these problems for two years that

I

I

had

realized that

even for the CAR flow the intertwining semigroup U(t) is not unique. In fact, there are a whole family of intertwining semigroups. believe (because

I

have checked most but not all of the detail) that

I

if one uses one the other intertwining semigroups one obtains the same representation and the same index. Also

I

have found an example of a E0-semigroup of

for which

19(rl-j)

there is no intertwining semigroup (see [6) for details).

The

example is constructed by taking a certain generalized free state of the CAR algebra. because

I

I

will not discuss the details of the construction

do not feel the detail are particularly important.

What is

important is that an intertwining semigroup need not exist. When I arrived in Australia to visit Derek Robinson in October

of 1986

I

knew that intertwining semigroups U(t) need not exist and

if they did exist they were most probably not unique.

it was clear

that we needed an approach to index theory that was independent of an intertwining semigroup. I found.

Here is an approach that Derek Robinson and

The theorems that follow can be found in [7].

The basic idea is the following.

Suppose A and B are hermitian

operators of deficiency indices (O,n) and (O,m) and you want to determine if n = m.

Consider the operator C = A ® (-B).

if C has a

deficiency indices (m,n) and C has self adjoint extension if and only if n = m.

We use the analogous construction for Eo-semigroups.

We begin by introducing the notion of paired EQ-semigroups.


19 8

Definition 2.

Suppose at and

0t are E0-semigroups of M1 and M2.

We say at and $t are paired denoted

e P if there exists a

continuous one parameter group rt of *-automorphisms of M1 ® M2 so that r't(A ® i) = at(A) ® I and r-t(I ® B) = I ® 0t(B) for all

A e M1,

B e M2 and t 4 0. From the definition it follows that. Theorem 3.

Suppose at,@t,Yt and vt are E0-semigroups of von

Neumann algebras.

If (at,dt) e P and (Yt,vt) e P then (at®Yt,at®Qt) a P. If (at,]3t) e P and at and Yt are inner conjugate then (Yt,Bt) a P.

If (at,gt) e P and vt is a continuous one parameter group of *-automorphisms then (at® (TV

,Bt)

a P.

One can show Theorem 4.

Suppose at is an Eo-semigroup of M = 93($).

Then there

exists an Eo-semigroup pt of N = 93(Y5) so that (at,gt) e P.

All E0-semigroups of 9B($> that are known to us we have the

property that they are paired with themselves.

it would be

interesting to know if every E0-semigroup of 98(55) is paired with itself.

Theorem S.

Suppose mt,gt,Yt and of are EQ-semigroups of M1,M2,M3

and My and each of the M. are *-isomorphic with 98(53). (at,8t) a P,

(Bt,Yt) E P and (Yt,vt) a P.

Suppose

Then (at,o't) a P.

With this theorem we can define the notion of an index for E0-semigroups of B(l).

199

Powers: An index for semigroups Definition 6.

We say that two E0-semigroups at and gt of +9(5)

have the same index if there is an Eo-semigroup (r t of 8(i5) so that

(at0Tt) e P and ($t,vt) e P (i.e., at and gt have the same index if they can both be paired with the same E0-semigroup). It follow immediately from theorem 5 that if at and $t are

E0-semigroups of the same index and pt and Vt are of the same index then at and vt are of the same index.

The notion of index then

divides the set of Eo-semigroups of M(5) into equivalence classes.

The obvious question is can this new index distinguish the CAR flows of rank n from one another. Theorem 7.

The answer is it can.

The CAR flow of rank n and the CAR flow of rank m have

the same index if and only if n = m.

The CAR flows are

The proof of this theorem runs as follows. constructed using the CAR algebra.

There is an analogous

construction using the CCR algebra, the algebra of the canonical commutation relations.

in [7] we show that the Eo-semigroups

constructed from the CCR algebra are conjugate to the CAR flows.

When the above theorem is expressed in terms of the CCR algebra the the result can be obtain using the theory of Araki and Wood of continuous tensor products of Hilbert space [1]. Theorem 8.

suppose at is an Eo-semigroup of 93(ii) and there exists

a strongly continuous semigroup of isometries U(t) so that U(t)A = at(A)U(t) for all A e !B(15) and t A O. Eo-semigroup of !05) and (at,gt) a P.

Suppose $t is an

Then there is a strongly

continuous semigroup of isometries V(t) so that for V(t)A =

for all A e005) and t A 0.

t(A)V(t)

200


Since there is an E0-semigroup at of 93(53) for which there is no

strongly continuous intertwining semigroup of isometries (i.e., there is no strongly continuous semigroup of isometries U(t) so that U(t)A

= at(A)U(t) for all A e 9305) and t A 0). Since for CAR flows there exists a strongly continuous semigroup of intertwining isometries it follows that any flow which can be paired to a CAR flow has a strongly continuous semigroup of intertwining isometries by the previous theorem.

And, therefore, any flow which is of the same

index as a CAR flow must necessarily have a strongly continuous semigroup of intertwining isometries.

Hence, at is not of the same

index as any CAR flow. we see then that the set of indices for Eo-semigroups of +B($)

corresponds to a set which properly contains the non negative integers plus a point at infinity.


201

References

1. Araki, H. and Woods, E.T.: Complete Boolean Algebras of Type I Factors, Pubs. Res. Inst. Math. Sciences Kyoto University, Series A Vol

II (1966) 157-242.

2. Bratteli, O. and Robinson, D.: Operator Algebras and Quantum Statistical Mechanics I. Springer - Verlag, 1979. 3. Dunford, N. and Schwartz, J.: Linear operators part II. Interscience, 1963.

4. Haag, R. and D. Kastler, An algebraic approach to quantum field theory, J. Math Phys. 7 (1964),848-861

5. Powers, R. T.; An index theory for semigroups of *-endomorphisms of l8(Fj) and type 111 factors, to appear in Canadian Jour. of Math.

6. Powers, R. T.: A non spatial continuous semigroup of *-endomorphisms of E(F)), to appear in Pubs. Res. Inst. Math. Sciences Kyoto University. 7. Powers, R.

T. and Robinson, D. W.: An index for continuous

semigroups of *-endomorphisms of E()i), to appear in Tour. of Functional Analysis.

Coactions and Yang-Baxter Equations for Ergodic Actions and Subfactors by Antony Wassermann (University of Liverpool and University of California, Berkeley)

I. Introduction The aim of this paperf is to bring into evidence the usefulness of considering not only actions of compact groups on operator algebras, but also the dual notion of coaction. A lot of what we shall say is contained in much greater detail in three series of papers, due to be published in the near future: four papers on ergodic actions, two on product type actions and two on equivariant K-theory. These all have their rather primitive origins in the three chapters of my thesis. The other main proponent of coactions of compact groups is Adrian Ocneanu, and we shall make frequent reference to his still unpublished work. We now briefly summarise the contents of the rest of this paper. In Section II we recall the basic definitions of coactions of compact groups on von Neumann and C* algebras. We present two examples of C* algebras which arise perhaps unexpectedly as crossed products by coactions, and show how this observation can be used to explore their structure. The basic idea here is an old one: to use symmetry properties to simplify and elucidate computations. In Section III we exhibit two general principles in equivariant KK-theory, namely Frobenius Reciprocity and Dirac Induction. When combined with the equivariant Thom isomorphism, these lead to a generalisation of a spectral theorem of Hodgkin (for the K-theory of spaces) which in principle provides a homological machine whereby ordinary KK-theory (of a pair of algebras) can be deduced from equivariant KK-theory. In particular if C is a compact Lie group which is both connected and simply connected, this leads one to suspect that if A is a G-algebra for which KG(A) is just Z (with the augmentation action of R(G)), then A is KK-equivalent to C(G). Examples of such actions arise naturally in the theory of ergodic actions. In Section IV we outline the general theory of ergodic actions and show how they can be understood better by exploiting a link between equivariant K-theory and spectral multiplicities. This leads to the concept of the `multiplicity map' and its associated diagrams. We explain how these ideas can be used to classify the ergodic actions of SU(2). To achieve this we need two additional tools: the first exploits the theory of coactions, while the second hinges on the presence of gaps in the spectra of certain homogeneous spaces of SU(2). We reach the perhaps disappointing conclusion that SU(2) has no exotic ergodic actions

t This is an expanded version of a talk given at the U.S.-U.K. joint conference on Operator Algebras at Warwick in July, 1987.

204

Wassermann: Coactions and Yang-Baxter equations

on operator algebras, i.e. all its ergodic actions are necessarily on Type I von Neumann algebras. On the other hand, looking on the bright side, we recapture the classification of the closed subgroups of SU(2) up to conjugacy by a new if somewhat long-winded method. In Section V we explain how it is possible to develop a theory which parallels in every respect the easy classification of ergodic actions of compact Abelian groups, but this time for non-commutative groups. The natural restriction here is that the crossed product by the ergodic action should be a factor. One finds that such actions are classified by cocycles and bicharacters of the group dual, plus analogues of the usual non-degeneracy criteria for the action to be on a factor. The non-commutativity, however, introduces a quite novel feature, the symmetric quantum Yang-Baxter equation. This equation (with parameters) has been much studied in Russia in connection with completely integrable Hamiltonian systems and exactly solved models in quantum field theory. It also turns out to be the key to understanding the second cohomology of the dual of a simple compact group. In the remainder of this section we illustrate how these cocycles can be classified for certain classical groups of low rank. In Section VI we study actions of compact groups on von Neumann algebras for which the crossed product is a factor. We consider in particular actions where there is an invariant subalgebra satisfying the same hypothesis. We then examine the position of the fixed point algebra of the subalgebra in the fixed point algebra of the whole algebra using the framework provided by the theory of subfactors. We find that there is a particularly simple ìnvariance principle' operating here. Finally in Section VII, we bring together the ideas of the two preceding sections. Every solution of the symmetric quantum YangBaxter equation gives rise to a factor representation of the infinite symmetric group in the hyperfinite III factor, and hence we naturally obtain a subfactor. We conjecture that the position of this subfactor is more or less equivalent to the information provided by the bicharacter and go some way to lending credibility to this idea, by showing the analogy with the theory developed in Section VI. The style of this paper has been kept deliberately terse, primarily because of limitations of space and time, but also because I preferred to emphasise broad principles rather than concentrating on fussy details. The same restrictions also apply to the list of references provided, which has deliberately been kept short and consequently no doubt has many glaring omissions. More complete references can be found in the bibliographies of the papers referred to at the beginning of this introduction. The debt of gratitude I owe to other mathematicians would be hard to repay, in particular to Bill Arveson, Dick Kadison, Claude Schochet and Masamichi Takesaki; I would also like to thank Vaughan Jones, Adrian Ocneanu, Picu Voiculescu and Hans Wenzl for their help and encouragement.

II. Coactions of Compact Groups Let G be a compact group with C* algebra C*(G) = ®a E& End(V,,) and right von Neumann algebra 1Z(G) = ®End(V,r). If A is a C* or W* algebra, a coaction of G (or


205

action of G) is described by a *-isomorphism 8: A -+ A®R(G) such that (t®SG)S = (S®t)S, where SG: R(G) -+ R(G) ® R(G) is the canonical coaction p(g) i--+ p(g) ® p(g). If G is

Abelian, this clearly amounts to giving an action of G on A. Otherwise if vr1, ..., 7r,, are a faithful family of irreducible representations of G, then S is uniquely specified by 67, 1... ,5.ir,, , the parts of S going into the irk components of A ® R(G). We do not wish to

enter into the duality theory for coactions and actions (which is anyway rather easy for compact groups). Suffice it to say that for every action on A there is a canonical coaction on the crossed product A >4 G, which is given by (A ® 1C(L2(G)))G or (A ® B(L2(G)))G according to whether A is a C* or W* algebra. It is induced by the natural isomorphism L2 (G) ® V=+dim(V) L2(G) of G-spaces. Moreover one defines the crossed product A A 6 G by the coaction to be the subalgebra of A®B(L2(G)) generated by 5(A) and C(G) with the natural dual action of G given by Ad(A), where A denotes the left regular representation of

G on L2(G). Later we shall be concerned with coactions of G on von Neumann algebras, but first let us give two non-trivial examples arising as crossed products by coactions. The main criterion we shall use to recognize that an algebra has this form, that is supports

an action of G which is dual, will be that of M. Landstad, namely the presence of an equivariant copy of C(G) in the algebra. In both examples that we give there is a common

underlying principle, namely that the structure of an algebra can be determined more easily by exploiting its symmetry properties to the full.

Example 1: Zeroth Order Pseudodifferential Operators on a Compact Lie Group G. Let U(G) be the ring of left invariant differential operators on G, the universal enveloping algebra of Lie(G). If X1,...,X is an orthonormal basis of Lie(G) with respect to a G-invariant inner product, then we may define the Laplacian A = -(Xi +... + X2). It is central in U(G) and therefore yields a positive scalar in every non-trivial irreducible representation of G. Regarding all operators in U(G) as sitting inside ]I End(V.V), we may convert the first order differential operators Xl,..., X (vector fields) into zeroth order pseudodifferential operators Tk = Xk/(I + p)1/2 . The C* algebra of zeroth order pseudodiferential operators 'Yo(G) on G is by definition the C* algebra generated on L2(G) by these bounded operators and C(G). It is therefore by definition the crossed product by a coaction on %p (G), the C*algebra of left invariant zeroth order pseudodifferential operators generated by just the Tk's. This enables one to establish the fundamental symbol sequence

0 -- PC(L2(G)) --+ 'Io(G) ---+ C(G x S(Lie(G)*)) -+ 0

by considering the much simpler invariant sequence 0 -+ C*(G) --+ %p (G) --+ C(S(Lie(G)*)) --+ 0

and then forming the crossed product with G. The invariant sequence is easy to analyse using asymptotic properties of the representation theory of G and is particularly easy when G is a torus (and the invariant algebra is Abelian). For general G, the asymptotics have close links with results of V. Guillemin, W. Lichtenstein and H. Widom.

206


Example 2: Toeplitz Operators on the Unitary Groups. The unitaries U. form the Silov boundary of the open unit ball D. = {x : x*x < 1} in MM(C). The topological boundary ODn of Dn breaks up naturally into n distinct parts corresponding to the number of eigenvalues of x*x equal to 1. Each part is an orbit Ok for the natural action of U(n, n) on Dn and is a union of components isomorphic to Dk with 1 < k < n, whose Silov boundaries are naturally contained Un. Thus Ok = UU,,,EU uDkv* with the Silov boundary of uDkv* given by uUkv*. Let HZ(Un) C LZ(Un) be the Hardy space given by the LZ boundary values of holomorphic functions in this ball. Thus HZ(Un) is just the closure of the space of polynomials p(zll, , znn) restricted to Un. Let P: L2 -4 H2 be the Hardy space projection and define T(Un) to be the C* algebra generated by the Toeplitz operators T(¢) = Pm(¢)P where ¢ E C(Un) and m(¢) is the corresponding multiplication operator on LZ(Un). The analogue of this algebra was considered in the unbounded (upper half-plane) realisation of the unit ball by Paul Muhly and Jean Renault. Their approach was to analyse the algebra using `groupoid' techniques, which essentially boiled down to using crossed products by Rn. The Hardy space projection in their case was given by convolution by a certain distribution on Rn which in Fourier corresponded to multiplication by the characteristic function of a cone. The structure of the algebra was then obtained by a microlocal analysis of this distribution, an analysis which simply amounted to compactifying the cone according to its asymptotics. Our approach to unravelling the structure of T(Un) exactly mirrors that of Muhly and Renault, except that the euclidean group Rn is replaced by the compact group Un (or more precisely Un x Un). The structure can be described quite simply in terms of the boundary components of the unit ball and the analysis proceeds by using coactions and a sort of microlocal analysis on Un. In fact from the Cauchy formula expressing the values of a holomorphic function in terms of its boundary values on Un

Az) = JUn

f(u)du det(1- u*z)n'

one can write P(f) = 6n* f where On is the central distribution

det(1-u)-n

= lim1i det(1-

eu)-ndu on Un. Let An be the C* algebra generated by the convolution operators 0 On (0 E C°O(Un). This could be called the `microlocal algebra' of On, since its asymptotics (i.e. its irreducible representations) are very much related to the microlocal strucure of en. In fact by simple representation theoretic arguments one can show that the irreducible representations of An come from the natural homomorphisms 0 On

09 dk (09 = 0 conjugated by g)

associated with Hardy space distributions for smaller boundary components. The existence of these homomorphisms lies at the heart of our analysis and follows from the fact that if

-if is the highest weight matrix coefficient of Un with signature f : fl >- f2 >_ then N

On = 1ndUk?unOk

> fn,


207

This limit formula is established by using some elementary representation theory of U1. The structure of T(UU) (which is a U x Un-algebra) can then be deduced immediately once one establishes that it is isomorphic to P(An x Un)P. This identification, however, is a direct consequence of the classical fact that the natural representation of T(Un) on H2(Un) is irreducible. Having performed this analysis, one then finds that the irreducible representations of T(Un) are completely described by Toeplitz representations on the var-

ious boundary components of the unit ball. Thus one fixes 1 < k < n and u, v E Un and defines an irreducible representation 7fk,u,,, of T(Un) by T(O) i--+ T(Oluukv.), where as above uUkv* is regarded as the Silov boundary of the unit ball uDkv*. As a consequence

of this analysis one can apply an argument of Coburn to show that if n > 2 and f is a holomorphic function on Dn extending to a continuous function on Dn, then the set of values taken by f on the topological boundary OD is equal to the closure of the set of values taken by f in Dn. This of course is a considerable strengthening of the usual maximal modulus principle. Before leaving this example, we should point out that another line of attack on T(Un)

using Jordan triple systems was pursued independently by Harald Upmeier, who succeeded in obtaining a uniform classification for the Toeplitz algebras corresponding to all of E. Cartan's bounded symmetric domains. It is not clear whether the remaining domains, namely those not of type Inn, can be dealt with by an adaptation of our 'non-commutative groupoid' approach, although this does not seem totally implausible.

III. Equivariant Kasparov Theory Frobenius Reciprocity, Dirac Induction and Hodgkin's Spectral Sequence The C* algebra T(Un) we considered in the last example has a sequence of Un x Uninvariant ideals

T(Un)DJ1DJ2D...DJ.D{0} with successive quotients Jk/Jk+1 = indN(uk)IU xun1C(L2(Uk)) as Un x Un-algebras. It is therefore of great interest to study the extensions

0 --' Jk+1/Jk+2 - Jk/Jk+2 -' Jk/Jk+1 - 0 as elements of the equivariant Kasparov group KK1 Unxun(Jk/Jk+11Jk+1/Jk+2) or even as

elements of the ordinary Kasparov group KK1(Jk/Jk+l,Jk+1/Jk+2) Upmeier has used the index theorems of Boutet de Monvel to identify these elements explicitly, but another method is available if one exploits the symmetries a little. In fact there are two basic principles in equivariant KK-theory: 1. Frobenius Reciprocity: KKG(A,indHIGB) = KKH(resG1HA,B) whenever H is a closed subgroup of G with A a G-algebra and B an H-algebra. 2. Dirac Induction: KKT(A, B) - R(T) ®R(G) KKG(A, B) whenever T is a maximal torus in the compact Lie group G and A and B are G-algebras. We must assume in addition that G is connected and that 111(G) is torsion-free.

208


The first of these results is an example of the metamathematical principle that anything functorially true for HomG(A, B) should extend automatically to KKG(A, B). This is because elements of the latter group are morally just differences of homomorphisms. A precise form of this vague statement has been supplied by J. Cuntz's alternative description of KK-theory via quasi-homomorphisms and (more recently) his algebra qA. Our proof uses an imprimitivity theorem for induced Hilbert C* modules, but it would be interesting to supply a more direct proof based on an equivariant version of qA. The second result on Dirac induction is much harder to prove. The main ingredients of the proof are as follows: (i) Steinberg's Basis Theorem and the formalism of the discrimant (ii) Bott's Index Theorem for homogeneous elliptic operators on G/T (iii) Borel's T-equivariant cell decomposition of G/T (iv) Weyl's Character Formula

Let us give a brief sketch of the proof. By (i) one knows that R(T) is a free R(G)module of rank JWI where W = N(T)/T is the Weyl group. Moreover Steinberg actually constructs an explicit basis given by characters in T. Thus the tensor product R(T) ®R(G) KKG(A, B) may be identified with the direct sum of JW copies of KKG(A, B) as an R(G)-module. Our task is therefore to construct maps a: KKT(A, B) -> KKG(A, B)1 W1 and 0: KKG(A, B) I n'I -- KKT(A, B) such that a/3 and ,(3a are the identity maps. To do so, we use Frobenius Reciprocity to identify KKT(A, B) with KKG(A, B ® C(G/T)). We then produce elements xl,...,x1wI in KKG(C,C(G/T)) and elements yl,.. ,ylw1 in KKG(C(G/T),C) which form `dual bases' in the sense that (1) x; ®c yj = &j and (2) w x; z c Here we take Kasparov products and the element

on the right hand side of the second equation is just the identity element of the ring KKG(C(G/T),C(G/T)). The isomorphisms a and ,3 can then be defined using Kasparov and J3(yj) _ E; yJ®cxj where f E KKG(A, B(&C(G/T)) and ('j) E KKG(A,B)IWI. Thus everything hinges on constructing the dual bases and in fact the main work lies products as

in producing a solution of (1). By taking the xi's as homogeneous line bundles E; over G/T and the yd's as Dirac operators DF, on G/T with coefficents in appropriate homomogeneous line bundles F1, the proof that the first equation can be satisfied reduces to the computation

of a determinant. Indeed the E;'s and Ft's essentially correspond to characters of T. Moreover the Kasparov product Xi ®C(G/T) yi is given by the R(G)-valued index of DE; ®F; .

On the other hand, by (ii) and (iv), the index of DE is equal to A(E®Eo)/A(Eo) where Eo is a certain fixed character of T and A(E) denotes the alternating sum °Eyy(-1)°v(E) over the Weyl group. Thus to solve (1) it suffices to find characters El, ... , EI tiy1 such

that the determinant A = det(A(E; ® E;)) is equal to ±A(Eo)In'I. Let o ,...,oiyyl be the elements of W and consider the matrices M = (o,(E;)) and N = ((-1)°iv;(E1)). We then have det(M)2 = ±det(N1)det(M) = ±det(N'M) = fA so that up to a sign A is given by the square of det(M). On the other hand the Steinberg basis of R(T) satifies det(M) = A(Eo)IH'I/2, so we are through.


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Let us now indicate how (1) and (iii) together imply (2). We have to show that the

idempotent e = Ei yi ®c xi is the identity in the ring R = KKG(C(G/T), C(G/T)). It is easily verified that e is a two sided unit for elements of the form yi ®c x j, which themselves form a linearly independent set over R(G). Let Ro be the R(G)-submodule of R that they generate. We claim that (over R(G)) R is torsion free and R/Ro is a torsion module. To see this we use Frobenius Reciprocity to identify R with KKT(C(G/T),C) (as an R(G)-module). The Borel cell decomposition for G/T and the T-equivariant Thom isomorphism then imply that R = KKT(C(G/T),C) is torsion-free as an R(T)-module and that if k(T) is the field of fractions of R(T), then the dimension of k(T) ®R(T) R over k(T) is exactly IWI. (This idea goes back to J. McLeod.) Returning to the R(G)-module structure of R, we deduce that R is torsion-free over R(G). Moreover some simple Galois theory shows that k(G) ®R(G) R has dimension JWI2 over k(G), where k(G) is the field of fractions of R(G). Since Ro is free of rank 1W 12, it follows that R/R0 is a torsion module. Let us now show that e is the identity element 1 of R. Since R/Ro is a torsion module, we

can certainly find a non-zero element a E R(G) such that al E Ro. Hence e(al) = al so that a(e - 1) = 0. Since R has no torsion, we infer that e = 1 as required. (From this one can deduce that R = Ro using the easily verified relation RORR0 C R and hence that the elements yi ®c xj form an R(G)-basis of KKG(C(G/T),C(G/T)). A similar argument shows that the y;'s form a basis of KKG(C(G/T), C). In this way we obtain an alternative combinatorial approach to Poincare duality for the homogeneous space G/T.)

Note that as a result of Frobenius Reciprocity one can identify the group

KKu

with the equivariant K-homology group KKN(U,t+l) (ind(1C(H2(Uk))),C) which is much easier to understand. We shall now indicate a much more useful application of the two principles. We start with the extension of Z-algebras

0 - Co(R\Z) - Co(R)->Co(Z)->0 where Z acts on R by translation. Passing to the n-fold tensor product and taking the crossed product by Z", one obtains a natural filtration on the algebra Co(R") >4 Z" which is compatible with the dual action of Ti'. This filtration appears implicitly in Kasparov's conspectus as a special case of a far more general construction. If T = T" is a maximal torus in G then we obtain a corresponding filtration on the induced algebra indT?GCo(R") )4

T. On the other hand, the Z-equivariant Thom isomorphism theorem implies that this induced algebra is actually KKG-equivalent to C(G). Combining this observation with the spectral sequence corresponding to the filtration of the induced algebra immediately yields a spectral sequence converging to KKG(A, B ® C(G)) = KK(A, B). In this way we obtain a generalisation of Hodgkin's Spectral Sequence to equivariant KK-theory. Routine computations similar to those of Fack and Skandalis allow one to work out the E2 term explicitly, the end result being a spectral sequence for arbitrary (Z2-graded) G-algebras A and B converging to KK*(A, B) with E2 term TorR(G)(KKG(A, B), Z). Symbolically we have TorR(G) (KKG(A, B), Z) - KKP+q(A, B).

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Thus in principle ordinary KK-theory can be deduced from equivariant KK-theory just by using homological algebra. Not only is the method of derivation we have outlined above significantly shorter than the original method of Hodgkin and Snaith, but it also yields a more general result. It is fair to say that this is largely due to the systematic use of Kasparov products. These already appeared implicitly in Atiyah's use of Dirac induction to obtain a splitting for the forgetful map KG(X) -a KT(X), a result which was one of the key ingredients in Hodgkin's method. His method, however, relied strongly on techniques from algebraic topology for constructing spectral sequences. This had the disadvantage of restricting the class of spaces to which the result applied, a restriction which disappears if one works in the more flexible framework of Kasparov theory and uses èlliptic topology' instead of àlgebraic topology'.

For the remainder of this paper, we shall not be concerned with general Kasparov theory which we have not really defined. Instead we shall only consider the groups KKG(C, A) = KG(A), that is the equivariant K-groups of A. Somewhat unexpectedly perhaps these will play an important role in the next seeemingly unrelated topic. By definition Ko (A) is the Grothendieck of finitely generated projective A-modules E with compatible actions of G. It has a natural R(G)-module structure described by the maps £ --* V ® £ where V is a finite-dimensional representation of G. Now a basic result of P. Green and P. Julg states that KG (A) and Ko(A A G) are canonically isomorphic. One can also see this by using Kasparov's equivalent Hilbert module formulation of KKG(C, A) together with the natural identification of Ana G with (A®K(LZ(G)))' ®Ada. More generally Baaj and Skandalis have obtained (as a special case of a result on actions of Hopf algebras on C* algebras) a canonical isomorphism KKG(A, B) = KKG(A > G, B i G) where KKG

is defined by a straightforward extension of Kasparov's definitions for Abelian G. Here one has to view the coaction as a module action of the Fourier algebra A(G) of G, or even its G-finite part, i.e. the algebra of matrix coefficients of G. Their proof has two main ingredients: the Landstad-Takesaki duality theory for actions and coactions; and a simplified version of Higson's proof of Kasparov's technical lemma that is well adapted to coactions.

IV. Ergodic Actions of Compact Groups Equivariant K-theory and the Multiplicity Map Let M be a von Neumann algebra and G a compact group acting on M. This action is said to be ergodic if the fixed point algebra MG is trivial, i.e. MG = C. A complete classification of ergodic actions of compact Abelian groups was obtained first by Albeverio and Hoegh-Krohn and later by Olesen, Pedersen and Takesaki. The idea was very simple. We may assume the action of G faithful. If uX # 0 is an eigenvector in the spectral subspace MX of M corresponding to x E G, then uXUX and uXUX are both invariant under G so must be scalars, necessarily equal. Thus all non-zero spectral subspaces MX are one-dimensional spanned by a unitary eigenvector uX. Since the set of


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x for which Mx # {0} is evidently a subgroup of G, the faithfulness of the action of G forces it to be the whole of G. It follows that the unique G-invariant state of M (given by averaging over G) is a trace. The assignment x ' -4 ux defines a projective representation of G corresponding to a unique class of cocycle in H2 (G, T). Due to the presence of the dual action of G, this is necessarily the regular representation of G on LZ(G) (or LZ(G) in Fourier) corresponding to the particular cocycle. This process can be reversed: in other words every faithful ergodic action of G is the natural action on the von Neumann algebra generated by some regular cocycle representation of G. Thus one gets the natural generalisations of the non-commutative tori of Marc Rieffel. Physicists, however, would prefer G to be non-Abelian. Thus the basic open question is whether any non-Abelian compact Lie groups can admit non-trivial ergodic actions on von Neumann algebras, that is on the hyperfinite type II, factor. (The reduction to the factor case follows by using induced actions.) In the remainder of this section and the next section we will develop techniques which allow us to tackle this problem for groups of low rank, such as SU(2) and SU(3). Let us now outline the basic elements in the general theory of ergodic actions and explain its position in the general programme for classifying actions of compact groups on von Neumann algebras. The first and most crucial result in the subject is the finiteness theorem of Hoegh-Krohn, Landstad and Stormer which asserts that the unique G-invariant state 0 on M (obtained by averaging over G) is in fact a trace and that every irreducible representation of G occurs with multiplicity less than or equal to its degree in the spectral

decomposition M = ®,EGM7r of M. This can be proved by completely elementary means by comparing the two inner products cb(xy*) and O(y*x) on finite-dimensional G-submodules of M and using Hermann Weyl's dimension formula. One has to make constant use of eigenmatrices in the proof: for 7r E G, we define an eigenmatrix for 7r to

be an element M E M 0 End(V,) such that ay(M) = Mir(g) for all g E G. (One can also define the notion of an eigenvector for 7r as an element of M 0 V * satisfying the same condition as an eigenmatrix; thus the rows of an eigenmatrix M form eigenvectors for -7r

and each eigenvector corresponds to a copy of it in M.) The basic ideas in our proof of the 'finiteness' theorem nevertheless are exactly the original ones, although no direct appeal to Tomita theory is necessary. Once one has the result of the theorem, it follows that (M ® End(V))G is finite-dimensional for any finite-dimensional representation V of G. The same is true if M is replaced by the C* algebra A generated by the spectral subspaces of M. Thus M A G and A x G have approximate identities consisting of finite

rank projections. (We say that a projection is of finite rank if the corner it defines is finite-dimensional.) Thus A A G = ®1C(7-l,) and M A G - ®13(7.1;) are both direct sums of algebras of compact operators or bounded operators. The main idea is then to arrange for a confrontation of the two distinct approaches suggested by the von Neumann theory and

the C* theory. Before doing so, we can enlarge the problem slightly and notice that the study of ergodic actions is equivalent to the study of actions for which the crossed product M A G is type I with atomic centre and such that MG contains no non-scalar central

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elements, i.e. M' fl MG = C. (This is equivalent to the fact that G acts ergodically on the centre of M.) This class of actions is stable under perturbations. Thus we are really classifying coactions of G on atomic type I algebras. Adrian Ocneanu has tackled the problem of classifying prime or minimal actions, namely those for which the crossed

product M x G is a type III factor and for which the fixed point algebra has trivial relative commutant, M fl (MG)' = C. This work remains unpublished, since Ocneanu has reason to believe that his machine for classifying subfactors of finite depth can be extended to yield a classification of prime actions. The end result is that all prime actions are conjugate.

Since it will be needed in the last two sections in this paper, we pause here to give the quintessential example of a prime action of an arbitrary compact Lie group G on the hyperfinite III factor. Let it be a (projective) representation of G on a finite-dimensional vector space V such that the homomorphism Adir: G -> PU(V) = Aut(End(V )) is faithful. Let A be the UHF algebra ®°°_IEnd(V) with its canonical trace T and let R be the von

Neumann algebra irr(A)". G acts on both these algebras by the infinite tensor action a9 = ®Ad(ir(g)). Now r restricts to an extremal (or factor) trace on the fixed point algebra AG. This follows essentially by an old observation of D. Voiculescu: the extremal traces of AG correspond to the order-preserving homomorphisms of the ordered ring Ko(AG) into R. Hence RG must be a subfactor of R. Now the group S,,, of finite permutations of the positive integers has a natural representation 7r in A given by the canonical embeddings of the finite symmetric groups S. in ® IEnd(V) = End(V®n). Moreover 7r(S M,, (Z) such that M(-7r)c = dim(7r)c. Our main result is then that M is a multiplicity matrix. Thus Mij(-7r) is the multiplicity of 7r E G in ei(M ® B)e, and each of these multiplicities is bounded by the dimension of 7r. Thus by passing from the original ergodic action on M = e1 (M ®B)e1 (say) to the associated (or conjugate) ergodic actions on ei(M ® B)ei together with the intertwiner spaces ei(M ® B)ej, one finds the finds the remarkable property of multiplicativity of multiplicities. Data of the multiplicity matrices can be conveniently recorded on multiplicity diagrams. These are directed graphs with the M(7r) as the incidences matri-

ces (7r need not be irreducible here). The condition MG fl M' = C is equivalent to the connectivity of the graph provided that 7r is faithful. It is a straightforward exercise in com-

binatorics to check that the only possible multiplicity diagrams for the two-dimensional identity representation of SU(2) correspond exactly to the McKay diagrams for the closed subgroups of SU(2) (that is - with the exception of the trivial subgroups - the extended

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Dynkin diagrams of type An (1 < n < oo), Dn (2 < n < oo), E6i E7, and Ej with their canonical weights). Thus from the point of view of spectral multiplicities, every ergodic action `looks like' an action on indHISU(2)End(V) where V is a (projective) irreducible representation of a closed subgroup H of SU(2). Broadly speaking there are two strategies for showing that this isomorphism of G-spaces can be made to respect the algebraic structure, thereby proving the main result on ergodic actions of SU(2) - a `coaction' strategy and a `gap' stratgegy. In the first strategy one has to show that a certain action is dual, usually the action on a suitable reduction of M ® B. This reduction is chosen so that the fixed point algebra looks like a copy of the von Neumann algebra of the corresponding subgroup H, and in this case there are always unitary eigenmatrices. This works fine for the maximal torus and its subgroups. One shows that the action is dual by applying a general èigenmatrix criterion', which in this case amounts to exhibiting a a* b 1 unitary eigenmatrix for the two-dimensional representation having the form a*

l

For a general connected compact Lie group, one knows that G is algebraic (in a particular faithful representation 7r) and the duality criterion is then that G should have a unitary eigenmatrix for 7r with commuting normal entries such that the algebraic equations defining G are satisfied by the entries of the matrix. The cases corresponding to the normaliser of the maximal torus and its binary dihedral subgroups are again handled by the coaction

strategy, but this time it is necessary to pass to the fixed point algebra of the centre of SU(2). Here one has an action of SO(3) and one now looks for three-dimensional eigenmatrices with commuting self-adjoint entries and determinant one. The basic methods of producing such special eigenmatrices are illustrated in the first and third examples of the next section. The remaining subgroups H are treated by the gap strategy. One has an algebra M with the same multiplicities as L°O(G/H) and in each case L°°(G/H) has enough gaps in its `G-spectrum' to force it to be Abelian. Thus for example one might have some irreducible representation it such that no subrepresentation of A2ir appeared in L°°(G/H). The vectors in any copy of 7r in M would have to form a commuting set, as can be seen by considering the maps A2V -> M, v A w H vw - wv, where V is some G-invariant subspace of M. The gap strategy thus juxtaposes the internal product structure of the algebra with the external tensor product structure of representations. As an example let us consider an algebra M having the same spectral decomposition as L°O(SU(2)/H) where H is the normaliser of a maximal torus. Thus the five-dimensional representation 1r2 of SU(2) ap-

pears with multiplicity one and, since it is real, it can be represented by a self-adjoint basis. Since no subrepresentation of 1r2 occurs in M, this basis generates an Abelian von Neumann algebra which is SU(2)-invariant. It is necessarily of the form L°O(SU(2)/K) for some closed subgroup K and since 1r2 is the smallest non-trivial representation to occur in it, one can easily conclude that K and H are conjugate subgroups of SU(2). It follows

therefore that M and L°°(SU(2)/H) are equivariantly isomorphic as required.


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V. Full Multiplicity Ergodic Actions of Compact Groups Bicharacters and Examples As we have seen the classification of ergodic actions of compact Abelian groups reduced

immediately to the study of faithful ergodic actions. It turns out that this condition is satisfied precisely when the crossed product is a factor. Analogously for non-Abelian G, there is an important class of ergodic actions to consider first before tackling the general case. These actions are characterised by the following equivalent conditions:

1. dim(M,) = dim(ir) for all x E G (i.e. the action has full multiplicity) 2. each 7r E G has a unitary eigenmatrix in M 3. the W* crossed product is a factor (of type I) 4. the C* crossed product is isomorphic to an algebra of compact operators.

Thus in this case the multiplicity map is trivial because KG(A) is equal to Z with the augmentation action of R(G). In view of Hodgkin's spectral sequence, it is not unreasonable to conjecture that A and C(G) should be KK-equivalent or at least that their K-theories should coincide. Indeed one would expect the generators of the K-theory to appear in K1(A) as the unitary eigenmatrices of the `fundamental' representations of G. This is a path we have not explored. Instead we have extended the cocycle and bicharacter classification from the Abelian case to this case. We find that if we are very careful, then everything carries over perfectly from the Abelian case, except for one curious but decisive twist coming from the non-commutativity of the group. The celebrated quantum Yang-

Baxter equations rather unexpectedly appear as a consequence of the usual bicharacter relations. As we shall indicate in the final section, the role they play is crucial and in a certain sense they seem to determine the cocycle and ergodic action completely, which surely must come as something of a surprise. It would not have been suspected three years ago that the computation of H2(G, T) would turn out to be more or less equivalent to solving the so-called `constant symmetric quantum Yang-Baxter equations', which are currently being studied by the Russian school of Fadeev, Sklyanin and, more particularly, Drinfeld, Gurevich and Lyubashenko; and that the key to understanding this link would come from the theory of subfactors (described in the penultimate section) and the study of certain representations of S... It will be convenient to give a general definition of cocycles of G and the corresponding cocycle representations before explaining how they figure in the classification of full multiplicity ergodic actions. A cocycle of G is a unitary w E R(G) ® R(G) satisfying the cocycle identity: SG ® t(w)w12 = c ®6G(w)w23. Two cocycles w,w' are called cQhomologous

or equivalent provided there is a unitary v E R(G) such that w' = ° - SG(v*)w(v (9 v) and the set of equivalence classes is denoted by H2(G, T ). It is easily verified in the case of an abelian group that this is just the usual definition of 2-cohomology of G: one uses the identification between R(G) and £°°(G). Now let M be a von Neumann

algebra. A cocycle representation of 6 in M is a unitary W E M 0 R(G) such that

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f(W) = t®SG(W*)W12W13 lies in R(G)®R(G). Thus f(W) = I®w where w is a cocycle of G and we say that W is an w-representation. Any such W defines a coaction S of G on M via the formula S(x) = W(x ® I)W* and we shall say that S is implemented by W. In case M = B(n), we shall refer to W as an w-representation of G on H. It turns out that every coaction of G on B(1-1) is implemented by a cocycle representation of G on W. Now suppose that in addition to the operator W E M ® R(G) we have an action ay of G on

M such that a, 0 t(W) = (10 p(g)-1)W for all g E G: we say that W is an equivariant w-representation of G. Let us establish the existence of such representations by defining the regular w-representation on L2(G) via W,,, = WGow where WG E L°°(G) ® R(G) is defined by WG(g) = p(g). The operator WG is fundamental in the theory of coactions. Its well known properties, together with the easily verified fact that ow(= w21) also defines a cocycle dual to w, show that W,, defines an w-representation, equivariant with respect to the action Ad.\ of G on B(L2(G)). The theory of w-representations will be clarified below once we have constructed a C* algebra universal for w-representations of 6. This will immediately permit us to speak of the usual notions from representation theory such as commutants, equivalence, intertwining operators, etc. These notions can, however, be studied directly without reference to the C* algebra. For example if W is an w-representation on 1-l, then the commutant of W is just the von Neumann algebra consisting of all operators T on 7-l satisfying the commutation relation [W, T 0 I] = 0. The bicommutant is then the commutant of this algebra and in the case of the regular w-representation will be suggestively denoted by ir,,,(G)". The fact that W,, is equivariant means that G acts on the von Neumann algebra 7r,,,(G)" and this action is ergodic: the action of G is given by the restriction of Ad\. It turns out that the commutant of 7r,(G)" in the B(L2(G)) is just 7r,,,(G)": this explains why we referred to ow as the cocycle `dual' to w. A cocycle is self-dual (or symmetric) if and only if it is trivial. Interesting as these observations are, for the purposes of ergodic actions there is a far more significant way in which w and aw are related. One can show that the crossed product M A G by a coaction implemented by an w-representation is isomorphic to M 0 -7r,,,(G)" where the dual action of G is given by t®AdA. This result allows one to prove the following classification theorem for ergodic actions of full multiplicity.

Classification Theorem

All ergodic actions of full multiplicity arise from the

natural actions of G on

Two ergodic actions are equivalent if and only if the correponding cocycles are cohomologous, so that there is a natural bijection between H2(G,T) and ergodic actions of full multiplicity. To obtain the explicit link between ergodic actions and cocyles, we note that if a: G -> Aut(M) is an ergodic action of full multiplicity, then we may choose an unitary eigenmatrix U, for each 7r E G. Defining U by ®nEGUn as an element in MOR(G), we obtain a unitary U such that ay®t(U) = U(I®p(g)) for all g. The ergodicity of a then implies that W = U* is an equivariant cocycle representation of d in M, so that all the theory we have developed above becomes applicable.


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Many of the facts so far outlined can be understood much better after the introduction of the L' and C* algebras associated with the cocyles. In order to give clean definitions of these algebras, however, it is necessary to say a few words about normalisation of cocycles. Let us first recall that in the Abelian case a cocycle w is said to be normalised if w(!;, -1) = 1 for all E G; and that an w-representation UU is normalised provided that U{1 = U£-1 and U1 = I. In the non-Abelian case one must find substitutes for these conditions. Thus we say that a cocycle w is normalised provided that SG(el)w = SG(el) where el is the minimal central projection in R(G) corresponding to the trivial representation of G; and a cocycle representation is normalised provided that t®a(W) = W* and W(I (gel) = I®el, where a is the antipodal *-antiautomorphism p(g) --> p(g)-1 of R(G). We then have the following result.

Normalisation Theorem 1. Every cocycle is equivalent to a normalised cocycle. 2. A cocycle is normalised if and only if the corresponding regular cocycle representation is normalised, or equivalently if any (and hence all) of its cocycle representations is normalised. In the Abelian case, for a normalised cocycle, one has the additional relations w(C,1) =

1 = w(1, ) together with the alternating condition w(e, 77) =

-1). The non-

commutative analogues for a normalised cocycle are that w(el ® I) = el ® I, w(I ® el) = I ® e1 and ow = aw* (where the antipode is applied to both factors). Moreover w° will also be normalised if and only if v satisfies the normalisation conditions av = v* and vet = el. To prove all these assertions, one has to make constant use of the elementary but nevertheless fundamental properties of the `diagonal' element A = SG(el) Of R(G) 07Z(G):

1. 0(I®x)=A(ax(9 I). 2. 0(I®x)=0 if and only ifx=0. 3. A(R(G) ® R(G)) = 0(I ® R(G)). We are now in a position to define the algebras L'(G) and C,*,(G), where w is a normalised cocycle of G. We define V(G) to be R(G), its usual Banach space structure and involution but with multiplication perturbed by w. Thus (4) o zb, x) = (¢ (9 0, SG(x)w) for 0, 0 E R(G)* and x E R(G). This makes Ll (G) into a unital Banach *-algebra with its usual unit e, where e is the trivial character. (We recall that when w is the trivial cocycle, we just recover the usual Fourier algebra of G, which may be identified with an algebra of functions on G using the map ¢ '-r ¢ where 4)(g) = 4)(p(g)).) One checks that the formula irw(4)) = t ® 4)(W) establishes a one-one correspondence between w-representations of G and unital *-representations of L'(G). In the course of establishing this it is important to consider the special case of the regular w-representation. Here one starts by noting that G acts on the algebra L' (6) according

to the formula a9(4)) = 0 p(g-1) and that, since it coincides with the action by left translation on functions, this action is ergodic. The unique G-invariant state on L,1, (G)

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given by averaging is therefore just r(o) = l(ei). Its associated GNS-representation 7r, is equivariantly equivalent to the regular representation 1rw,. This dictionary between wrepresentations and unital *-representations of L'(6) can of course be extended to cover general equivariant representations, commutants, intertwining operators, and so forth. There are no surprises. We can then define C;,,(G) to be the enveloping C* algebra of LJ(G). The action of G on extends to an ergodic action of G on C ,,*(d) of full multiplicity. It is a nuclear C* algebra, since C;*,(C) >4 G = X is nuclear, and can be identified with the C* algebra generated by the spectral subspaces of ir,,,(C)". As we shall see its spectrum always has the form G/H as a G-space, so that C,*,(C) is simple if and only if 7r,,,(G)" is a factor. The justification for this lies in the observation of Magnus

Landstad that if a compact group acts on a primitive C* algebra, then every non-zero ideal in the algebra contains a non-zero invariant ideal. Landstad's proof of this was extracted from some ingenious computations of Olesen and Pedersen. Let us give here an alternative more conceptual proof. Any non-zero ideal corresponds to a closed subset of the primitive ideal space, missing the zero ideal. By a result of Glimm, the compact group acts continuously on the primitive ideal space so that the saturation of the closed set is automatically closed. It is also by definition invariant and misses the zero ideal. The corresponding ideal therefore provides the required non-zero invariant ideal. The question naturally arises as to when -7r,,,(C)" can be a factor. In the Abelian case, this question can be answered by introducing the bicharacter associated to a cocycle. Let us briefly indicate how this is effected. Firstly when G is Abelian, there is an isomorphism

of H'(6, T) into the group of alternating bicharacters on G given by [w] H /3 where rl) = 77)w(77, C). Thus /3 is separately multiplicative in each variable and satisfies the alternating condition /30',17) = 6(77, £). The bicharacter is said to be non-degenerate or totally skew if the usual condition holds: /3(C,77) = 1 for all 77 if and only if = 1. It is also usual to interpret this condition in terms of the homomorphism A of G into G defined by A(C) = One obtains the following criteria for 7r,,,(G)" to be a factor:

Factoriality Theorem

If w is a cocycle of G with corresponding bicharacter 13, the following conditions are equivalent: 1. 7r4,(C)" is a factor. 2. A is injective. 3. A has dense image.

4. /3 is non-degenerate. 5. C,*,(G) (or L'(G)) has a unique trace. 6. CW(C) (or L',(C)) has trivial centre. 7. CC,(G) is primitive. 8. Cw((j) is simple. These results are all fairly straightforward to prove in the Abelian case, and our task now will be to show that by appropriately defining bicharacters /3 and associated maps A everything carries over to the non-Abelian setting. As we have already hinted, however,


219

the non-commutativity of the group will nevertheless give rise to a fundamental and crucial point of departure from the Abelian prototype. Before giving the analogous definitions, let us pause to explain Landstad's method of establishing the last condition from the others. The customary method proceeded by observing that the subgroup A(G) of G always acts

by inner automorphisms on C,*,((j). Thus when A is dense, the action of G has to be approximately inner and therefore any ideal of C*((j) is automatically G-invariant. Since the crossed product is simple, the original algebra has to be simple. All we needed to make the last part of the argument work was some non-trivial G-invariant ideal in CW(G): this could have been produced equally easily, however, by assuming C,*(G) not to be simple

and then using Landstad's observation. This reasoning of course has the advantage of extending to the case of actions of non-commutative groups. We are now ready to introduce the formalism of bicharacters. This is closely related to Drinfeld's use of "triangular Hopf algebras" for studying the quantum Yang-Baxter equations although we have avoided using the language of Kac (or Hopf-von Neumann) algebras too much. (We shall give a fuller explanation for the link with the work of Drinfeld and his school in Section VII.) If w is a normalised cocycle of G, we define the bicharacter /3u, of w to be the unitary 8,,, = (ow*)w = w21w12 in R(G) ® R(G). Thus if W is an w-representation, then /3. = Wi2Wi3W12W13. We define a new comultiplication 5,,, on R(G) by 5,,,(x) = w*bG(x)w in order to reveal the bicharacter nature of /3.. This induces a new unital Banach *-algebra structure on the predual R(G)*, which we shall denote by A,,,(G) when it has this structure. The perturbed comultiplication now satisfies o,6,,,(x) = /36,,(x)/3*, so that the multiplication on A,,,(G) is no longer commutative, unlike the case of the usual Fourier algebra A(G); it is evident, however, that the multiplication on the space of class functions remains unaltered so that it forms a commutative subalgebra of A,,,(G). It will not in general be central. One can also show that any faithful set of

matrix coefficients will generate A,,,(G) as a *-algebra. We summarise the properties of the pair (,(3, 8) (which we shall simply write as (/3, 6)): 1. (Alternating) /3* = o/3, a,8 = ,3, t ® a(/3) = ,(3* = a ® c(,3)

2. (Normalised) (e1 ® I)[3 = el ® I, (I (9 el)/3 = 10 e1, b(el)/3 = 6(e1)(= A) 3. (Cocycle relations) (3126 0 t(/3) =023t 0 b(i3) q 4. (Bicharacter relations) 6 ® t(/3) = /313/323, c ® b(/3) = /3133/312 5. (Quantum Yang-Baxter equations) ,312,313/323 = /323/313/312

Two pairs (/31,61) and (/32,62) are said to be equivalent if there is a normalised unitary v E 1(G) such that ,31 = (v* (9 v*),32(v (9 v) and bl = Ad(v* ® v*) 62 Ad(v). This is consistent with the notion of equivalence on cocycles, although if G is not connected the notions need not be equivalent. In general the equivalence relation on pairs when pulled back to H2((j,T) via the map w i-4 (/3k,,5k,) induces an extra equivalence relation. To explain this weaker equivalence relation we must recall a definition due to Burnside. Let Aut,(G) be the group of automorphisms of G acting trivially on G. Such automorphisms can equivalently be described by the triviality of their action on class functions or by the

220


fact that they may be implemented by conjugation by a unitary in R(G). This normal subgroup of Aut(G) clearly contains the group Inn(G) of inner automorphisms. The quotient Aut°(G)/Inn(G) is trivial if G is connected and otherwise is a projective limit of finite solvable groups (by work of Sah). Now for an element y E Aut,(G), let it E R(G) be a (normalised) unitary such that Ad(u)p(g) = p(y(g)) for all g E G. Then the map w ra Ad(u ® u)w defines an action of Aut,(G)/Inn(G) on H2(G,T). The main result is that two pairs are equivalent if and only if the corresponding cocycles lie in the same orbit

of Aut°(G)/Inn(G) in H2(6,T). Let us now define the remaining ingredients required for the Factoriality Theorem in the non-commutative case. We shall say that /3 is non-degenerate provided that (x ®I)/3 =

x ® I implies that x is a scalar multiple of el. The role of the map A: G -+ G is played (dually) by a *-homomorphism A: A,,,(G) -1 R(G) defined by A(4) = (q5 0 i)/3. The theorem then holds without change (except that the image of A should now be dense in the ultraweak topology on R(G)). One has to use the new tensor product on representations of G defined by the perturbed comultiplication 5,,,; the formulas for decomposing tensor products remain the same as before, but the operator /3 must now be used to exhibit the isomorphism between V ® W and W ® V. The map A appears in the picture because the ultraweak closure of its image A(A,(G)" is a Kac subalgebra of R(G): it is always of the form v*R(H)v for some closed subgroup H of G and the primitive spectrum of CM(G) is then just G/H. The only tricky part of the theorem lies in finding a purely Hopf-algebraic proof of the assertion that if the restriction of every irreducible representation of R(G) to some Kac subalgebra never contains the trivial representation, then the subalgebra must be the whole of R(G). Of course the `classical' proof would use Frobenius Reciprocity.

We shall close this section by showing how the bicharacters can be classified for four examples, namely SU(2), SO(3), SU(2) x SU(2) and SU(3), although the techniques are applicable to other groups such as U(2), PU(3), SO(4), etc. In these cases, however, it is often easier to work with a covering group and use the theory of the multiplicity matrix.

Example 1: SU(2). We will prove that if a: SU(2) -+ Aut(M) is an ergodic action in which at least one copy of the two-dimensional irreducible representation it of SU(2) occurs, then M = L°°(SU(2)) as an SU(2)-algebra. This will show that SU(2) has no non-trivial bicharacters. So suppose that we can find non-zero elements a and b in M such that ag(a) = as - (3b ag(b) _ /3a + ab,

for all g - -7r(g) = C a

in SU(2). Let M = C b*

a /

. Then M is an eigenmatrix

for 7r, in that ag(M) = M7r(g) for all g E SU(2). Hence ag(MM*) = MM*, so that the entries of MM* are scalars by the ergodicity of a. Thus aa* + bb*, b*b + a*a and ab - ba are scalars. Taking traces and normalising so that aa* + bb* = 1, we find that MM* = I. Since M is finite, this forces the M to be unitary, i.e. M*M = I. Hence

221


a*a + bb* = 1 and a*b = ba*. Therefore a and b generate an Abelian *- algebra which is SU(2)-invariant. We can then either use Frobenius Reciprocity or our general eigenmatrix argument to deduce that the ultraweak closure of this algebra is isomorphic to L°°(SU(2)) as an SU(2)-algebra. Since every representation of SU(2) must therefore occur with full multiplicity in the algebra generated by a and b, it follows that a and b generate the whole of M and our claim is established.

Example 2: SO(3). We shall next establish that SO(3) has just two bicharacters, the trivial one and the one induced from the non-trivial bicharacter of the Klein 4-group. We shall do this by showing that any ergodic action of SO(3) of full multiplicity is necessarily on a Type I algebra; this is clearly equivalent to the statement on bicharacters. So suppose that a: SO(3) -+ Aut(M) is an ergodic action of full multiplicity. By choosing self-adjoint orthogonal bases ai, bi, ci for the three copies of the three-dimensional representation 7r1, we can produce a unitary eigenmatrix M =

a1

b1

cl

a2 a3

b2

c2

b3

C3

for 1r1. We next introduce the

non-commutative determinants µijk defined by ai

µijk =: det

bi

ci

aj bj

cj

ak

Ck

bk

_ (aibj - biaj)Ck + (bicj - cibj)ak + (ciaj - aic3)bk

= 6Tr(aibjck).

That these determinants are scalars follows from two easy facts: namely, if a, b, c and a', b', c' are two copies of 1r1 in an SO(3)-algebra, then on the one hand aa' + bb' + cc' lies in the fixed point algebra; while on the other hand, the wedge products a" = bc' - cb', b" = ca' - ac', c" = ab' - ba' give another copy of 7r1. It follows immediately from the properties of the trace, that the determinant µijk is invariant under cyclic permutations of the indices (which need not be distinct) and also satisfies µijk = -µkji. To establish further algebraic properties of these determinants, we introduce the matrices Ai and Bi j defined as follows

Ai =

0

-ci

ci

0

-ai

-bi

ai

0

bi

and Bij =

aiaj aibj

biaj

ciaj

bibj

cibj

aicj

bicj

cicj

We note that if X is any of these matrices, then a9(X) = lr1(g)*Xlr1(g) for g E SO(3) and hence MXM* has scalar entries. In particular the (i, j) entry of MAkM* is just µikj. Let us list some of the properties of the matrices Ai and Bij. By definition Ai is skew-adjoint

with trace zero. Moreover Bij = A;Aj + SijI, >i A, _ -21 and tr(A;Aj) = -2Sij. Now the skew property µ,jk = -µkji and cyclic property µijk = µjki of the determinant µijk show that it has the form µijk = Qeijk + ZPijk

222


where a is real, eijk is the alternating tensor and Pijk is (completely) symmetric and real. Note that pijk = Im(6ir(aibJck)). We claim that we may choose the bases 'ai, bi, ci so that pill = 0 = P211. Now in any subspace formed by two copies of x1, we can find a basis a, b, c satisfying Im(tr(abc) = 0, since this just amounts to finding a real root of a cubic with real coefficients. So we can choose al, b1, cl so that pill = 0. The wedge products of this basis with itself are then orthogonal to it. If they are zero, there is nothing to prove. Otherwise, we take the second basis a2, b2, c2 to be orthogonal to the first basis and its wedge product with itself. We now write out the matrices (pijk)i,k explicitly imposing all the conditions mentioned above. We find that they become 0

0

0

b

e

f

e

f

-b

,

0

b

f

b

y

z

f

z

e

,

-y

f

f z

-b

-y

-b -y -e-Z)

where eb = 0 and 1 - OZ = e2 + b2 + f 2 = b2 + f 2 + y2 + z2 = e2 + z2 + f 2 + ez + y2 + V.

These relations imply that z = -e and y = 0 and that either e or b is zero. We have just one relation remaining after these deductions, namely a2 + e2 + f2 +b2 = 1. Now if e = 0, then we find that

M(al ®I)M* =

1

0

0

0 0

A

2Qib

2oib

A*

al (&I

where A = -b2 + (if - O)2. (This follows by rewriting MBlk in the form (MB1kM*)M.) The self-adjointness of a1 implies that ub = a f = 0 so that M(a® ® I)M* = a2 ® I. Thus al, and hence all products from the first basis, is central. Similarly if b = 0, we see that all products of elements in the third basis a3, b3, c3 are central. Now an easy extension of the eigenmatrix duality criterion allows us to deduce that the (central) algebra generated by either of these sets of products is equivariantly isomorphic to L°°(SO(3)/N(T)), where N(T) is the normaliser of a maximal torus, i.e. the infinite dihedral group D*00 . Thus the ergodic action of SO(3) must be induced from an ergodic action of N(T), and hence the bicharacters of SO(3) are as claimed.

Example 3: SU(2) x SU(2). We will show that all bicharacters of SU(2) x SU(2) are induced from bicharacters of a maximal torus and therefore are classified by a parameter in the circle group. So suppose that we have a full multiplicity ergodic action of G =

SU(2) x SU(2) on M. Let the two factors of G be Gl and G2. Then we have full multiplicity ergodic actions of G1 and G2 on MG2 and MG, respectively. By the result of

Example 1, for each factor Gi we can find unitary eigenmatrices Mi = (

)

b*

a*

lying

in the fixed point algebra of the other factor. Each Mi is unique up to multiplication by an element of SU(2). By uniqueness it follows that Mi(a2 (9 I)M, = Pat - Qb2 where P \and

Q are scalar matrices. The unitarity of M2 then implies that the matrix

(-Q*

Q

f

is

unitary, so that P and Q are in particular normal and commute. Premultiplying M1 by


223

an element of SU(2) has the effect of conjugating P and Q by that same element. We may therefore assume that P and Q are in diagonal form and a simple manipulation shows that (P 0

their diagonal entries are complex conjugates so that P =

0) and Q = P

(q

0

0 q

)

with IPI2 + Igi2 = 1. The final step consists of checking what happens if we adjust the eigenmatrix M2 by premultiplying by some element

we just have to conjugate

P*

(-q

q p*

a

aa of SU(2). It turns out that

/

by this element. Thus we reach a final canonical

/ form by conjugating by an element so that q = 0. The Riedel rotation p E T is thus the only remaining invariant. Now consider the *-algebra generated by the products xy* where x, y E jai, bi } (i = 1, 2). It is Abelian because the canonical form shows that these products commute with both Ml and M2. As in the previous example, we may use an argument similar to that employed in the proof of the eigenmatrix criterion for duality to show that this algebra is equivariantly isomorphic to L°O(G/T) where T is a maximal torus in G. Thus the ergodic action must be induced from a maximal torus, as claimed.

Example 4: SU(3). We will sketch a proof that all bicharacters (or equivalently full multiplicity ergodic actions) of SU(3) are induced form those of a maximal torus. Actually

we will content ourselves with proving the slightly weaker statement that any algebra admitting a full multiplicity ergodic action of SU(3) has a non--trivial (and fairly large) centre. Our proof has two ingredients: namely the formalism of determinants, generalising the techniques of Example 2; and the existence of points of inflexion for plane cubic curves defined over C. We already implicitly considered a real cubic cubic curve when working with SO(3), namely the curve defined by µ11l = 0. To be more precise, this was the curve F(A) = tr(aabaca) = 0 where xa = Alxl + A2x2 + A3x3 in the real projective plane with coordinates (A,: A2: A3).

Let us start by introducing the appropriate versions of the matrices Ai and Bid. al bl ci Let M = a2 b2 c2 be a unitary eigenmatrix for the three-dimensional (identity) a3

b3

C3

representation -7r of SU(3), so that a9(M) = Mir(g) for all g E SU(3). The entries of M are of course no longer self-adjoint and M is now unique up to multplication by a unitary, rather than an orthogonal, matrix. We take the same definitions for the determinants µijk and matrices Ai as before. This time, however, the representation 7r is no longer self-conjugate and therefore the definition of the Bid's must be modified. It is still true, however, that the matrix obtained by applying the involution to the entries of M is a unitary eigenmatrix for the conjugate representation W. As a consequence the matrix Mt will also be unitary and satisfy a9(M2) = ir(g)tMt. We define Bi3 =

aia aib* aic!

bias

cia;

bibs bic

cib* cc*

224


We then find that ag(Ai) = ir(g)*Ai(7r(g)t)* and ag(Bij) = ir(g)*Bija(g), so that MAiMt and MBijM* are scalar matrices. (Note that analogous matrices to the Bij may be defined for unitary eigenmatrices corresponding to an (irreducible) representation 7r of any compact Lie group.) The corresponding scalar tensor I3ijkl = (MBijM*)k, can easily be identified with the (ir, 7r) component of the corresponding bicharacter and in particular satisfies the quantum Yang-Baxter equation. As before the (i, k) entry of the matrix MAjMt is the

determinant µijk = 6tr(aibjck). Furthermore we have the relations Bij = 6ijI EiBii = I, tr(Bij) = Sij, tr(B) = 1, and (M(ak ®I)M*)ij = Elf3kjitat. We also have the symmetry condition for the Yang-Baxter equation, which may be stated in terms of the operator A =

B12 B22 B32

B11 B21

B31

B13 B23

. Thus we require that A be a self-adjoint unitary.

B33

We shall only use the fact that the Bij's arise from the determinants µP9,.. These may be regarded as providing a `square root' of the Yang-Baxter solution. In general we can modify p by replacing M by AM where A E U(3). By definition the tensor µijk is cyclic, i.e. µijk = µjki, and hence it may be written uniquely as the sum of an alternating tensor a and a symmetric tensor a Pijk = 6ijk + aijk where aijk = µijk unless {i, j,k} = {1,2,3} when it is given by 2(µ123 + µ213); and where aijk = 2(p123 - 11213)eijk Let us consider the cubic curve in CP2 defined by µ11l = 0 as A varies. Thus if (al, 02, a3) is the first row of A, we are considering the curve whose homogeneous coordinates (01:02:03) satisfy F(a) = tr(a«b«c«) = 0, where x« =

alxl +02x2 +03x3. The equation of the cubic may be rewritten as F, µijkaiajak = 0 or yet again as E aijkaiajak = 0. Exactly as for SO(3) we may arrange for /,till = 0 = µ112; however, if we demand in addition that µl13 = 0, then this is equivalent to the cubic having a singular point which we cannot be sure happens. So let us suppose first that the contrary holds and our cubic is non-singular. Now a non-singular cubic is known to have exactly

nine points of inflection. Since U(3) acts transitively on the sphere bundle of CP2, we can always arrange for (0:1:0) to be the point of inflection and 03 = 0 to be the tangent there. Thus we can find A so that 0'112 = 6221 = 0'222 = 0. Thus F(a) has the form F(a) = 633303+u111a1 +3a332a3a2+3u31la3al +3a223a3a2+3a331a3a1 +a123a1a2a3 Let Ci = MA;Mt. These matrices have the form

C1=

(a 0 0

0

P9

p

f q

,

C2=

0 0

0

f

s

g

0 s, C3= Q

(p f g q

q

I I r

s

Let X = C2C2. Then det(X) = 0 and Tr(X) = Tr(X2) = 2. Thus X has to be a projection. Similarly Y = C2 C2 must be a projection. Let us suppose first that g = 0. Then the identity X2 = X forces IsI = 1 and I = f = 0. Similarly if f = 0, then y2 = Y 1

implies that Isl = 1 and 8 = g = 0. X and Y must then both be equal to

0 0

0 0 0

0 0 1

, so


225

that Ial2 + IpI2 = 1 = IpI2 + Igl2 and ap+ pq = 0. Using the identity >2 C;CL = 21, it is easy to see that q, r and a must vanish and we may arrange that p = s = 1. Exactly as

-1

in the calculations for SO(3), we can prove that M(x ® I)M* =

-1

0 0

0

1

0

0 0

(x 0 I)

when x is an element in the third basis. It follows that a3, b3 and c3 generate an Abelian C* algebra such that all elements of the form xy or xy* are central, where x and y are any elements of the basis. It is not too hard to see that the C* algebra generated by these products is of the form C(SU(3)/H) where H is the subgroup SU(3) fl (U(2) x U(1)) of SU(3). Thus at least one knows that the action is induced from a full multiplicity ergodic action of H = U(2). (To show that it is in fact induced from an action of a maximal torus, one can make use of the extra information that the two new bases a1 + ia2, b1 ± ib2, c1 +ice satisfy exactly the same conditions as the third basis a3, b3 and c3. One then verifies that the central C* algebra generated by all elements of the form xy or xy*, where x and y lie in the same basis, has the form C(SU(3)/T) for some maximal torus T in SU(3).) Now we shall assume that g and f are non-zero, but the cubic is still non-singular. From the equations X2 = X and Y2 = Y we deduce that t = 0, If I = IgI and Ig12 + I3I2 = 1. We may then arrange that s = sin(g), g = cos(g) # 0 and f = (cos(O) where I(I = 1. We

then look at the relations implied by the identity >t C,C; = 2I. We find in particular that p(f + g) + (f + g)s = 0. On the other hand, if f + g = 0, then it is easy to check that the cubic has a singular point, contrary to assumption. Thus f + g cannot vanish and

therefore p = -( sin(g). The remaining relations then show that a = q = r = 0 so that finally we find that the Ci's have the canonical form 0

0

-(se

0

0

(co

-(se

co

0

C1 =

C2 =

,

( _(:89

0

0

0

0

se Co

(ce

se

0

,

C3 =

0

ce

se (Co

0

0

0

1

where ce = cos(g) and se = sin(g). Exactly as in the first case we may show that all elements of the form xy*, where x and y are drawn from the third basis, must be central.

Indeed we find that for any element x from the third basis we have M(x 0 I)M* _

a

/3

0

0

0

1

a0

(x ®I) where a = -se - (ce and 3 = cese(1 - (). Thus we find that

M commutes with all elements of the form xy* as required. Consequently our claim is established when the cubic is non-singular. When the cubic has a singular point, we can assume that /L111 = µ112 = µ113 = 0 This is equivalent to choosing a basis a1, b1, c1 whose elements commute. The matrices C; then assume the following form:

(0 C1=

0

(0

0

0

af

0

g

b

,

C2=

a f

a

g

x c, C3= c

y

0 g b

f

b

c

y

y

z

226


Let X = C1Ci. Then det(X) = 0 and Tr(X) = Tr(X2) = 2. From this it follows that C1C1 =

0 0 0

0

0

1

0 j so that C1 has the form C1 =

0

1

0 0 0

0

0

se

co(

, where co =

-cep se 1. Here we have used the equations a = 6Tr(a2bjc2) and

cos(9), se = sin(O) and b = 6Tr(a3bic3) to make a priori homotheties of the second and third bases to ensure that a, b > 0. Now the condition Ei C2C, = 21 immediately implies that x = c = y = z = 0. Thus the matrices Ci can be put in the canonical form C1 =

0

0

0

se

COO

0

-C9

s

0

'

C2 = I

0

se

-Cep

sg

0

0

C9C

0

0

,

C3 =

0

CoC

-C9C

0

sB

0

se 0 0

By employing arguments similar to those used in the previous cases, one can then show all elements of the form xy*, with x and y coming from the first basis, must be central

in M. Indeed for such elements one has M(x 0 I)M* =

1

0

0

0 0

a

Q

-/j a

(x ®I) where

a = (2c0 - se and ,Q = cese(( + ), so that xy* 01 commutes with M. This completes our discussion for SU(3).

VI. An Invariance Principle for Subfactors One of the main problems in the theory of subfactors (initiated by Vaughan Jones) is the determination of the two towers of higher relative commutants associated with the extensions of a factor by a subfactor obtained by iterating the Jones basic construction. These towers have certain remarkable properties which have been put into evidence by the work of Adrian Ocneanu, Mihai Pimsner, and Sorin Popa. These properties have been formalised by Ocneanu starting from the basic observation that the structure of the higher relative commutant towers of the factor and subfactor can be understood independently of the Jones basic construction by considering the natural operations of induction, restriction and conjugation between the four classes of bimodules associated with the factor and subfactor. For subfactors of finite index and finite depth (i.e. for which the Bratteli diagrams for the inclusions in the relative commutant towers eventually become periodic), Ocneanu now appears to have proved a wonderful rigidity theorem. Roughly speaking this states that the inclusion of factors obtained by taking the limiting algebras in the relative commutant towers is isomorphic to the original inclusion of factors. In particular, this leads to a complete classification of subfactors of index less than four, that is for the special values 4cos2(-7r/n). His methods, however, appear not to extend to the case of infinite depth where very little is known at present, even for integer indices. Our aim here is to show how a very simple invariance principle permits the painless determination of higher relative commutants for various subfactors of infinite depth associated with compact groups.


227

Let Al be a subfactor of the hyperfinite III factor M of finite index [M: N] and suppose furthermore that the compact Lie group G acts on M leaving N invariant. Let M n = (M, e1, e2, .... en)" be the tower of factors obtained by iterating the Jones basic construction. From the fact that the Jones basic constructions are canonical it follows that the action of G extends to an action on Un>1 Mn leaving each Mn invariant and in fact fixing each basic projection en. We first want to find conditions under which the basic tower for the inclusion of the fixed point algebras NG C MG is obtained by taking fixed points of the tower for the inclusion N C M. In fact we have the following ìnvariance principle':

Lemma Suppose that M x G and N x G are both factors. Then MG and NG are both finite factors with [MG: NG] = [M: N] and the basic tower for the inclusion NG C MG is isomorphic to the one obtained by successively adjoining

the basic projections e1, e2, ...to MG. The neh factor in the tower (MG ,e1, ez, ...,en)

1

can be naturally identified with Mn .

Proof

Since the fixed point algebra is always a reduction of the crossed product, the first assertion is immediate. Since the crossed product N x G is a factor, we know by Connes'

2x2 matrix trick that H1(G,Mn(N)) is trivial and hence that every 7r E G admits a unitary eigenmatrix M(a) in N ® End(V,). In view of the ampliation result of Pimsner and Popa and their result on the uniqueness of the basic construction, it suffices to prove the result for the first basic construction, since the general result follows by induction (or iteration). We are thus reduced to proving that the factor MG is generated as a von Neumann algebra by e1 and MG. Now on the one hand we know that elements of the form aelb* with a and b in M are total in M1. Thus if E denotes the conditional expectation onto the fixed point algebra of G obtained by averaging over G, it follows that elements of the

form E(aelb*) are total in M. On the other hand we know that elements of the form xM;j(7r) with x E MG are total in M, since the M(ir)'s form a complete set of unitary eigenmatrices in N and hence M. Therefore taking a = xMzj(7r) and b = yMre(r) with x, y E MG, we have E(aelb*) = E(xMqj(7r)e1MTe(o)*y*) = xE(Mij(7r)Mre(o)*)eIy*

which is patently in the von Neumann algebra generated by MG and e1. This completes the proof.

Let us now give three examples where the invariance principle may be applied to compute higher relative commutants.

Example 1. Suppose that a: G -3 Aut(R) is a minimal action of G on R in the sense of Section IV. Let a: G - U(W) be any finite dimensional (possibly reducible) unitary

228


representation of G and define N = R and M = R ® End(W), endowed with the action a ® Adv. We now observe that the canonical Jones tower for the inclusion of finitedimensional factors C C End(W) is just C C End(W) C End(W)®2 C End(W)®3 C with the basic projection given by the formula

en = (dim(W))-1 E end ® 23e 1 t.7

where a are matrix units in the mth factor. Note that if we make G act on the infinite tensor product ®°°EEnd(W) by the action Ado, ® Ado-, (& Ado, ® Ado ® = ®Ad(o, ®o), then the projections en are all fixed by G. Consequently, if we now tensor this tower by R with the obvious tensor product action, we obtain the tower and basic projections for the

equivariant inclusion R C R ® End(W). We may now apply the lemma. Thus the index of N = R' in M = (R (9 End(W))a®Ado is dim(W)2 and the tower obtained by iterating the basic construction for the inclusion Ra C (R ® End(W))a®Ado is given by Ra C (R ®End(W))a®Ado C (R ® End(W) 0 End(W))a®Ado®Ado C (R ® End(W) 0 End(W) ® End(W))a®Ado®Ado®Ado C...

Bearing in mind that R n (RG)' = C, we see that the tower of relative commutants of the common subfactor Ra is given by End(W)Ada C (End(W) ® End(W))Ado®Ab0 C (End(W) 0 End(W) ® End(W))Ado®Ad-®Ad°

C .. In particular N has trivial relative commutant in M if and only if W is an irreducible representation of G. The computation of the tower of relative commutants of MG will be left as an exercise for the interested reader in this and the subsequent examples.

Example 2. Let us observe that it was unnecessary to assume that M and N were factors in the proof of the lemma, but merely that their crossed products by G were. So suppose now that a: G -> Aut(R) is an outer action of the finite group G and that H is a subgroup of G. Let N = R and M = R 0 e°°(G/H) with the obvious diagonal action of G and the unique G-invariant trace. It is not hard to see that if X is a finite set, then the canonical Jones tower for the inclusion of finite dimensional algebras C C £°°(X) (with the canonical traces) is as follows C C £°° C End(12) C End(12) 0 1°° C End(12)®2 C End(12)®2 ® @°° C


229

where $2 = £2(X), too = £°°(X) and the projections for the successive basic constructions are the obvious ones, namely alternately FXT E exy and E exx ® exx. Of course when X = G/H all these projections are G-invariant, and by the immediate extension of Jones' basic construction to inclusions of finite direct sums of type II, factors (again defined by inclusion matrices), one sees that the tower and basic projections for the inclusion of R in R®£°°(X) are just obtained by taking the tensor product of the corresponding objects for the inclusion of C in £°°(X ). Now observe that if M is a G-algebra then (M (& .f°°(G/H))G admits a natural identification with MH (compatible with the inclusion of MG). Now let

G be a finite group with subgroup H and let a: G -> Aut(R) be an outer (so minimal) action of G. Then the index of 7ZG in RH is [G: H], the tower obtained by iterating the basic construction is given by

RGCRH C (R ® End(12(G/H)))G

C (R ® End(!2(G/H)))H C (R ® End(t2(G/H))®2)G C (R ® End(22(G/H))®2)11

C .. and the tower of higher relative commutants is given by

C C (End(22(G/H)))G C (End(P2(G/H)))H C (End(f2(G/H))®2)G C (End(12(G/H))112)H

C .. This example was independently computed by Ocneanu using his bimodule picture of the higher relative commutants. He considered the equivalent inclusion of crossed products

N= R x H C M= R x G and observed that any choice of coset representatives for H in G provides an explicit Pimsner-Popa basis for M over N. Thus the tensor products M OA (M ®,v ... (&A( M become easy to compute. On the other hand, our method can be extended to compact groups (by considering the inclusion C C L°°(G/H)) where there is no really convenient basis available. We shall illustrate this in the next example by taking H to be the trivial subgroup.

Example 3. We shall now show how to compute the tower of higher relative commutants

for an inclusion MG C M where G is a compact group with a minimal action on the factor M. Indeed we may identify M with (M 0 L°°(G))G and reduce the computation to that of the (equivariant) inclusion C C L°°(G). The tower obtained by iterating the

23 0


basic construction here is the exact analogue of what we got in the case that G was finite. Thus we obtain C C L°°(G) C 13(L2(G)) C B(L2(G)) ® L°°(G) C 13(L2(G)) 013(L2(G)) c

It follows that the tower for the iterated basic constructions for the original inclusion MG C M is just

MG CM C M A GCM®13(L2(G))c..., and from this it is easy to work out the higher relative commutants. One finds that the higher relative commutants of MG are stably isomorphic to either R(G) or C; while those of M are stably isomorphic to either L°°(G) or C.

VII. Yang-Baxter Representations of S(oo) and Ergodic Actions In this final section, we shall explain in a rather vague and speculative way how our theory of bicharacters runs parallel to the theory of Quantum Groups, in the sense that it may be regarded as a bounded version of Drinfeld's theory. We will indicate in particular how the quantum Yang-Baxter equation for bicharacters - which seemed almost to arise almost by accident - in fact plays a crucial role, in that any `local' solution of this equation automatically gives rise to a bicharacter. Using the fact that any homomorphism of compact groups 8: G1 -+ G2 gives rise to a functorial map B*: H2(G1, T) -* H2(G2, T), we lose no generality by restricting our discussion to the case when G is the unitary group U(n) or even just SU(n) or PU(n). We find that a bicharacter /3 E R is uniquely determined by some (7r, 7r) -component B = /3(a, 7r) E End(V, 0 V,) if it is a faithful irreducible representation of G, where B just satisfies the quantum Yang-Baxter equation and certain extra unitarity conditions. In principle it is possible to use the method of Drinfeld, axiomatised by Lyubashenko, to recover the whole bicharacter from the matrix B. This approach is very close in spirit to the strategy of S. Doplicher and J. Roberts for recovering a compact group from its associated monoidal C* category. It should also be added that in their previous work on quantum field theory, these authors had already noticed the appearance of a version of the QYBE as a type of obstruction in their theory and that it assumed a formal similarity with the classification of ergodic actions of compact groups. In their approach, however, the equation assumed no physical significance, so was not investigated further. We propose an alternative method of recovering the bicharacter from the local solution of the QYBE. We shall associate with any such solution a finite 'Yang-Baxter' factor representation it of the infinite symmetric group S.. in the hyperfinite II, factor R. We then consider the subfactor N = 7r(S,,)" of M = R. The existence of such a representation of S,,. had already been noted by the Russian school, although they only considered its restriction to the finite subgroups S,, rather than looking at the àsymptotic' properties of the whole representation it. It turns out that N has trivial relative commutant in M and has infinite index. It seems very likely that the corresponding relative commutant towers have depth two and that for example AP fl M1 is isomorphic to R(PU(V.,)). On


231

the other hand Ocneanu's `paragroup' structure on the relative commutant towers should allow one to recover the Kac algebra structure on this algebra and its dual. For example it is already clear that one has a tensor product structure on the irreducible representations of N' n M1 - it is just the usual tensor product on N-bimodules. One would also expect the algebra A,(G) to appear in the picture. What is not so clear at present is whether the homomorphism A or for that matter the algebra au,(G)" will figure. Before entering into the realm of wild speculation, let us now explain what we can actually do at present. Let 0 E R® R be a bicharacter of G corresponding to some cocycle w of G. We fix it E G and set V = V,. Let B E End(V ® V) be the (ir, a) -component of 3. Then the Yang-Baxter equation for B is just B12B13B23 = B23B13B12. Now we note that by definition /3 = o w*w. Let Il E End(V ® V) be the (a, -7r)-component of co and let S E End(V ® V) be the usual flip, S(vl ® vz) = v2 ®v1. Evidently B = SW SQ. As is customary, we next introduce the associated R-matrix R = SB = St*SSt. Since it is conjugate to the flip, R is a unitary involution. Moreover the Yang-Baxter equation takes a particularly simple form in terms of the R-matrix: R12R23R12 = R23R12R23. Since R has square one, this condition simply says that (R12R23)3 = I. This shows that the matrices R12 and R23 satisfy the standard relations for the generators of the symmetric group S3, with R12 and R23 corresponding to the permutations (1,2) and (2,3) respectively. Thus we get a representation of S3 in End(V) ® End(V). Now let A be the infinite tensor product ®°n°--1End(V). We can define a representation aR of the infinite symmetric group in A by making the R-matrix `propagate'. Indeed if we define 7rR((n,n + 1)) = Rn,n+l for n > 1, then this extends uniquely to a representation of S", since the matrices an = Rn,n+1 satisfy the defining relations for Sam, namely an = 1 = (anon+l)3 for all n and anam = 0-,n0'n whenever In - ml > 1. Such representations of S. will be called Yang-Baxter representations. If we take the natural trace r on A then it r restricts to a trace on ir(S,,,). It follows that if Ìi and 72 are disjoint permute in Sam, then T(irR('Y1'Y2)) = T(irR(-y1))T(irR('y2)) To see this we note that we may conjugate

the permutations so that yl only moves 1,2,... , N and rye fixes this set. The traces do not change when we do this and the multiplicativity becomes manifest. It follows that T defines an extremal trace of S,,, so that NR = is a subfactor of M = 7r,(A)". Let us now show that 7rR and as are quasi-equivalent representations of S., that is the natural map -7rR(a) r-a irs(a) extends (uniquely) to an isomorphism of NR onto Ns. For this we only have to check that the corresponding traces agree. By multiplicativity (dim(V))-n+l for every n > 2. Now this reduces to showing that the conditions a ® e(/3) = 6* and 8G(e1)f3 = SG(el) imply that the matrix R satisfies Tr ® t(R) = I = , ® Tr(R). In other words Rii,ab = Sab = i

Rab,iz.

(*)

i

(Here if ei is a basis in V, we define the matrix Rab,ed of R by the formula R(ei 0 ej) =

23 2


Ekl Rki,Pjek (9 et.) On the other hand (using the summation convention)

T(irR((1, 2,..., n))) = T(R12R23 ... Rn-l,n) = Rii,abRba,cdRdc,ef "' Ryx,jj (dim(V))-n+1

= (dim(V))-1b+1'

by repeated application of the identity (*). Thus as far as representations go -7rR and irs are indistinguishable; however, if the isomorphism of NR onto NS extends to an automorphism of M onto M then it is not too hard to see that R has to have the form (v (9 v)S(v* ® v*)

for some unitary v of V. Thus R will be trivial. It is therefore of paramount importance to examine the precise position of JVR in M, i.e. to work out the towers of higher relative commutants. We would like as far as possible to immitate the methods used for xs in the previous sections. For the higher relative commutants this could be facilitated by exhibiting a dual action of the Kac algebra 9 = (1 (PU(V)), &) on M with NR as fixed point algebra.

We would therefore also need to find an equivariant copy of the dual 0 = A,,,(PU(V)) in M and this would need some extension of Weyl duality for the actions on V®n of S. (via 7rR) and C9' (via the tensor product induced by 5,,,). Using Connes' 2 x 2 matrix trick for producing eigenmatrices together with Ocneanu's announced result on the vanishing of 2-cohomology for centrally free cocycle actions of discrete Kac algebras, we have at least some moral foundation for this copy of 0 to exist. We could then try to construct the Jones tower for the inclusion AIR C M by an analogue of the invariance principle of the last section. It would be equally interesting to do the analogous computations for nonsymmetric solutions of the QYBE using some version of the invariance principle. These subfactors correspond to factor representations of the infinite braid group or equivalently the infinite Hecke algebra H,,(q) and the trace has in general to be replaced by a Powers state to ensure unitarity. Consequently the machinery of Kosaki would have to be used. Unfortunately although this method would seem likely to work for the Pimsner-Popa subfactors (when q is real), the really interesting values - namely when q is a root of unity - would apparently remain inaccessible by this method. Using a quite different method based on a priori bounds on the dimensions of the higher relative commutants, however, Hans Wenzl has succeeded in computing the higher relative commutant towers even in these exceptional cases. It remains a tantalising task to obtain his results by some form of the invariance principle. At present we have only checked that the relative commutant NR fl M is trivial. In order to explain this computation, we start by observing that the unitarity properties of /3(ir, w) lead to the following new identities for the R-matrix: E Rij,abRji,cd = aadsbc = E Rab,ijRcd,jiij

(**)

ij

Exactly as in Section V, we will show that the relative commutant is trivial by proving that the conditional expectation onto it is scalar-valued. This we accomplish essentially by


233

exhibiting the conditional expectation as a weak limit of averages over the finite symmetric groups, although we shall need an extra `doubling' trick to make the computation efficiently.

Let us start with a simple case by showing that if x is an arbitrary element of finite support in A and y is an element in the first copy of End(V), then limn-o,,'r(xAvs (y)) = T(x)T(y). This will show that the conditional expectation E,&j nM(y) of y is just the scalar T(y). Suppose then that the element x lies in the first m copies of End(V). Since the subgroup of permutations fixing 1 also commutes with y in the representation 7rR, we may replace Avs by a sum over convenient coset representatives of this subgroup. Let ryt = (1,2,..., t) = (1, 2)(2, 3) . . . (t - 1, t) so that -yt 1 (1 < t < n) constitutes a complete set of coset representatives. We therefore have n

lim r(xAvs (y)) = lim 1 E 7(x7rR(Yt 1)yxR('Yt))

noo

n-. n t=1

= hm

1

ET(XirR((t-1,t)...(12))yirR((1,2)...(t-1,t)))

t=1 n

= LIT, 1

T(x7rR((t -1, t) ... (12))ylrR((1, 2) ... (t - 1, t))) t=m+1

= T (x 1 R (-Ym+l )Y-R (Ym+1) )

= T(x(Rm,m+1 ... R12)y(R12 . . Rm,m+1))

On the other hand, m successive applications of formula (**) show that this last expression is equal to T(x)T(y), as claimed.

As soon as we try to extend the above reasoning to cover elements x and y with arbitrary finite supports, we encounter a difficulty characteristic of the Yang-Baxter representation (or the analogous representations of Wenzl of the Hecke algebras of Type A.

generated by the Jones basic projections ei). The operators 7rR((i, j)) with i and j far apart are not local, i.e. they are not given by the naive expressions Rij. Instead they must be expressed in terms of the `simple' transpositions (i, i + 1). This is because the natural order on the positive integers is crucial in the definition of 7rR, whereas it was irrelevant for 1rs. The reader will recall that exactly the same problem arose for V. Jones in writing down an explicit basis for the Hecke algebras. To overcome this difficulty we shall make use of a simplified version of the Murakami-Wenzl `cabling' trick, originally used to facilitate the computation of higher link invariants from solutions of the QYBE. In our context this means that for fixed k, we regard the set of positive integers as being a disjoint union of k copies of itself in a manner consistent with the natural order. Thus we have the k identification maps Bi(n) = k(n -1) + j where 1 < j < k, which have the effect of reconstructing the set of positive integers by ìnterleaving' k copies of itself. Clearly On B,n = Onm. Moreover, each map On induces a homomorphism 9n of Soo into itself. Thus for n = 2 the transposition (i, i + 1) is mapped to (2i -1, 2i + 1)(2i, 2i + 2) under 82, so that 62 just amounts to a `doubling' of the permutation representation; and if we double

23 4


twice, we just quadruple the representation, and so on. It will be important to know what happens to the Yang-Baxter representation if we compose it with the doubling operation. In fact the expression (13)(24) = (23)(12)(34)(23) immediately shows that 1R 82 = 7rR(2), where R(2) is the R-matrix R23R12R34R23 on (V ® V) ® (V ® V). That the doubled Rmatrix satisfies the same conditions (*) and (**) as the original matrix follows by noting that the corresponding matrix B is just the (-7r ® 7r, -7r ® ir)-component of SW ®S ,(/3) (in an

obvious sense). Furthermore if we successively double the R-matrix, these conditions are still satisfied. Now what we have shown so far is that if y comes from the first copy of End(V) in

A, then E,vknM(y) = T(y). Suppose now that y has finite support, so lies in the first 2k copies of End(V) in A. Then y lies in the first copy of (End(V))®2k = End(V®Zk) in A. So applying our simple result to the k-fold doubling R' = R(2') of R, we find that EJp nM(y) = r(y). Since by definition 7rR'(S ,) C 7rR(S,,.), we see that NR, n M D NRnM; since conditional expectations are functorial with respect to inclusions, we deduce

that Ep nM(y) = T(y). By continuity the same holds true for any y E M, and so NR does indeed have trivial relative commutant. We conclude with some general remarks. Let G be a compact Abelian group. Now the exponential maps C -+ C* and Fl -> T induce natural maps H2(G,C) -* H2(G,C*) and H2(G, R) -+ H2(G, T) which we shall view as providing `quantisation' maps from

an additive theory to a multiplicative theory. This terminology is not as fanciful as it might appear, since as we have seen the multiplicative theory essentially measures the `global' deformations of the Hopf algebras U(G) and R(G) (or their duals), while the additive theory is precisely the infinitesimal linearised version of these deformations. In other words it is defined by the equations of the tangent space at the identity of the multiplicative theory. These defining equations can be obtained by differentiating the cocycle relations for a one-parameter family of cocycles emanating from the trivial identity cocycle. All this makes sense if one replaces G by a non-commutative compact group and

takes the appropriate Hopf algebras for the unbounded or bounded theory. In the case of the classical groups, the unbounded theory amounts precisely to Drinfeld's theory of quantum groups. The corresponding linearised theory is just the classical Yang-Baxter equation. There remains the problem of quantising the classical solutions to obtain the quantum solutions: Drinfeld has shown that there is no (formal) obstruction to doing this. It should be apparent that the bounded unitary theory that we have developed shares many similarities with the theory of Drinfeld and his school; in particular our theory has contact with the work of Gurevich and Lyubashenko on the symmetric QYBE. Our approach through ergodic actions and subfactors - might possibly lead to some new insights. For example the so-called finite groups of central type, namely those admitting full multiplicity ergodic actions on matrix algebras, provide new solutions of the symmetric QYBE. (These groups are not particularly well understood, although they are known to be solvable, by the classification of finite simple groups.) Furthermore the methods of Section V might possibly extend to higher rank classical groups to give a classification of the solutions of the


235

symmetric QYBE. The solutions we are considering are similar to the flip and therefore, as Gurevich points out, have the right signature to be interesting from the point of view of supersymmetry. One fairly plausible conjecture about cocycles and bicharacters (or the corresponding full multiplicity ergodic actions) is that if they are sufficiently near to the trivial cocycle, then they should be induced from a maximal torus. As support for this let us verify this conjecture on the infinitesimal level, using an argument presumably familiar to Bellavin and Sklyanin but unrecorded in the literature. The classical version of the QYBE is just obtained by taking a C1 family B(t) such that B12(t)B13(t)B23(t) = B23(t)B13(t)B12(t) for all t and B(O) = I and defining b E End(V ®V) by b = B(0). Thus b is skew adjoint, b12 = -b21 and b satisfies the classical Yang-Baxter equation: [b12, b13] + [b12, b23] + [b13, b23] = 0.

Now we define a map r: End(V) -> End(V) via Tr(r(X)Y) = Tr(b(X ®Y)). Actually we shall only consider the restriction of r to the Lie algebra L of skew-adjoint matrices with its tracial inner product. r is then a skew-symmetric endomorphism of L such that [r(X),r(Y)] = r([r(X),Y] + [X,r(Y)]) for all X,Y E L. Thus if r were invertible, its inverse would define a derivation of L. On the other hand, the only compact Lie algebras with invertible derivations are Abelian. So we end by observing that the image of r is a Lie subalgebra Lo of L. Moreover the inverse of the restriction of r to Lo defines an invertible

derivation of Lo, so that Co must therefore be Abelian. It is then easy to see that b is induced from this subalgebra (in an obvious sense) and hence from a maximal torus, just as we claimed.

References 1. V. G. Drinfeld, Quantum Groups, Seminar on Supermanifolds 4, ed. D. Leites, Stockholm, 1987. 2. D. J. Gurevich, Quantum Yang-Baxter equation and a generalization of the formal Lie theory, Seminar on Supermanifolds 4, ed. D. Leites, Stockholm, 1987. 3. L. Hodgkin, The equivariant Kenneth theorem in K-theory, in "Topics in K-theory," Lect. Notes in Math., 496, Springer-Verlag, 1975. 4. R. Hoegh-Krohn, M. Landstad and E. Stormer, Compact ergodic groups of automorphisms, Ann. Math. 114 (1981), 75-86. 4. V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1-25. 5. G. G. Kasparov, K-functor and extensions of C* algebras, Iz. Akad. Nauk. SSSR. 44 (1980),571-636.

6. G. G. Kasparov, K-theory, group C* algebras and higher signatures, Conspectus, Chernogolovka, 1983. 7. V. V. Lyubashenko, Vectorsymmetries, Seminar on Supermanifolds 14, ed. D. Leites, Stockholm, 1987. 8. A. Ocneanu, "Actions of Discrete Amenable Groups on von Neumann algebras," Lect. Notes in Math., 1138, Springer-Verlag, 1985. 9. A. Ocneanu, Classification of subfactors of finite index and finite depth, in preparation.

23 6


10. M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. Ec. Norm. Sup. 19 (1986), 57-106. 11. M. Pimsner and S. Popa, Iterating the basic construction, preprint, Increst, 1985. 12. H. Upmeier, Toeplitz C* algebras on bounded symmetric domains, Ann. of Math. 119 (1984), 549-576. 13. H. Upmeier, Index theory for Toeplitz operators on bounded symmetric domains, Bull. A.M.S. 16 (1987), 109-112. 14. A. J. Wassermann, "Automorphic Actions of Compact Groups on Operator Algebras," Ph. D. Dissertation, University of Pennsylvania, 1981. 15. A. J. Wassermann, Algebres d'operateurs de Toeplitz sur les groupes unitaires, C. R. Acad. Sc. Paris 299 (1984), 871-874. 16. A. J. Wassermann, Ergodic Actions of Compact Groups on Operator Algebras: I: General Theory, submitted for publication; H. Classification of Full Multiplicity Ergodic Actions, to appear in Can. J. Math.; III: Classification for SU(2), to appear in Invent. Math. 17. A. J. Wassermann, Equivariant K-theory II: Hodgkin's spectral sequence in Kasparov's bivariant theory, preprint, Liverpool University, 1987. 18. A. J. Wassermann, Product type actions of compact Lie groups I, II, to appear in J. Operator Theory.

Derived link invariants and subfactors Hans Wenzl University of California, Berkeley

We construct from given representations of braid groups new representations. This can be used to construct new link invariants, new subfactors and new R - matrices from given subfactors, link invariants and R"matrices respectively.

As usual, let B,o be the infinite braid group, given by generators 0'1,0'2, ...

and

relations (B1) (132)

aiai+lai = ai+1aiai+1 for i = 1,2, ... and aiaj = alai if Ii - jI > 2.

Furthermore we denote the subgroup of B,,, which is generated by a,., ar+1 ... a,_2,

a,_1 by Br,,. We will just write Bf for Bi,f. Let now the braid a;f) be given by the picture (see (Birman) for details)

(i-1)f (i-1)f+1 (i-1)f+2

.f

if+1

ifr2

(i+.1)f

G+1)f+1

We note here that our theory also works if we vary a, f) by either conjugating it by other braids or by multiplying it by central elements of B(i_I)f+1,if or Bif+1,(i+l)f (see for instance (Murakami)). The results, however, will be essentially the same as for our special choice. It is easy to check by pictures that ai aif) induces an injective homomorphism from B into itsself. If p is an arbitrary representation of B,,, we obtain a new representation PU) by P(f)(ai) =

In the sequel, we always assume that p is an approximately finite dimensional (= AFD ) is finite dimensional for all n E N and we also assume representation of B,,, i.e. that these finite dimensional representations are semisimple.

Wenzl: Derived link invariants and subfactors

23 8

We are now going to decompose these representations. Let p be a minimal idempotent in p(CB f). Observe that conjugation by interchanges of and of+f for j = (i - 1) f + 1, (i - 1) f + 2, ..., if - 1. We use this to define inductively p1 = p

pi =

and (p(f)(oi-1))pi-1(p(f)(oi-i))-1

Furthermore let p(n) = PIP2 ... pn. Then we have (a) pi E p(CB(a-1)f+l,if), (b) pipj = pjpi for all i, j E N and (c) p(f)(B,,) commutes with p(') for all m > n.

We will moreover assume that p(m)p(CBnf)p(m) - p(n)p(CBnf)p(n)

(*)

for all m > n.

We obtain from this injections p(f)(CBnf)p(n) -4 p(f)(CB(n+l)f)p(n+l)

p(f)(CB(n+2)f)p(n+2) -..

...

Taking the inductive limit, we obtain an AFD representation p(f,P) of B.. As an application of these techniques, we can construct new solutions of the quantum Yang-Baxter equations (QYBE) from existing ones. We start with a classical Lie algebra acting on a vector space V in its standard representation. Using their quantization (following work of Kulish, Reshetikhin and Sklyanin) M. Jimbo obtained matrices R-, depending on 2 parameters q and x, which act on V 0 V. We use them to define matrices Ri on an infinite tensor product ®°° V by

Ri =

... 1 OR(q,0) 0 1 ...

,

where R (q, 0) acts on the i-th and (i + 1)-th factor of ®°° V. It follows directly from the QYBE that these matrices satisfy the braid relations. As soon as we know the structure of the algebra generated by these matrices, we obtain new representations of the braid groups by the method above. If we take in our definition of Ri the full R -matrix, we similarly obtain new solutions of the QYBE. They are solutions of the QYBE corresponding to higher representations of the given Lie algebra. In case of the classical Lie algebras of type A, it has already been observed by Jimbo that the algebras generated by the Ri's are quotients of the Hecke algebras of type A. For types B, C and D, one obtains quotients of a new algebra discovered in connection with Kauffman's link invariant for type B, C and D (see (Birman&Wenzl) and (Turaev)). The main ingredient for getting a link invariant via the approach of Jones is a Markov trace, i.e. a trace tr on CB,, such that for all n E N and ,3 E B,, we have

tr(on 1Q) = tr(on)tr(N) Recall that any such trace defines a link invariant Lt, via Markov's theorem by

Ltr(Q) = tr(o1)1-ntr(P),


239

where Q E B. and / is its closure (see (Birman)). We remark here that for representations of braid groups as R -matrices, one can define Markov traces easily by a deformation of the usual trace on ®°° Gl(V) by a special density matrix. This was first observed in this context in (Pimsner&Popa) for the Jones polynomial, for which the matrices come from the deformation of s12 in standard form. It was only noticed after generalizing Pimsner and Popa's matrices to the s1k case (which was also done independently by Pimsner and Popa) that their matrices are the same as the ones of Jimbo. Let now tr be a Markov trace factoring over p. Under mild assumptions on tr (essentially just faithfulness) the condition (*) for the construction of p(f,P) holds. Let tr(P) be the normalized trace which is obtained from tr by restriction to p(f,P)(B,,) and renormalization. It is easy to see that this restriction is well-defined independently of n. To get a link invariant we only need to prove that tr(P) induces a Markov trace on CB., via p(f,P). This can be shown for the Hecke algebra representations of the braid groups in the following way: If we set the parameter q of the Hecke algebras equal to e2"`/t, tr factors over special unitary representations 7r(k,i) of B,,. for k = 1, 2, ... 1 (see (Wenzl)). In this case, the Markov property of tr(P) can be shown easily using detailed results about the centralizer of the subfactor 7r(k,l)(( o2f+1, °2f+2

))Il

of.7r(k,l) (CB,,,)". The general case follows from

this immediately using elementary properties of analytic functions.

Theorem There exists for each Young diagram A with f boxes an AFD representation p(a) of B,,, which depends on a parameter q. Moreover, there exists a Markov trace tr(A) depending on q and an additional parameter l which factors over p(A) and which is faithful on its image

for almost all values of the parameters (i.e. except for a set of measure 0). In particular, we obtain for each Young diagram A as above a 2-variable link invariant. In case of the Young diagram with one box, the corresponding representation is the Hecke algebra representation of the braid group and the link invariant is the 2-variable polynomial of (Freyd et al.).

Remarks: 1. If q is not a root of unity, the structure of the images of CB, under these representations can be determined using the Littlewood-Richardson rule. This provides a method of proving that our newly constructed traces are distinct from the original one. 2. Similar results have been obtained independently by Murakami for another choice of

QLf)

3. As in the original case, we obtain for special values of the parameters unitary repesentations of B,, and examples of subfactors of the hyperfinite III factor. 4. V. Jones independently defined new link invariants using Jimbo's solutions of the QYBE corresponding to higher representations of s12. They can be deduced as special cases from the link invariants in our theorem similarly as his one variable polynomial can be deduced from the 2-variable polynomial in (Freyd et al.). Moreover, all link polynomials coming from solutions of QYBE corresponding to any finite dimensional irreducible representation of slk for any k > 2 can be obtained as special cases of the invariants in our

240


theorem. 5. Similar results can also be obtained for the Kauffman polynomial using the algebras in (Birman&Wenzl) (see also (Murakami)). More precisely, we can again define for each Young diagram a 2-variable link invariant where the invariant corresponding to the diagram with one box is the Kauffman polynomial. As a consequence of Turaev's result, all link invariants which can be derived from solutions of the QYBE corresponding to finite dimensional representations of orthogonal or symplectic groups can be obtained from specializations of these polynomials.

References Birman, J., Braids, links and mapping class groups, Ann. Math. Studies 82, Princeton Univ. Press, 1974. Birman, J., Wenzl, H., Braids, link polynomials and a new algebra, preprint, Columbia University. Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., Ocneanu, A., A new polynomial invariant of knots and links, Bull. AMS 12, No.2, April 1985, p. 239-246. Jimbo, M., Quantum R-matrix for the generalized Toda system, Com. Math. Phys. 102,4 (1986), 537-547.

Jones, V.F.R., A polynomial invariant for knots via von Neumann algebras, Bull. AMS 12, No. 1, Jan. 1985, p. 103-111. Kulish, P.P., Sklyanin, E.P., On the solutions of the Yang-Baxter equation, J. Soviet Math. 19 (1982).

Murakami, J., The r-parallel version of link invariants, preprint. Pimsner, M., Popa, S., Entropy and index for subfactors, Ann. scient. Ec. Norm. Sup. 4e serie, t. 19, (1986), p. 57-106. Turaev, V., The Yang-Baxter equation and invariants of links, preprint, Steklov Institute. Wenzl, H., Representations of Hecke algebras and subfactors, thesis, University of Pennsylvania (1985).

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