Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Assumption of the Initial Values

We begin by showing that u(t) !u

, weakly in

(), as t !0

. Of course, ku(t)k remains bounded

as t !0

, in virtue of [48], and the eigenfunctions

} are complete in J(). Writing

uðtÞu

; a



¼ uðtÞu

ðtÞ; a





ðtÞ

 u

ð0Þ; a



þ u

ð0Þu

; a



note that the first and third terms on the right-hand

side can be made small by choosing n large. The

second can be written as

ðtÞu

ð0Þ; a



; a



d

which will be small if t is small, in view of [34].

Thus, (u(t)  u

, a

) !0, as t !0

, which implies

the desired weak convergence.

The strong convergence u(t) !u

in L

() follows

from the weak convergence if lim sup

t !0

ku(t)k

k. The energy estimate [48] for the approxima-

tions implies this also.

To conclude that u(t) !u

strongly in J

(), it only

remains to be shown that lim sup

t !0

kru(t)k

kru

k. This readily follows from [29],providedthe

bounding function M(t)satisfiesM (t) !ru

,as

t !0

. The bounding functions provided by our basic

estimates [24], [26], and [27] all have this property.

We may conclude that u(t) !u

weakly in W

(),

provided u(t)

()

remains bounded as t !0

.To

see this, remember that Pu

and u

are of

essentially the same order. Thus the term u

(t)

the left-hand side of [34] can be accompanied by a

term u(t)

()

Finally, to prove that u(t) !u

strongly in W

(),

we need only show that lim sup

t !0

Pu(t)



Pu

, since P

and 

()

are equivalent

norms on J

() \ W

(). To this end, multiply [46]

by 

=dt and sum to get

Pu

þru



¼ u

ru

; Pu



ru

; Pu

ðÞ

 u

ru

þ u

ru

; Pu



Integrating this gives

Pu

ðtÞ

 Pu

ð0Þ

Pu

¼ 2 u

ru

; Pu

ðÞj

2 u

ru

; Pu

ðÞj

 2



ds  2

ru



þu

ru

; Pu



ds ½53

For the terms under the last integral we have

ru

; Pu



þ u

ru

; Pu





ru



þc ru

1=2

Pu

3=2

Therefore, [53] implies

Pu

ðtÞkk

 Pu

ð0Þkk

þ2 u

ru

; Pu

ðÞj

 2 u

ru

; Pu

ðÞj

þKt

uniformly in n,ast !0

, where K is a constant

depending on the estimates [32] and [34]. Letting

n !1, gives

Puðt Þ

 Puð0Þ

þ2 u ru; PuðÞj

 2 u ru; PuðÞj

þKt

Since u ru !u

ru

strongly in L

(), and

Pu !Pu

weakly in L

(), we get the desired

result. The continuous assumption of the initial values

in W

() also implies their continuous assumption in

the classical sense, and hence that u 2 C(

  [0, T)).

Conclusion

Years ago, mathematical questions concerning the

Navier–Stokes equations were usually considered in

the context of generalized or weak solutions, which was

a technical barrier to many in the scientific community.

Nowadays, realizing that solutions are at least locally

classical, fundamental questions such as that of global

regularity can be studied within the classical context. If

the estimate [29] is proved for classical solutions, with

T = 1, and without a restriction on the size of the data,

this particular matter will be settled.

This work has been supported by the Natural

Sciences and Engineering Research Council of Canada.

See also: Compressible Flows: Mathematical Theory;

Elliptic Differential Equations: Linear Theory;

Incompressible Euler Equations: Mathematical Theory;

Interfaces and Multicomponent Fluids; Leray–Schauder

Theory and Mapping Degree; Non-Newtonian Fluids;

Partial Differential Equations: Some Examples;

Stochastic Hydrodynamics; Turbulence Theories;

Wavelets: Application to Turbulence.

Further Reading

Beira˜o da Veiga H (1997) A new approach to the L

-regularity

theorem for linear stationary nonhomogeneous Stokes sys-

tems. Portugaliae Mathematica 54(Fasc. 3): 271–286.

Cattabriga L (1961) Su un probleme al contorno relativo al

sistema di equazioni di Stokes. Rendi conti del Seminario

Matematico della Universita` di Padova 31: 308–340.

378 Viscous Incompressible Fluids: Mathematical Theory

Heywood JG (1976) On uniqueness questions in the theory of

viscous flow. Acta Mathematica 136: 61–102.

Heywood JG (1980) The Navier–Stokes equations: on the

existence, regularity and decay of solutions. Indiana Univer-

sity Mathematics Journal 29: 639–681.

Heywood JG (1990) Open problems in the theory of the

Navier–Stokes equations for viscous incompressible flow. In:

Heywood JG, Masuda K, Rautmann R, and Solonnikov VA

(eds.) The Navier–Stokes Equations: Theory and Numerical

Methods, Lecture Notes in Mathematics, vol. 1431, pp. 1–22.

Berlin–Heidelberg: Springer Verlag.

Heywood JG (1994) Remarks concerning the possible global

regularity of solutions of the three-dimensional incompressible

Navier–Stokes equations. In: Galdi GP, Malek J, and Necas J

(eds.) Progress in Theoretical and Computational Fluid

Dynamics, Pitman Research Notes in Mathematics Series,

vol. 308, pp. 1–32. Essex: Longman Scientific and Technical.

Heywood JG (2001) On a conjecture concerning the Stokes

problem in nonsmooth domains. In: Neustupa J and Penel P

(eds.) Mathematical Fluid Mechanics: Recent Results and

Open Problems, Advances in Mathematical Fluid Mechanics

vol. 2, pp. 195–206. Basel: Birkhauser Verlag.

Heywood JG (2003) A curious phenomenon in a model problem,

suggestive of the hydrodynamic inertial range and the smallest

scale of motion. Journal of Mathematical Fluid Mechanics 5:

403–423.

Heywood JG and Rannacher R (1982) Finite element approxima-

tion of the nonstationary Navier–Stokes problem, Part I:

Regularity of solutions and second-order error estimates for

the spatial discretization. SIAM Journal on Numerical

Analysis 19: 275–311.

Heywood JG and Xie W (1997) Smooth solutions of the vector

Burgers equation in nonsmooth domains. Differential and

Integral Equations 10: 961–974.

Ladyzhenskaya OA (1969) The Mathematical Theory of Viscous

Incompressible Flow, 2nd edn. New York: Gordon and

Breach.

Prodi G (1962) Teoremi di tipo lacale per il sistema de Navier–

Stokes e stabilita` delle soluzione stazionarie. Rendi conti del

Seminario Matematico della Universita` di Padova 32:

374–397.

Solonnikov VA (1964) On general boundary-value problems for

elliptic systems in the sense of Douglis–Nirenberg. I. Izvestiya

Akademii Nauk SSSR, Sariya Matematicheskaya 28: 665–706.

Solonnikov VA (1966) On general boundary-value problems for

elliptic systems in the sense of Douglis–Nirenberg. II. Trudy

Matematiceskogo Instituta Imeni V.A. Steklova 92: 233–297.

Solonnikov VA and Sc

adilov VE (1973) On a boundary value

problem for a stationary system of Navier–Stokes equations.

Proceedings. Steklov Institute of Matematics 125: 186–199.

Xie W (1991) A sharp pointwise bound for functions with L

Laplacians and zero boundary values on arbitrary three-

dimensional domains. Indiana University Mathematics Journal

40: 1185–1192.

Xie W (1992) On a three-norm inequality for the Stokes operator in

nonsmooth domains. In: Heywood JG, Masuda K, Rautmann R,

and Solonnikov VA (eds.) The Navier–Stokes Equations II:

Theory and Numerical Methods, Lecture Notes in Mathematics,

vol. 1530, pp. 310–315. Berlin–Heidelberg: Springer Verlag.

Xie W (1997) Sharp Sobolev interpolation inequalities for the

Stokes operator. Differential and Integral Equations 10:

393–399.

von Neumann Algebras: Introduction, Modular Theory,

and Classification Theory

V S Sunder, The Institute of Mathematical Sciences,

Chennai, India

Introduction

von Neumann algebras, as they are called now,

first made their appearance under the name

‘‘rings of operators’’ in a series of seminal papers –

see Murray and von Neumann (1936, 1937, 1943)

and von Neumann (1936) – by F J Murray and J von

Neumann starting in 1936. Murray and von

Neumann (1936) specifically cite ‘‘attempts to

generalize the theory of unitary group representa-

tions’’ and ‘‘demands by various aspects of the

quantum-mechanical formalism’’ among the reasons

for the elucidation of this subject.

In fact, the simplest definition of a von Neumann

algebra is via unitary group representations:

a collection M of continuous linear operators on a

Hilbert space H (in order to avoid some potential

technical problems, we shall restrict ourselves to

separable Hilbert spaces throughout this article) is

a von Neumann algebra precisely when there is a

representation  of a group G as unitary operators

on H such that

M ¼fx 2 LðHÞ : xðtÞ¼ ðtÞx 8t 2 Gg

As above, we shall write L(H) for the collection of

all continuous linear operators on the Hilbert space H;

recall that a linear mapping x : H!H is continuous

precisely when there exists a positive constant K such

that kxkKkk8 2H. If the norm kxk of the

operator x is defined as the smallest constant K with

the above property, then the set L(H) acquires the

structure of a Banach space. In fact L(H

)isaBanach



-algebra with respect to the composition product, and

involution x 7!x



given by

hx; i¼h; x



i8;  2H

The first major result in the subject is the

remarkable ‘‘double commutant theorem,’’ which

establishes the equivalence of a purely algebraic

requirement to purely topological ones. We need

von Neumann Algebras: Introduction, Modular Theory, and Classification Theory 379

two bits of terminology to be able to state the

theorem.

First, define the commutant S

of a subset S 

L(H)by

¼fx

2 LðHÞ : x

x ¼ xx

8x 2 Sg

Second, the strong (resp., weak) operator topology is

the topology on L(H) of ‘‘pointwise strong (resp.,

weak) convergence’’: that is, x

!x precisely when

  xk!0 8 2H (resp., hx

  x, i!0 8,

 2H).

Theorem 1 The following conditions on a unital



-subalgebra M of L(H) are equivalent:

(i) M = M

(= (M

)

(ii) M is closed in the strong operator topology.

(iii) M is closed in the weak operator topology.

The conventional definition of a von Neumann

algebra is that it is a unital



-subalgebra of L(H)

which satisfies the equivalent conditions above. The

equivalence with our earlier ‘‘simplest definition’’ is

a consequence of the double commutant theorem

and the fact that any element of a von Neumann

algebra is a linear combination of four unitary

elements of the algebra: simply take G to be the

group of unitary operators in M

Another consequence of the double commutant

theorem is that von Neumann algebras are closed

under any ‘‘canonical construction.’’ For instance,

the uniqueness of the spectral measure E 7!P

(E)

associated to a normal operator x shows that if u is

unitary, then P

uxu



(E) = uP

(E)u



for all Borel sets E.

In particular, if x 2 M and u

2U(M

), then

(E)u

0

= P

0

(E) = P

(E), and hence, we may

conclude that P

(E) 2U(M

)

= (M

)

= M (we will

write U(N) (resp., P(N)) to denote the collection of

unitary (resp., projection) operators in any von

Neumann algebra N); that is, if a von Neumann

algebra contains a normal operator, it also contains

all the associated spectral projections. This fact,

together with the spectral theorem, has the conse-

quence that any von Neumann algebra M is the

closed linear span of P(M).

The analogy with unitary group representations is

fruitful. Suppose then that M = (G)

, for a unitary

representation of G. Then the last sentence of the

previous paragraph implies that (G)

= C precisely

when there exist no nontrivial -stable subspaces

(here and in the sequel, we identify C with its image

under the unique unital homomorphism of C into

L(H); and we reserve the symbol Z(M )todenotethe

center of M), that is, when  is irreducible. In general,

the -stable subspaces are precisely the ranges of

projection operators in M. The notion of unitary

equivalence of subrepresentations of  is seen to

translate to the equivalence defined on the set P(M)

of projections in M,wherebyp  q if and only if

there exists an operator u 2 M such that u



u = p and



= q. (Such a u is called a partial isometry, with

‘‘initial space’’ = range p,and‘‘finalspace’’= range q.)

This is the definition of what is known as the

‘‘Murray–von Neumann equivalence rel M’’ and is

denoted by 

. The following accompanying defini-

tion is natural: if p, q 2P(M ), say p 

q if there

exists p

2P(M)suchthatp 

 q –whereof

course e  f ,range(e)  range(f ).

The Murray–von Neumann

Classification of Factors

We start with a fact (whose proof is quite easy) and

a consequent fundamental definition.

Proposition 2 The following conditions on a von

Neumann algebra M are equivalent:

(i) for any p, q 2P(M), it is true that either p 

or q 

(ii) Z(M) = M \ M

= C.

The von Neumann algebra M is called a ‘‘factor’’ if

it satisfies the equivalent conditions above.

The alert reader would have noticed that if G is

a finite group, then (G)

is a factor precisely when

the representation  is ‘‘isotypical.’’ Thus, the

‘‘representation-theoretic fact,’’ that any unitary

representation is expressible as a direct sum of

isotypical subrepresentations, translates into the

‘‘von Neumann algebraic fact’’ that any -subalgebra

of L(H) is isomorphic, when H is finite dimensional,

to a direct sum of factors. In complete generality,

von Neumann (1949) showed that any von

Neumann algebra is expressible as a ‘‘direct integral

of factors.’’ We shall interpret this fact from

‘‘reduction theory’’ as the statement that all the

magic/mystery of von Neumann algebras is con-

tained in factors and hence restrict ourselves, for a

while, to the consideration of factors.

Murray and von Neumann initiated the study of a

general factor M via a qualitative as well as a

quantitative analysis of the relation 

on P(M).

First, call a p 2P(M) infinite if there exists a p

 p

such that p 

and p

6¼ p; otherwise, say p is

finite. They obtained an analog, called the ‘‘dimen-

sion function,’’ of the Haar measure, as follows.

Theorem 3

(i) With M as above, there exists a function

: P(M) ![0, 1] which satisfies the following

380 von Neumann Algebras: Introduction, Modular Theory, and Classification Theory

properties, and is determined up to a multiplicative

constant, by them:



p 

q ,D

(p)  D

(q)



p is finite if and only if D

(p) < 1



If {p

: n = 1, 2, ...} is any sequence of pairwise

orthogonal projections in P(M) and

p =

, then D

(p) =

)

(ii) M falls into exactly one of five possible cases,

depending on which of the following sets is the

range of some scaling of D



) {0, 1, 2, ..., n}



) {0, 1, 2, ..., 1}



(II

) [0, 1]



(II

) [0, 1]



(III ) {0, 1}

In words, we may say that a factor M is of:

1. type I (i.e., of type I

for some 1  n 1)

precisely when M contains a minimal projection,

2. type II (i.e., of type II

or II

) precisely when M

contains nonzero finite projections but no mini-

mal projections, and

3. type III precisely when M contains no nonzero

finite projections.

Examples L

(, ) may be regarded as a von

Neumann algebra acting on L

(, )asmulti-

plication operators; thus, if we set m

() = f ,

then m : f 7!m

defines an isomorphism of L

(, )

onto a commutative von Neumann subalgebra of

L(L

(, )). In fact, ‘‘up to multiplicity,’’ this is how

any commutative von Neumann algebra looks.

It is a simple exercise to prove that M L(H)isa

factor of type I

,1 n 1, if and only if there exist

Hilbert spaces H

and K and identifications H= H



K, M = {x  id

: x 2L(H

)} where dim H

= n;and

so M ﬃL(H

To discuss examples of the other types, it will be

convenient to use ‘‘crossed products’’ of von

Neumann algebras by ergodically acting groups of

automorphisms. We shall now digress with a

discussion of this generalization of the notion of a

semidirect product of groups.

If  : G !Aut(M) is an action of a countable group

G on M,whereM L(H) is a von Neumann

algebra, and

H= ‘

(G, H), there are representations

 : M !L(

H)and : G !U(L(

H)) defined by

ðxÞðÞðsÞ¼

1 ðxÞðsÞ;ðtÞðÞðsÞ¼ðt

1

sÞ

These representations satisfy the commutation rela-

tion (t)(x)(t

1

) = (

(x)), and the crossed pro-

duct Mo



G is the von Neumann subalgebra of L(

defined by

M = ((M) [ (G))

Let us restrict ourselves to the case of

M = L

(, ) acting on L

(, ). In this case, it is

true that any automorphism of M is of the form

f 7!f  T

1

, where T is a ‘‘nonsingular transforma-

tion of the measure space (, )’’ (= a bijection

which preserves the class of sets of  -measure 0). So,

an action of G on M is of the form 

(f ) = f  T

1

for some homomorphism t 7!T

from G to the

group of nonsingular transformations of (, ). We

have the following elegantly complete result from

Murray and von Neumann (1936).

Theorem 4 Let M, G,  be as in the last section,

and let

M = Mo



G. Assume the G-action is ‘‘free,’’

meaning that if t 6¼ 1 2 G, then ({! 2  :

(!) = !}) = 0. Then:

(i)

M is a factor if and only if G acts ergodically on

(, )–meaning that the only G-fixed functions

in M are the constants.

(ii) Assume that G acts ergodically. Then the type of

the factor

M is determined as follows:



M is of type I or II if and only if there exists

aG-invariant measure  which i s mutually

absolutely continuous with respect to ,

meaning (E) = 0 ,(E) = 0; (the ergodicity

assumption implies that such a  is necessa-

rily unique up to scaling by a positive

constant;)



M is of type I

precisely when the  as ab ove is

totally atomic, and  is the disjoint union of n

atoms for ;



M is of type II precisely when the  as above is

nonatomic;



M is of finite type – meaning that 1 is a finite

projection in

M – precisely when the  as

above is a finite measure;



M is of type III if and only if there exists no 

as above.

Thus, we get all the types of factors by this

construction; for instance, we may take:

)G = Z

acting on =Z

by translation, and

 =  = counting measure

)G = Z acting on =Z by translation, and

 =  = counting measure

(II

)G = Z acting on =T = {z 2 C : jzj= 1} by

powers of an aperiodic rotation, and  =  =

arclength measure

(II

)G = Q acting on =R by translations, and

 =  = Lebesgue measure

(III )G = ax þ b group acting in the obvious manner

on =R,  =  = Lebesgue measure.

Such crossed products of a commutative von

Neumann algebra by an ergodically acting countable

von Neumann Algebras: Introduction, Modular Theory, and Classification Theory 381

group were intensively studied by Krieger (1970,

1976). We shall simply refer to such factors as

‘‘Krieger factors.’’ The term ‘‘Krieger factor’’ is

actually used for factors obtained from a slightly

more general construction, with ergodic group

actions replaced by more general ergodic equiva-

lence relations. Since there is no difference in the

two notions at least in good (amenable) cases, we

will say no more about this.

Abstract von Neumann Algebras

So far, we have described matters as they were in

von Neumann’s time. To come to the modern era, it

is desirable to ‘‘free a von Neumann algebra from

the ambient Hilbert space’’ and to regard it as an

abstract object in its own right which can act on

different Hilbert spaces – for example, L

(, )is

an object worthy of study in its own right, without

reference to L

(, ).

The abstract viewpoint is furnished by a theorem

of Sakai (1983); let us define an abstract von

Neumann algebra to be an abstract C



-algebra

(this is a Banach algebra with an involution related

to the norm by the so-called C



-identity kxk



xk) M which admits a pre-dual M



– i.e., M is

isometrically isomorphic to the Banach dual space



)



. It turns out that a predual of such an abstract

von Neumann algebra is unique up to isometric

isomorphism. Consequently, an abstract von

Neumann algebra comes equipped with a canonical

‘‘weak



-topology,’’ usually called the ‘‘-weak topol-

ogy’’ on M. The natural morphisms in the category

of abstract von Neumann algebras are -homo-

morphisms which are continuous with respect to

-weak topologies on domain and range. It is

customary to call a linear map between abstract

von Neumann algebras ‘‘normal’’ if it is continuous

with respect to -weak topologies on domain and

range.

The equivalence of the ‘‘abstract’’ definition of

this section, with the ‘‘concrete’’ one given earlier

(which depends on an ambient Hilbert space), relies

on the following four facts:

1. L(H) is an abstract von Neumann algebra, with

the predual L(H)



being the so-called ‘‘trace class’’

of operators, equipped with the ‘‘trace norm.’’

2. A self-adjoint subalgebra of L(H) is closed in the

strong operator topology, and is hence a ‘‘con-

crete von Neumann algebra’’ precisely when it is

closed in the -weak topology on L(H).

3. If M is an abstract von Neumann algebra, and N

is a



-subalgebra of M which is closed in the

-weak topology of M, then N is also an abstract

von Neumann algebra, with one candidate for N



being M



(where N

= { 2 M



: n() =

0 8n 2 N}).

4. Any abstract von Neumann algebra (with separ-

able predual) is isomorphic (in the category of

abstract von Neumann algebras) to a (concrete)

von Neumann subalgebra of L(H) (for a separ-

able H).

With the abstract viewpoint available, we shall

look for modules over a von Neumann algebra M,

meaning pairs (H, ) where  : M !L(H) is a normal



-homomorphism.

A brief digression into the proof of fact (4)

above – which asserts the existence of faithful

M-modules – will be instructive and useful. Suppose

M is an abstract von Neumann algebra. A linear

functional  on M is called a normal state if:



(positivity) (x



x)  08x 2 M;



(normality)  : M !C is normal; and



(normalization) (1) = 1.

(Normal states on L

(, ) correspond to non-

negative probability measures on  which are

absolutely continuous with respect to .) It is true

that there exist plenty of normal states on M.

In fact, they linearly span M



. This implies that if



is separable, then there exist normal states

on M which are even ‘‘faithful’’ – meaning

(x



x) = 0 ,x = 0.

Fix a faithful normal state  on M. (Consistent

with our convention about separable H’s, we shall

only consider M’s with separable preduals.) The

well-known ‘‘Gelfand–Naimark–Segal’’ construction

then yields a faithful M-module which is usually

denoted by L

(M, ) – motivated by the fact that if

M = L

(, ), and (f ) =

f d, with  a probabil-

ity measure mutually absolutely continuous with

respect to , then L

(M, ) = L

(, ) with L

(, )

acting as multiplication operators. The construction

mimics this case: the assumptions on  ensure that

the equation

hx; yi¼ðy



xÞ

defines a positive-definite inner product on M; let

(M, ) be the Hilbert space completion of M.It

turns out that the operator of left-multiplication by

an element of M extends as a bounded operator to

(M, ), and it then follows easily that L

(M, )is

indeed a faithful M-module, thereby establishing

fact (4) above.

Since we wish to distinguish between elements of

the dense subspace M of L

(M, ) and the operators

of left-multiplication by members of M , let us write

x for an element of M when thought of as an

382 von Neumann Algebras: Introduction, Modular Theory, and Classification Theory

element of L

(M, ), and x for the operator of left-

multiplication by x; thus, for instance,

x = x

1, and

y =

xy, hx

1i= (x), etc.

Modular Theory

While type III factors were more or less an enigma

at the time of von Neumann, all that changed with

the advent of Connes. The first major result of this

‘‘type III era’’ is the celebrated ‘‘Tomita–Takesaki

theorem’’ (cf. Takesaki (1970)), which views the

adjoint mapping on M as an appropriate operator

on L

(M, ), and analyzes its polar decomposition.

Specifically, we have:

Theorem 5 If  is any faithful normal state on M,

consider the densely defined conjugate-linear opera-

tor given, with domain {

x: x 2 M}, by S

(0)



(

x) =



Then,

(i) there is a unique conjugate-linear operat or



(the ‘‘closure of S

(0)



’’) whose graph is

the closure of the graph of S

(0)



; if we write



= J





1=2



for the polar decomposition of the

conjugate-linear closed operator S



, then

(ii) J



is an antiunitary involution on L

(M, )(i.e.,

it is a conjugate -linear norm-preserving bijec-

tion of L

(M, ) onto itself which is its own

inverse);

(iii) 



is an injective positive self -adjoint operator

on L

(M, ) such that J



f (





f (

1



) for all

Borel functions f : R !R, and most crucially

(iv)



¼ M

and 



M

it



¼ M 8t 2 R

(Here and elsewhere, we shall identify x 2 M

with the operator of ‘‘left-multiplication by x’’

on L

(M, ).)

Thus, each faithful normal state  on M yields a

one-parameter group {



: t 2 R} of automorphisms

of M – referred to as the group of ‘‘modular

automorphisms’’ – given by





ðxÞ¼



x

it



The extent of dependence of the modular group on

the state is captured precisely by Connes’ Radon

Nikodym theorem (Connes 1973), which shows that

the modular groups associated to two different

faithful normal states are related by a ‘‘unitary

cocycle in M.’’ This has the consequence that if

 : Aut(M) !Out(M) = Aut(M)=Int(M) is the quoti-

ent mapping – where Int(M) denotes the normal

subgroup of inner automorphisms given by unitary

elements of M – then the one-parameter subgroup

{(



):t 2 R} of Out(M) is independent of .

Connes’ Classification and

Injective Factors

Given a factor M, Connes defined

SðMÞ¼

fspecð



Þ :  a faithful normal state on Mg

which is obviously an isomorphism invariant. He

then classified (Connes 1973) type III factors into a

continuum of factors:

Theorem 6 Let M be a factor. Then,

(i) 0 2 S(M) , M is of type III; and

(ii) if M is a type III factor, there are three mutually

exclusive and exhaustive possibilities:



(III

)S(M) = {0, 1}



(III



)S(M) = {0} [ 

, for some 0 <<1



(III

)S(M) = [0, 1)

Example 7 Consider the compact group =

n = 1

where G

is a finite cyclic group of order 

for

each n. Let  =

n = 1



, where 

is a probability

measure defined on the subsets of G

which assigns

positive mass to each singleton. Let G = 

n = 1

the dense subgroup of  consisting of finitely

nonzero sequences. It is not hard to see that each

translation T

, g 2 G (given by T

(!) = g þ !)isa

nonsingular transformation of the measure space

(, ). The density of G in  shows that this action

of G on L

(, ) is fixed-point-free and ergodic,

with the result that the crossed product L

(, )o G

is a factor.

Krieger showed that in the case of a Krieger factor

M = L

(, )o G, the invariant S(M) agrees with the

so-called ‘‘asymptotic ratio set’’ of the group G of

nonsingular transformations, which is computable

purely in terms of the Radon–Nikodym derivatives

d(  T

)=d. Using this ratio set description, it is

not hard to see that the Krieger factor M given by

the infinite product 



is a factor of type III



if 

= 2and

{0} = =(1 þ )

for all n;



is a factor of type III

if 

= 2 and 

{0} =

=(1 þ ), 

2nþ1

{0} = =(1 þ ), for all n, pro-

vided that {, } generates a dense multiplicative

subgroup of R



;



can be of type III

Among all factors, Connes identified one tractable

class – the so-called injective factors – which are

ubiquitous and amenable to classification. To start

von Neumann Algebras: Introduction, Modular Theory, and Classification Theory 383

with, he established the equivalence of several

(seemingly quite disparate) requirements on a von

Neumann algebra M L(H) – ranging from injec-

tivity (meaning the existence of a projection of norm

1 from L(H) onto M) to ‘‘approximate finite

dimensionality’’ (meaning M = ( [

)

for some

increasing sequence A

 A

A

 of

finite-dimensional -subalgebras). In the same

paper, Connes (1976) essentially finished the com-

plete classification of injective factors. Only the

injective III

factor withstood his onslaught; but

eventually even it had to surrender to the technical

virtuosity of Haagerup (1987) a few years later!

In the language we have developed thus far, the

classification of injective factors may be summarized

as follows:



Every injective factor is isomorphic to a Krieger

factor.



Up to isomorphism, there is a unique injective

factor of each type with the solitary exception of

III



Injective factors of type III

are classified (up to

isomorphism) by an invariant of an ergodic-

theoretic nature called the ‘‘flow of weights’’;

unfortunately, coming up with a crisp description

of this invariant, which is simultaneously acces-

sible to the nonexpert and is consistent with the

stipulated size of this survey, is beyond the scope

of this author.

The interested reader is invited to browse through

one of the books (Connes 1994, Sunder 1986,

Dixmier 1981) for further details; the third book is

the oldest (a classic but the language has changed a

bit since it was written), the second is more recent

(but quite sketchy in many places), and the first is

clearly the best choice (if one has the time to read it

carefully and digest it). Alternatively, the interested

reader might want to browse through the encyclo-

pediac treatments (Kadison and Ringrose) or

(Takesaki).

See also: Algebraic Approach to Quantum Field Theory;

Bicrossproduct Hopf Algebras and Noncommutative

Spacetime; Braided and Modular Tensor Categories;



-Algebras and Their Classification; Ergodic Theory;

Finite-Type Invariants; Hopf Algebra Structure of

Renormalizable Quantum Field Theory; Hopf Algebras

and q-Deformation Quantum Groups; The Jones

Polynomial; Knot Theory and Physics; Noncommutative

Geometry and the Standard Model; Noncommutative

Tori, Yang–Mills and String Theory; Positive Maps on



-Algebras; Quantum 3-Manifold Invariants; Quantum

Entropy; Tomita–Takesaki Modular Theory;

von Neumann Algebras: Subfactor Theory.

Further Reading

Connes A (1973) Une classification des facteurs de type III.

Annales Scientifiques de l’Ecole Normale Superieure 6:

133–252.

Connes A (1976) Classification of injective factors. Annals of

Mathematics 104: 73–115.

Connes A (1994) Non-commutative Geometry. San Diego:

Academic Press.

Dixmier J (1981) von Neumann Algebras. Amsterdam: North

Holland.

Haagerup U (1987) Connes’ bicentralizer problem and uniqueness

of the injective factor of type III

. Acta Mathematica 158:

95–148.

Kadison RV and Ringrose JR (1983/1986) Fundamentals of the

Theory of Operator Algebras, vol. I–IV. New York: Academic

Press.

Krieger W (1970) On the Araki–Woods asymptotic ratio set and

non-singular transformations of a measure space. In: Con-

tributions to Ergodic Theory and Probability, pp. 158–177.

Lecture Notes in Mathematics, vol. 160, Springer-Verlag.

Krieger W (1976) On ergodic flows and the isomorphism of

factors. Mathematische Annalen 223: 19–70.

Murray FJ and von Neumann J (1936) Rings of operators. Annals

of Mathematics 37: 116–229.

Murray FJ and von Neumann J (1937) On rings of operators, II.

Transactions of the American Mathematical Society 41:

208–248.

Murray FJ and von Neumann J (1943) On rings of operators, IV.

Annals of Mathematics 44: 716–808.

Sakai S (1983) C



-algebras and W



-algebras. Berlin–New York:

Springer-Verlag.

Sunder VS (1986) An Invitation to von Neumann Algebras.

New York: Springer-Verlag.

Takesaki M (1970) Tomita’s Theory of Modular Hilbert Algebras

and its Applications, Lecture Notes in Mathematics 128,

Berlin–Heidelberg–New York: Springer-Verlag.

Takesaki M (1979/2003) Theory of Operator Algebras,

vols. I–III. Heidelberg: Springer Verlag.

von Neumann J (1936) On rings of operators, III. Annals of

Mathematics 37: 111–115.

von Neumann J (1949) On rings of operators. Reduction theory.

Annals of Mathematics, 50: 401–485.

384 von Neumann Algebras: Introduction, Modular Theory, and Classification Theory

von Neumann Algebras: Subfactor Theory

Y Kawahigashi, University of Tokyo, Tokyo, Japan

Introduction

Subfactor theory was initiated by Jones (1983) and

has experienced rapid progress beyond the frame-

work of operator algebras. Here we start with a

basic introduction in this section.

A factor is a von Neumann algebra with a trivial

center. A von Neumann algebra M is an algebra of

bounded linear operators on a Hilbert space H,

which contains the identitiy ope rator and is closed

under the -operation and weak operator topology,

and its center is the intersection of M and its

commutant

¼fx 2 BðHÞjxy ¼ yx for all y 2 Mg

where B(H) denotes the set of all the bounded linear

operators on H. (We are mostly interested in

separable, infinite-dimensional Hilbert spaces. A

von Neumann algebra is automatically closed also

in the norm topolog y and thus it is also a C



-algebra.)

By definition, a factor M acts on a certain Hilbert

space H, but we also consider its action on another

Hilbert space K, that is, a -weakly continuous

homomorphism preserving the -operation from M

into B(K). A subfactor is a factor N which is

contained in another factor M and has the same

identity. A factor is classified into types

(n = 1, 2, 3, ...), I

,II

, and III. In most of

the interesting studies of subfactors, the two factors

are of both type II

or both type III. A factor M is

said to be of type II

if it is infinite dimensional

and has a finite trace tr : M !C. By definition, a

finite trace tr is a linear functional on M satisfying

tr(1) = 1, tr(xy) = tr(yx) for all x, y 2 M,and

tr(x



x)  0 for all x 2 M. When a factor M, not

isomorphic to C, acts on a separable Hilbert space, it

is of type III if and only if for any two nonzero

projections p, q 2 M, we have an operator v 2 M

with vv



= p and v



v = q. One obviously cannot have

a trace on such a factor. (See Takesaki (2002, 2003)

for a general theory on factors.)

Let M be a type II

factor acting on a Hilbert

space H. We then have the coupling constant of

Murray and von Neumann, which is denoted by

dim

H and belongs to (0, 1]. This measures the

relative dimension of H with respect to M.Note

that the factor M acts on M itself by the left

multiplication. We introduce an inner product on

M by (x, y) = tr(y



x) and denote the completion by

(M). Then M acts on this Hilbert space and we

have dim

(M) = 1.

Let N  M be a subfactor and suppose that both

N and M are of type II

. (We then simply say that

N  M is a type II

subfactor.) Suppose that M acts

on a Hilbert space H with dim

H < 1. Then we

define the Jones index of N in M by

½M : N¼

dim

2½1; 1

This number is independent of the choice of H,as

long as dim

H < 1, so we can take H = L

(M),

then we have [M : N] = dim

(M). The equality

[M : N] = 1 means M = N. The first major discovery

of Jones (1983) is that the value of the Jones index is

in the set

f4 cos

ð=mÞjm ¼ 3; 4; 5; ...g[½4; 1 ½1

and all the values in this set are indeed realized.

Suppose we have a II

factor M and an action of

an at most countable, discrete group G on M, that

is, a homomorphism  : G ! Aut(M), where Aut(M)

is the automorphism group of M. Then we have a

construction Mo



G, called the crossed product. If



is not an inner automorphism of M for any g 2 G

other than the identity element of G, then Mo



G is

also a type II

factor. (An automorphism  of M is

said to be inner if it is of the form (x) = uxu



for

some unitary operator u 2 M.) The index of a

subfactor M  Mo



G is the order of G, which can

be infinite. If we have a subgroup H of G, then we

obtain a subfactor Mo



H  Mo



G and its index is

given by the index [G : H] of the subgroup H. This

analogy to the index of a subgroup is the origin of

the terminology of the Jones index for a subfactor.

The Jones index is also analogous to the de gree of

an extension of a field. From the viewpoint of this

analogy, subfactor theory can be regarded as a

certain generalized analogue (or the ‘‘quantum’’

version) of the classical Galois theory for field

extensions. (The direct analog of the classical Galois

correspondence for subfactors was studied by

Nakamura–Takeda in the early days, and Izumi–

Longo–Popa gave the most genera l form.)

The tools Jones (1983) has introduced to study

subfactors are as follows. Let N  M be a subfactor of

type II

with finite Jones index. We consider the

actions of N, M on L

(M). The completion of N with

respect to the inner product given by the trace gives

(N), which is naturally regarded as a closed

subspace of L

(M). Let e

be the projection on

(M)ontoL

(N), which is called the Jones

von Neumann Algebras: Subfactor Theory 385

projection. We define M

to be the von Neumann

algebra generated by M and e

on L

(M). This is again

atypeII

factor and denoted by M

. This construction

is called the basic construction. We obtain [M

: M] =

[M : N]. Repeat the same procedure for M  M

acting on L

) this time. In this way, we have an

increasing sequence of type II

factors,

N  M  M

 M



which is called the Jones tower. We label the

corresponding Jones projections as e

= e

, e

= e

, .... We then have the following celebrated

Jones relations:

¼ e

; if jj  kj > 1

j1

½M : N

½2

Jones proved the above-mentioned restriction on

the possible values of the Jones index using these

relations. The realization of the index values below

4intheset[1] by Jones also relies on these

relations of the Jones projection. The basic con-

struction is also possible for the other direction.

That is, we can construct a subfactor N

 N so

that N  M is the basic construction of N

 N.

This is called the downward basic construction.

This N

is not unique, but is unique up to an inner

automorphism of N.

A subfactor N  M is said to be irreducible if the

relative commutant N

\ M is equal to C.Ifa

subfactor has Jones index less than 4, then it is

automatically irreducible. The original realization of

the Jones index values above 4 by Jones was through

reducible subfactors. Popa proved that all the values

above 4 are realiz ed with irreducible subfactors. A

factor is said to be hyperfinite if it has a dense

subalgebra given as the union of increasing sequence

of finite-dimensional -algebras. If M is a hy perfinite

type II

factor, then its subfactor is automatically

also hyperfinite by a deep theorem of Connes. For

hyperfinite, irreducible type II

subfactors, it is still

an open problem to determine all the possible values

of the Jones index.

For type II

factors N  M  P , the Jones index

[P : N] is equal to the product [P:M][M:N]. Thus for

the Jones tower, we have [M

:N] = [M:N]

kþ1

.In

general, if a subfactor N  M has a finite Jones

index, then the relative commutant N

\ M is

automatically finite dimensional. So, if we start

with a type II

subfactor N  M with finite Jones

index, we have an increasing sequence of finite-

dimensional algebras as follows:

\ M  N

\ M

 N

\ M

 N

\ M

 ½3

These finite-dimensional algebras are called higher

relative commutants of N  M. We draw the

Bratteli diagram for the higher relative commutants

as follows. Consider N

\ M

(with convention

1

= N, M

= M), then it is a finite-dimensional

-algebra; thus, it is of the form

(C), where

we have only finitely many direct summands. We

draw a dot for each summand. We similarly draw a

dot for each summand in

\ M

kþ1

Let  be the inclusion map from N

\ M

(C)toN

\ M

kþ1

(C)andp

the

identity of M

(C), which is a projection in N

kþ1

. We denote by 

the mul tiplicity of the

embedding map x 7!(x)p

from M

(C)toM

(C).

Then we draw 

edges from the jth dot for M

(C)

to the lth dot for M

(C). We repeat this procedure

for all k, and get a picture as in Figure 1, which is

called the Bratteli diagram of the higher relative

commutants of N  M .

It turns out that the edges connecting the kth and

(k þ 1)th steps of the Bratteli diagram consist of the

reflection of those connecting the (k  1)th and kth

steps, and a (possibly empty) new part. The ‘‘new’’

parts taken altogether in the above Bratteli diagram

constitute the principal graph of a subfactor N  M.

In the example of Figure 1, the principal graph is the

Dynkin diagram A

. In general, a principal graph

can be finite or infinite. If it is finite, we say that a

subfactor is of finite depth. If a subfactor has the

Jones index less than 4, it is automatically of

finite depth and the principal graph must be one of

the A–D–E Dynkin diagrams.

Pimsner and Popa (1986) obtained the character-

ization of the Jones index value in terms of the

Pimsner–Popa inequality for a conditional expec-

tation. This can be used as a definition of the index

for a subfactor of arbitrary type (and even for



-subalgebras). Kosaki obtained a definition of the

index for type III subfactors based on works of

Connes and Haagerup.

N ′ ∩ N

′ ∩ M

N ′ ∩ M

Figure 1 The Bratteli diagram of the higher relative commutants.

386 von Neumann Algebras: Subfactor Theory

Analytic Classification Theory

If M is a hyperfinite type II

factor, then it is unique

up to isomorphism. So any subfactor of such M is

isomorphic to M itself. We next consider the

classification problem of hyperfinite type II

subfac-

tors. We say that a subfactor N  M is isomorphic to

P  Q if we have an isomorphism of M onto Q

which maps N onto P. The following tower of finite-

dimensional algebras is a natural invariant for a type

subfactor N  M with finite Jones index and it is

called the standard invariant for N  M:

\ M  M

\ M

 M

\ M



\\ \

\ M  N

\ M

 N

\ M



½4

Each square

\ M

 M

\ M

kþ1

\ M

 N

\ M

kþ1

is a special combination of inclusions called a

commuting square. Under a fairly general condition

(called extremality of a subfactor, which automati-

cally holds for an irreducible subfactor), the above

sequence [4] is anti-isomorphic to the following

sequence of finite-dimensional algebras, including

the trace values:

\ M  N

\ M 

\\\

\ M

 N

\ M

 N

\ M



½5

where N

 N

 N  M is given by

repeated downward basic constructions. So, if the

closure of

\ M

) in the weak operator topology

is equal to M

for an appropriate choice of N

’s,

then the closure of

\ M) is also M, and the

isomorphism class of the subfactor N  M is recov-

ered from the standard invariant. In such a case, we

say that a subfactor has a generating property, and

then we have a complete classification of subfactors

in terms of the standard invariant. Popa (1994)

introduced a notion called strong amenability and

proved that a subfactor of type II

is strongly

amenable if and only if it has the generating property.

This is the fundamental result in the classification of

subfactors. A hyperfinite type II

subfactor with finite

Jones index and finite depth is automatically strongly

amenable, so such a subfactor is covere d by this

classification theorem of Popa. Popa also has a

similar result for subfactors of type III.

Constructions and Combinatorial

Classification

As mentioned in the above section, Jones con-

structed hyperfinite type II

subfactors for all

possible index values below 4. They have the

Dynkin diagrams A

as the principal graphs. It has

been an im portant problem to construct new

subfactors since then. Using the Hecke algebras,

Wenzl constructed a series of subfactors with index

values sin

(N=k)= sin

(=k) with N = 2, 3, 4, ...,

where the series for N = 2 coincide with the ones

constructed by Jones. Wenzl’s dimension estimate in

this work for the relative commutant has been an

important tool to study subfactors. It was soon

noticed that the subfactors of Jones and Wenzl are

related to the quantum groups U

(sl

) of Drinfel’d–

Jimbo, at the value of the deformation parameter q

at exp (i=k). Constructions of subfactors from other

quantum groups have been given by Wenzl.

Ocneanu (1988) has introduced a notion of a

paragroup and characterized the higher relative

commutants arising from a type II

subfactor with

finite Jones index and finite depth as a paragroup. If

we start with a subfactor N  No



G for a finite

group G, the corresponding paragroup contains

complete information on the group G and its

representations. In this sense, a paragroup is a

generalization of a (finite) group. The basic idea is

to regard the bimodule

(M)

as an analog of the

fundamental representation of a compact Lie group

and make finite relative tensor pro ducts



ðMÞ



Then one makes an irreducible decomposition and

studies various intertwiners arising from these

irreducible bimodules. In this way, we obtain a

certain combinatorial object and it is called a

paragroup. The vertices of the principal graphs

correspond to irreducible bimodules and the edges

correspond to basis vectors in the intertwiner spaces.

Note that by Popa’s theorem explained in the

previous section, a classification of subfactors of a

hyperfinite type II

factor with finite Jones index

and finite depth is reduced to one of paragroups.

Using this theory of paragroups, Ocne anu has

found that the Dynkin diagrams A

, D

, E

, and

are realized as principal graphs of subfactors, but

2nþ1

and E

are not. Furthermore, each of the

graphs A

and D

has unique realization and each

of E

, E

has two realizations. At the index value 4,

the principal graph must be one of the extended

Dynkin diagrams, A

(1)

2n1

, D

(1)

, E

(1)

, E

(1)

, E

(1)

, A

1,1

, and D

, and all are realized. (The last

three correspond to subfactors of infinite depth.)

von Neumann Algebras: Subfactor Theory 387