No . ( 198 ) I-9 CLASSIFICATION OF INJECTIVE FACTORS : THE WORK OF ALAIN CONNES

The fundamental results of A. Connes which determine a complete set of isomorphism classes for most injective factors are discussed in detail. After some introductory remarks which lay the foundation for the subsequent discussion, an historical survey of some of the principal lines of the investigation in the classification of factors is presented, culminating in the Connes-Takesaki structure theory of type III factors. After a discussion of injectivity for finite factors, the main result of the paper, the uniqueness of the injective II1 factor, is deduced, and the structure of II∞ and type III injective factors is then obtained as corollaries of the main result.


S. WRIGHT
II I factor and a complete classification of the factors of type I.As the years passed, there was vigorous development of the theory, but no results appeared that were as definitive as these early advances.However, in 1973 a new era dawned with the publication of the thesis [5] of Alain Connes.Basing his work on earlier results of Tomita and Takesaki [6], Connes initiated a program for the classifica- tion of factors that can be termed as nothing short of revolutionary.In [5], [7], [8], and [9], he obtained a classification of factors of type III and auto- morphisms of certain factors of type II I and II which culminated in the remarkable work of [I0], in which appear the first major advances beyond the classical theory in the classification of factors for the non-type I case.The purpose of these notes is to give a rather detailed discussion of the most important of these results.
We assume the reader has a familiarity with the theory of von Neumann algebras on the level of [II], say.We recall some basic facts that play an important role in the sequel.A factor is a von Neumann algebra with a trivial center, i.e., the only elements of the algebra which commute with every element are scalar multiples of the identity.We will be concerned primarily with von Neumann algebras of finite type, and we will use the tracial characterization of this (see [12], Section 111-8).Recall that a trace on the factor N is a positive linear functional of norm I, i.e., a state on N which satisfies (ab) (ba) V a b N (i.l) A factor is said to be finite if it has a trace.The trace on a finite factor is uniquely determined among the states by (i.I), and it is automatically faithful and ultraweakly continuous (Theorem 2.4.6 of [13]).These facts will be used frequently.
We will also employ the standard representation of a finite factor N Let be the (canonical) trace on N Then the representation in the Gelfand-Naimark-Segal construction ,H,} corresponding to is faithful and ultraweakly continuous, and the cyclic vector is separating for (N) (i.e., is cyclic for (N)' where called the standard representation of N and when we identify N with its image ,T(N) in H L2(N,T), we will say that N acts @ta.ndardly in H Tensor products of C*-algebras and yon Neumann algebras w..ll play an important role in what follows.Let Ai,i=l,2 be C*-algebras, which for convenience we will assume act on Hilbert spaces H i,i =1,2 Form the algebraic tensor product A 1 @ A 2 which can be viewed in a natural way as a *-subalgebra of B(H 1 (R) H 2) the algebra of bounded operators on H I (R) H 2 A seminorm p on A I A 2 is called a C*'-subcross seminorm if 2 (i) p(x*x) p(x) x A 1 A2, (ll) p(a I (R) a2) _< llalll lla211 a i A i i 1,2 is a C*-crossnorm if p is a norm satisfying (1), and (li) with -< replaced by If we take the supremum of all C*-subcross seminorms defined on A 1 8 A 2 we will obtain a C*-crossnorm v which by definition will be the lrgest C*-cross- norm definable on A 1 8 A 2 Another crossnorm can be defined on A 1 0 A 2 by recalling that AI e A2 acts on H I (R) H 2 and so each element of A I @ A 2 has a norm considered as a ohn-d ogerator These algebras will in general be distinct.If we further assume that A I and A 2 are von Neumann algebras, we may also take the closure of A 1 A 2 relative to the weak operator topology on B(H I (R) H 2) and thus obtain a yon Neumann subalgebra of B(H 1 (R) H 2)   the (spatial) W*-tensor product A 1 (R) A 2 of A 1 an__d A 2 An excellent source of information about tensor products is [14].
Suppose M. and N. are finite factors, i 1,2, and M i N i i 1,2 1 1 ('-denotes isomorphism).We claim that M (R) M2"-= N 1 (R) N 2 Let i:Mi Ni be isomorphisms, and let Ti,i denote the canonical traces on M. and N S. WRIGHT Now I (R) 2 is an isomorphism of M 1 @ M 2 onto N 1 (R) N 2 and by (1.2), (I 0"2 o (I (R) 2 i (R) 2 Since i (R) 2 and I (R) 2 are the canonical traces on N 1 (R) N 2 and M 1 (R) M 2 respectively, they are faithful and normal.
Thus by [15], Lemma I, (see Lemma 9.2 infra) I (R) 2 extends to an isomorphism Finally, we define the infinite tensor product of von Neumann algebras.Let We will preface our discussion of [I0] by a rather brief overview of the main lines of work which led up to it.Because of the limitations of space and time, we have concentrated on emphasizing only work which deals directly with the classification of factors, and have reluctantly suppressed discussion of many other interesting and important developments in the structure theory of von Neumann algebras.
Our story begins, as all stories about yon Neumann algebras do, with the work of Murray and von Neumann [I], [2], [3], and [4].The classification of factors was the motivating problem for this work, and all other progress on the problem was based on this pioneering effort.In [I], Murray and von Neumann introduced the fundamental notion of types (I, II I, II, and III), and in [i] and [2], they obtained a complete classification of the factors of.type I: if M is a factor of type I, then there exists a Hilbert space H such that M is isomorphic to B(H) the algebra of all bounded operators on H For our purposes, however, the most important advance that Murray and von Neumann made was their famous characterization of the hyperfinite II factor.Since this result will play a key role in the sequel, we will describe it in detail.
Let M be a yon Neumann algebra acting on a separable Hilbert space (from now 0mtil the end of this section, all von Neumann algebras will be assumed acting on a separable Hilbert space without explicit mention).M is said to be h_finite if there is a sequence {M n} of flnite-dimensional yon Neumann subalgebras of M totally ordered by inclusion, whose union is dense in M relative to the weak operator topology.If M is a finite factor, hyperfiniteness of M is equivalent to the following condition (condition C of Murray-von Neumann): for each finite subset {Xl,...,xn} of M and >0 there is a finite-dlmenslonal subfactor C of M and Vl,...,vn E C such that llx i -vill2 < , i l,...,n The following theorem will be used in a crucial way in our later discussion: 2. I. THEOREM.([4], Theorem XIV).Let M I and be fa,ztors of type II I.If M I and both satisfy condition C then M and M 2 are isomorphic.
We will denote by R the hyperfinite II 1 factor, unique up to isomorphism by Theorem 2. I.
Another important technique contained in the Murray-yon Neumann papers is the so-called group-measure space construction of factors.This technique was used to give the first examples of type II 1 and II(R) factors in [2], and the first example of a factor of type III in [3].We will not go into the details of this construction right now: it will emerge later as a special case of the Connes- Takesaki crossed product construction.
The next major advance in the structure theory of von Neumann algebras occured in 1949 with the publication of von Neumann's reduction theory [16].
This paper introduced and used the concept of direct integral of Hilbert spaces to decompose an arbitrary separably acting yon Nemann algebra into a direct integral of factors, and thereby reduced the study of    III factor in 3 ], but did not give another nonlsomorphic one.In 1956, Pukansky [17] constructed a pair of nonisomorphic type III factors using refinements of the group-measure space construction, and there the matter lay until 1963 when J.T.
Schwartz [18] found a new invariant, his well-known Property _P: a yon Neumann  [19].He showed that all hyperflnlte factors have property P and conjectured the converse.This converse was one of the many things established by Connes in [i0].
Since it had been a fairly difficult task to obtain such a small number of examples of nonisomorphic factors, there gradually emerged a hope that a somewhat complete classification of all factors just might be possible.In 1967, that hope began to fade when R.T. Powers [20] constructed the first uncountable family of nonisoorphic factors.Let   [0,I] and let M 2 denote the 2 x 2 complex matrices.Let % denote the state defined on defined by

+--+
The Powers factor RX is defined as the infinite tensor product of countably many copies of M 2 relative to the product state (R) n where each is X n n We note that R 1 is the hyperfinite II 1 factor.R is a hyperflnite factor of type III for each X (0,I) and Powers showed that R X is not isoorphlc to I RX if I and 2 are distinct elements of (0, I).We will see this class of 2 factors again.After Powers' examples appeared, the flood gates opened, and nonisomorphlc factors soon poured out.In 1969, McDuff  [21] exhibited an uncountable family of II factors, and in 1970, Sakai  [22], [23] found uncountable families of II and nonhyperfinite type III factors.This work dashed forever the hope of a complete classi=fication of all factors.Attention began to focus on particular classes of factors for which there did seem to be a possibility of a satisfactory classification with results that were to shortly revolutionize the theory.
Let {M The resulting factor is an example of what Araki and n x n Woods [24] dubbed an ITPFI factor (for "infinite tensor product of finite type I's").
In this paper, which appeared in 1968 and was motivated in part by Powers' work, these authors introduced an important new invariant for the ITPFI factors, the asymtotic ratio set, and succeeded in obtaining an almost complete classification of this class of factors.Both for its historical importance and for its fore- shadowing of even bigger things to come, we will briefly describe the asymptotic ratio set and the Araki-Woods classification.
Let M be a factor.The asymptotic rat___i_0_o se___t r(M) o__f M is the subset of [0, (R)) defined as follows: where RX is the Powers factor defined previously.This is not the definition of r(M) first given by Araki and Woods for the ITPFI factors.The first definition ([24], Definition 3.2) was expressed in terms of a complicated limiting procedure involving the ratios of the eigen-values of the density matrices of the states occurring in the infinite tensor product decomposition of the ITPFI factors, and Araki and Woods later showed that the definition given above was equivalent to the original one ([24], proof of Theorem 5.9).If M is an ITPFI factor, they deduced that r=(M) must have one of the following forms: Araki and Woods showed that S corresponds to only one isomorphism class of ITPFI factors, and it is of type III.We will denote this factor by R (M (R) M2)* equal i (R) M_?In 1967, M. romita [25], [26] answered this question affirmatively by a new and original analysis of the spatial relationship between avon Neumann algebra and its commutant.The exposition of [25] and [26] was somewhat obscure, however, and in 1970, M. Takesaki published his seminal monograph [6] which explained and extended Tomita's earlier work.Let M be a von Neumann algebra with a vector which is both cyclic and separating for M. Takesaki associated a closed, densely defined, self-adjoint operator A with M, the modular o er, which has two very useful properties The first is that {A it t (-,(R))} forms a one-parameter unitary group for which A-itMit M the modular @utomorphism roup of M and the second is that A induces a conjugate-linear, involutive isometry J of the underlying Hilbert space for which JMJ M .
This shows in particular that M and M are anti-isomorphic, and is the key to Tomita's proof of the commutation theorem for tensor products.In actuality, the existence of a cyclic and separating vector is not necessary for the definition of the modular operator, and in fact if is any faithful, normal, positive linear functional on M then the modular operator A and the modular auto- morphism group {At} corresponding to can be constructed in H relative to (M) where {n,H} is the representation of M arising from the GNS construction induced by The construction of the modular operator and the verification of its main properties are quite technical, and for that reason we will not go into the details.We instead refer the reader to Rieffel and van Daele's excellent development of the Tomita-Takesaki theory [27], and, of course, also to the original memoir [6].
The stage was now set, and in 1973 Connes' thesis [5] appeared.This work contained a classification scheme for factors of type III which was to have a profound influence on all subsequent work in this area.Taking his cue from the Araki-Woods classification scheme and the Tomita-Takesaki theory, Connes introduced the modular spectrum as an isomorphism invariant for type III von Neumann algebras.
Let M be a separable acting type III von Neumann algebra.The modular spectrum S(M) of M is the intersection of the Arveson spectra (see [29]) of the modular operators A corresponding to all faithful, normal, positive linear functionals on M Connes and van Daele proved the remarkable fact that S(M)\{0} forms a subgroup of (0,-) and that S(M) is a closed set which is an isomorphism invariant of M Connes then divided the type III von Neumann algebras into subtypes as follows: M is said to be This is a direct generalization of the Araki-Woods classification:for a type III ITPFI factor M, r(M) S(M) and so M is of type III I0 S. WRIGHT Connes then proceeded to prove a fundamental structure theorem for the factors of type lllk, k [0,i) in terms of discrete crossed products of von Neumann algebras.The case of type III was not treated, but Connes was not alone in this work.In the same year as Connes' thesis appeared, Takesaki also offered [28], which solved the III case by the introduction of continuous crossed products, and which also developed a duality theory for crossed products that was to be very influential.We now proceed to describe the structure theorems of Connes and Takesaki.
Let G be a locally compact abelian group, M a von Neumann algebra.
the group of all *-automorphisms of M such that for each x M the mapping g a (x) is *-strongly continuous.If G R (as will be the case in the sequel), Th__e crosse__d product W*(M,) of M b_ is the von Neumann subalgebra of B(L2(G;H)) generated by { (x) x M} and {k(g the crossed product W*(M,@) of M by 8 is the crossed product of M by the 8 n discrete action nn--0, +I, +2,...We will often refer to W*(M,8) as a discrete crossed product or a discrete decomposition.
The oldest, and in many ways the most important, example of a crossed product is the classical group-measure space construction of Murray and von Neumann.Let (X,) be a o-finite measure space with positive measure and call a bijective mapping T of X onto X an automorphism of X if T and T -I are Ii measurable and T maps sets ofmeasure zero to sets of g-measure zero.An automorphism T of X induces an algebra automorphism s T of L (X,) in the natural way Let H L2(X,) denote the Hilhert space of allsquare integrable functions on X L(X,) acts by pointwise multiplication on H and thereby forms a maximal abelian von Neumann subalgebra of B(H) and s T is a *-automorphism of this von Neumann algebra.The von Neumann algebra given by the group-measure space construction is simply the discrete crossed product W*(L (X,),T) W*(L (X,),T) is hyperfinite, and if T is ergodic, it is a factor.Two automorphisms T and T 2 of X are weakly equivalent if there exists an n (UT 2nu automorphism U of X such that {T I ( n Z} for gallmost all x In a famous paper which generalized the work of Araki and Woods, Krieger [30] proved that if T and T 2 are ergodic automorphisms of X then W*(L(X,),TI is isomorphic to W*(L(X,),T2 if and only if T and T 2 are weakly equivalent.For this reason, Connes calls a discrete crossed product of an abelian von Neumann algebra by an ergodic automorphism of the algebra a Krieer factor, and so therefore will we. (Incidentally, Connes cites the work of Krieger (along with Araki-Woods [24]) in the introduction to [5] as being one of the primary motivations for developing his classification scheme for type III factors).
We are now in a position to state what may appropriately be called the first, second, and third fundamental structure theorems for type III factors.
There exists a factor N of type II, an automorphism 8 of N and a faithful normal, semifinite trace of N such that Furthermore, any pair (N,8) satisfying (i) gives rise to a crossed product factor of type II, and any two such pairs (NI,8 I) (N2,82) give isomorphic factors if and only if they correspond to the same k and there is an isomorphism , of N 1 onto N 2 such that p(n 81-I) p(82) ,where p is the canonical quotient map of Aut(N2) onto the quotient of Aut(N2) by its subgroup of inner automorphisms.
2.4.THEOREM.(Connes, [5]).Let M be a factor of type III 0. There exists a von Neumann algebra N of type II with a diffuse center, an automorphism 8 of N which is ergodic on the center of N and a faithful, normal, semifinite trace on N such that N (i) for some k 0 < 1 (8(x)) k0T(x for all positive elements x of (li) M is isomorphic to W*(N,8).
Any pair (N,8) satisfying the above conditions gives rise to a factor of type

III o
These theorems in principle reduce the study of factors of type III to the study of von Neumann algebras of type II and their automorphisms, and Connes lost no time in beginning such a study.
If M is a II.factor acting on a separable Hilbert space H a theorem of Murray and von Neumann [4], Theorem IX) allows one to write M as the tensor product of a II 1 factor N and B(H) Thus the study of automorphisms of II.
factors can be effectively reduced to the study of automorphisms of II factors.
Now the hyperfinite II factor is in many ways the simplest of all the II 1 factors, and hence any program which aspires to a classification of automorphisms of II 1 factors must first of all handle the case of the hyperfinite II 1 factor R In [9], this is precisely what Connes did.In this deep and penetrat- ing study, he determined all outer conjugacy classes of automorphisms of R and in particular showed that the quotient of Aut(R) by its subgroup of inner auto- morphisms is a simple group with only countably many conjugacy classes.By our previous comments, all automorphisms of the II factor R 0,I be determined.

NB (H) can now
The factor R0,1 is also of interest for another reason.An old question of Murray-vonNeumann asked whether all hyperfinite II factors are isomorphic to R 0 ?This question arose naturally from their work on the hyperfinite II ,I factor, and for a long time was viewed as one of the most important open questions in the theory of von Neumann algebras.In [i0] it received an affirmative answer.
The key to this answer lies in the concept of injectivity, an idea introduced by Tomiyama in [31] and exploited by him and others in the study of tensor products of yon Neumann algebras (the terminology is due to Effros and Lance [14]).The great achievement of [10] was to identify injectivity and hyperfiniteness for separably acting factors, first in the II 1 case, then for the II case, and finally for the type III case using the fundamental structure Theorems 2.2, 2.3, and 2.4.It is to a detailed discussion of these ideas that we now turn.
A few words are in order concerning the organization of the remainder of the paper.The main theorem occurs in Section 8, and asserts that all injective II factors are isomorphic.In Sectio 3, we define injectivity for von Neumann algebras and give a tracial characterization of injectivity for finite factors.
Sections 4 and 5 are concerned with establishing a certain type of finite dimensional approximation property for standardly acting, injective, finite factors which plays a central role in the proof of the main theorem.Section 6 discusses semi- discreteness for injective finite factors.Automorphisms of factors are briefly treated in Section 7, and several important results of Connes on automorphisms of II factors are stated for use in the proof of the main theorem.Section 8 commences with the proof of the main theorem, Section 9 provides some necessary lemmas on embeddings in ultraproducts, and proof of the main theorem is completed S. WRIGHT in Section I0.The eleventh and final section uses the results of Sections 2 and 8, together with some results from [7] and [8], to obtain the classification of in- jective factors of type II(R) and III k [0,i) Conversely, suppose N is a finite factor acting standardly on H with hypertrace We will eventually show that this forces N to be injective, but before doing this we need to recall some facts about the Hilbert space L2(N,T) of Segal ([32]).An element x of L2(N,) is a closed, densely defined operator on H affiliated with N in the sense of Definition 2.1 of [32].A positive operator T affiliated with N is called .integrable when its spectral measure (E) (XE(T)) (E a Borel subset of (0, + )) satisfies o d(k) < and we set (T) o d(k) 2(N,) consists of all integrable x satisfying (x*x) < , and the norm of 2(N,) is given by x (x*x) 1/2 N is contained in 2(N,) as a dense submanifold, and for a N a (a*a) 1/2 If we restrict the hypertrace to N a tracial state on N results and so %1 N_ The Schwartz inequality for positive linear functionals therefore yields for each A B(H) l(aA) -< (A*A) 1/2 (a*a) 1/2 (A*A) 1/2 (a*a) A ll(a*a) 1/2 V a N (3.2) Thus each A B(H) determines a bounded linear functional A a (aA) on L2(N,) and hence a unique E(A) in L2(N,) such that (aE(A)) This verifies the claim.
We can now show that E(A) N for A >_ 0 Since E(A) is affiliated with N it suffices by the double commutant theorem to show that E(A) extends to a bounded operator on H--[2 (N,) It is now straightforward to deduce from (3.3) that A E(A) is a projection of norm of B(H) onto N.
The following proposition now obtains: 3.1.PROPOSITION.Let N be a finite factor.N is injective if and only if N admits a hypertrace in its standard representation.Ul,...,Un} of unitaries in N and e > 0, there exists a normal state @ on B(H) such that II@-( o aduj)ll < e j l,...,n (4.1) PROOF.The proof uses a separation argument first employed by Day [33] in the context of countable amenable discrete groups.
We consider the Banach space (B(H),) n formed by taking the Banach space direct sum of n copies of the predual of B(H) As mentioned in the previous section, Day used the technique of Proposition 4.1 to show that when G is a countable, amenable, discrete group, there exists a sequence F of normalized positive functions in gl(G) such that n llgF n Fnlll 0 for all g G Somewhat earlier Finer [35] had given a stronger result by finding a sequence S n of finite subsets of G such that S # for all n and n (S n A S n 0 V cj G where A # cardfnality of A c G and A denotes synnetric difference.
Let N be an injective finite factor acting standardly in H The next proposition gives an analog of Flnr's result for N and will play a crucial role in the sequel.Before giving its statement and proof, we need a rather technical lemma concerning certain approximations for positive Hilbert-Schmidt operators.
LEMMA.Let H be a Hilbert space.Let X denote the characteristic a function of the interval (a,+) a 0 Let hl,...,h n be positive Hilbert- Schmidt operators on H such that {lhj -h1{{HS < {IhI{{HS j I, n for some g > 0 where [[HS denotes the Hilbert-Schmidt norm.Then there exists a > 0 such that Xa(hl) # 0 and  functional on H is the trace norm of its density matrix, we conclude from (5.2) that lluj*puj -911Tr < s j l,...,n ( I, n set hj uj*huj where h 9 1/2 is a positive Hilbert- Schmidt operator.By (5.3) and the Powers-Strmer inequality [36], llhj hllHS -< llhj 2 h211Tr lluj*ouj 911Tr < j l,...,n 2 Thus by the lemma, there exists a > 0 such that X (h) # 0 and Assume for simplicity that in the Gelfand- Naimark-Segal construction (@,H,) corresponding to @ is separating for (A)" Let A A c denote the algebraic tensor product of A and A c and where & is the modular operator corresponding to the existence of which follows from Tomita-Takesaki theory.
always extends to a state on A (R) A c max if extends to a state on A (R).A c the spatial tensor product on H (R) H c mn we say that admits _a purification.
Let N be a factor acting on H An early result of Murray and von Neumann [I] asserts that the homomorphism defined on N @ N' by :a (R) a' aa' is an isomorphism.N is semidiscrete (in the sense of Effros and Lance, [14]) if this isomorphism extends to an isomorphism of N min N' i.e., if Connes characterizes semidiscreteness in terms of purification of states as follows ([37] Section 2.8): 6.1.PROPOSITION.Let N be a factor on H Then N is semidiscrete if and only if N has a faithful normal state which admits a purification.Now, let N be an injective finite factor, with denoting the canonical trace. is normal and faithful.We claim that admits a purification.To -< llj=IY, a.j (R) bjCllHC, V aj,bj E N.
It follows that is bounded on N @ N c relative to the spatial tensor product norm, and therefore has an extension to N %in N c We can now deduce from Proposition 6.1: 6.2.PROPOSITION.Injective finite factors are semidiscrete.

AUTOMORPHISMS OF FACTORS.
Let M be a von Neumann algebra.We set Aut(M) automorphism group of M (by automorphism, we will always mean *-automorphism)

Int(M)
inner automorphisms of M i.e., the set of all automorphisms of the form adu: x u*xu x M u a unitary operator in M.This is a normal subgroup of Aut(M) The weak topology on Aut(M) is the topology of point-norm convergence in the predual M, of M for the action 8() (8-I) i.e., a net We set Int M closure of Int(M) in the topology just described.
Let f be a fixed linear functional on M and let a M The linear functionals af and fa on M are defined respectively by af:x f(xa) and let {x n} be a centralizing sequence.We must show that (x n) x 0 *-strongly.Now 8(x n) x [u,x ]u* and setting Thus by (7.4) and (7.5) [u,x ]q)(y) 2) now follows from (7.6) and the fact that {x is centralizing.n 7.2.LEMMA.Let M be a von Neumann algebra with separable predual.
Then for each 0 PROOF.For M, denote the seminorm x (x*x) 1/2 by II@, Since is centrally trivial, for each positive integer n there is a neighborhood V of n in Aut(M) such that for all unitaries u M with ad u V n -n IlO(u) Let W be a decreasing basis of heighborhoods of e in Aut(M) such that n W W -1 c V and II@ o -I o -lll < 2 -2n Let u be a unitary in M such On the other hand, Connes proves the following remarkable theorem:

I)
Let M be a factor with separable predual, and let R denote the hyperfinite factor of type II are equivalent: Moreover, if (a) holds, then p(Ct(M)) commutant of p(Int M) We now list several theorems which allow us to manipulate Int N and Ct(N) for a factor N of type II 1 Regretfully we must omit all proofs; for details, Let NI,N 2 be II 1 factors.Then for 'i Aut (Ni) i 1,2 8. UNIQUENESS OF THE INJECTIVE 111 FACTOR.
In this section, we begin the proof of the main theorem.The proof will end in Section I0.PROOF.Let N denote an injective II factor acting on a separable Hilbert space, and acting standardly on H L2(N,) We will show that N is isomorphic to the hyperfinite lllfactor R.
By Proposition 6o2, N is semidiscrete, ioeo, the mapping D:N e N' B(H) n n given by rl: 7.. a. (R) b. 7. a.b. is isometric as a mapping from B(H (R) H) B(H) it follows that extends to an isomorphism of N (R) N' onto C*(N,N'). rain We claim that N and N' are simple, i.e., they contain no closed, two- sided ideals.Since N is a finite factor, by [38], Theorem 6.2 N contains no nonzero maximal ideals.Since every two-sided ideal of N is contained in a maximal one, we conclude that N is simple.Since N is conjugate-linearly isomorphic to N' ([13], Proposition 2.9o2), it follows that N' is also simple.
By a theorem of Takesaki ( [39] we conclude that N (R) N' is simple so that min C*(N,N') is simple.Hence if K denotes the (closed, two-sided) ideal of compact This shows that extends to an automorphism a of C*(M,M') such that aiM-8 [M'-identity on M' Thus by Theorem 7.6, 8 Int M.
We have hence shown that Aut M Int Mo S. WRIGHT Now by a theorem of Effros and Lance ([14], Proposition 5.6) N (R) N inherits semidiscreteness from N and standard facts about tensor products imply that N (R) N is a standardly acting II factor.We hence conclude that Aut (N (R) N) Int (N (R) N), and so o F 6 Int (N (R) N) Since N acts on a separable Hilbert space, so does N (R) N and therefore N (R) N has separable predual.By Lemma 7.2, p(Ct(N (R) N)) c commutant of p(Int(N (R) N)) whence by Theorem 7.7, p(8 (R) l)P(OF) p(oF)p(8 (R) i) V This completes the first major step of the proof of Theorem 8.1.In order to fully exploit this isomorphism, we must relate N more closely to R (after all, we are trying to show that N and R are in fact the same) This is done by embedding N in the ultraproduct R free ultrafilter.We take up the details in the next section and it is there that Proposition 5.1 plays a crucial role.

EMBEDDINGS OF N IN ULTRAPRODUCTSo
Let Z+ denote the positive integers with the discrete topology.Let C b (Z+) g denote the C*-algebra of bounded sequences.We identify the Stone-Cech compactification Z+ of Z+ with Mg tL= maximal ideal space of g Points n Z+ correspond to the homomorphism of evalutaion at n and free ultrafilters Z+\Z+ correspond to omomorphisms not of this form.

Mg
Then for a sequence a } we write lima a if ({ a a.
n n n Now, let ()kZ+ be a sequence of factors with finite normalized traces k and let be a free ultrafilter on Z+ On @ iNk the C*-direct sum of the Nk'S we define the trace :(xk) lim k(Xk k-Let Z kernel of {x @iNk:(x*x) 0 } Z is a closed, two-sided ideal in @iNk Let Then ([41], p. 451) NNk is a yon Neumann algebra with finite normalized trace 00 called the ultraproduct of the Nk'S corresponding __t Denote by M c the ultraproduct formed by countably many copies of the factor M. We proceed to construct an embedding of N into R Let Fn denote the free group on the generators gl,...,gn If m Fn we define the of m as the sum of the absolute values of all exponents -(Ul""' n-U) v _(Vl,... n-V) and is the normalized trace on q Q PROOF.We first strengthen Proposition 5.1 as follows: given Xl,...,x n N and s > 0 there exists a finite-rank projection E # 0 on H such that for j I,..., n If[E, xj]ll HS < s IIEIIHS (x .E,E) l(xj) , To see this, we first claim that there exist unitaries {Ul,..., Um} N and 6 > 0 such that for any state on N with ll[uj, ]II -< 6 j I,   IIxlIIIEII -I 211EIIHS by (iv) HS Since [uj, % ](x) (%-%u.) (xuj) x E N we conclude that 3 l[[uj, %][[ -< 6 j i,..., n as asserted.Let T 0 denote the normalized trace on R Since N acts separably, it contains a strongly dense sequence, and so we may find a sequence u } of n unitary operators generating N as avon Neumann algebra.Finally, let F k _c F be the set of all words involving only gl''''' gk and with length <_ k Using Lemma 9.1 and the hyperfiniteness of R choose for each k unitaries   (9.8), lira IIxj (R) 1R x u (I N (R) zj (U))xu,ii2 < j n (9.9) Since the evaluations at points in Z+ are weak *-dense in Mg it follows from (9.9) that X and z (u) (u) Recall from Section 2 that a finite factor M satisfies condition C if for each finite subset {Xl,...,x } of M and s > 0 there exists a finite- n dimensional subfactor C of M and Vl, o,Vn in C such that IIxj vjll 2 < s j n In order to show that N is isomorphic to R 0 I: N 02: < R So we may extend 81 (R) B 2 to an isomorphism of M k (R) < N (R) R which preserves the trace (see the introduction), and hence the L2-norms.Thus, by identifying M k with its image under the isomorphism a l(a) (R) R a 6 M k we may assume by (iv) that each x.' has the form m. (R)   in some factorization Since R is hyperfinite, there is a finite-dimensional subfactor Q of R and ql,...,q n Q such that llzj qjll 2 < With Theorem 8.1 and the fundamental structure Theorems 2.2, 2.3, and 2.4 now at our disposal, the precise structure of injective factors of type II and III, X E (0,I] can now be deduced in a fairly straightforward fashion.
Recall from Section 2 that R 0 is the hyperfinite II factor R (R) B(H where H is a separable Hilbert space. Ii.i.THEOREM ([I0], Theorem 7.4).All separably acting injective factors of type II are isomorphic to R 0,I PROOF.Let M be an injective II factor acting on a separable Hilbert space H By [4], Theorem IX, we can write M -N (R) B(H) for a separably acting II factor N M can thus be viewed as the algebra of -X-matrices factor of type III0, then M is a Krieger factor.
PROOF.Let W*(N,8) be a discrete decomposition of M as in Theorem 2.4.
As in the proof of Theorem 11.3, N is injective; however, it is not in general a factor.On the other hand, it is a direct integral of factors, and each factor appearing in its direct integral decomposition (except possibly for a set of measure 0) is injective by [I0], Proposition 6.5 and of type II and hence isomorphic to R0,1 But these are precisely the hypotheses of [8], Theorem II.I, which asserts that under these conditions M is a Krieger factor QED It follows from [30] that injective III 0 factors are classified up to isomorphism by ergodic non-transitive flows.For further details, the reader can consult [30] [I0], and [37].
The only known injective factor of type III 1 is the factor R of Araki and Woods (see Section 2).It is not known whether this is the only possible one.
ACKNOWLEDGEMENTS.I would like to thank Sze-kai Tsui for several stimulating and instructive conversations about the material contained herein.I would also like to thank Jim Agler, John B. Conway, and Andrew Lenard for very active participation in a seminar given during the summer of 1978 at Indiana University, during which many inaccuracies in an early version of this work were detected and corrected.
minimal C*-tensor products o__f A I and A 2

Mn}
be a sequence of von Neumann algebras, let n be a normal state on Mn n denote the infinite tensor product state on the infinite and let n C*-tensor product (R) M of the M's ([13], Section 1.23).Let denote the n n n elfand-Naimark-Segal representation of M determined by Then the yon n n Neumann algebra(n Mn)" is called the (W*-) HISTORICAL PERSPECTIVE ON CONNES' WORK least in principle, to the study of factors.This then was essentially the state of the art in the classification of von Neumann algebras at the start of the 1950's.Most of the work in the area consisted in a refining and strengthening of the tools bequeathed by Murray and S. WRIGHT von Neumann.The theory at that time suffered from a dearth of examples of nonlsoorphic factors, and consequently one of the sin lines of work was the construction of such examples.Von Neumann had constructed one example of a type parameter action on M Suppose M acts on a Hilbert space H Let denote Haar measure on G and let L2(G;H) denote the Hilbert space of all H-valued, (a,E(A)*)62(N,) (aA) V a N (3.3)Claim: If A >_ 0 then A is positive on N Let a N T B(H) Then (a*aT*T) (aT*Ta*) is a hypertrace) (aT* (aT*) *) >_ 0 (A) a, blHl l(b*E(A)a)l l(ab*E(A)) by [32], Corollary 11.2 l(ab*A) lA(ab*)l by (3.3) A(a*a) A(bb*) since A 0 _< [[A[[ (a a*) 1/4 (b b*) 1/4 by (3.2) IIAII l l a l 1 1 / 2 2 llbll Since N is dense in H [2(N,) it follows that E(A) extends to a bounded operator on H with IIE(A) II-< IIAII We conclude that E( 4o THE METHOD OF DAY FOR INJECTIVE FINITE FACTORS Let N be avon Neumann algebra, a linear functional on N PROPOSITION.Let N be an injective finite factor acting standardly in H Then for each finite set ANALOG OF F#LNER'S CONDITION FOR INJECTIVE FACTORS. such that E # 0 and II[E,xj]IIHS < glIEIIHS j l,...,n (For operators A and B we denote AB-BA by [A,B].)
]u* it suffices by (*) to show that n Xn lira ll[u,x ]il lira II[u Xn]Xn]* = [u*,x *] Thus to verify (7.1), it suffices to show that for n verifies by direct computation that for y M[UXnU*, ](y) [n' u*u] (u*yu) Xn](y) (yux n YXnU) u(yux u* YXn) n u(yux u* ux u'y) + u(UXnU*Y yx n) n n [UXnU*,U] (y) + u(UXnU*Y YXn)[Xn,U] (u*yu) + u(UXnU*Y-yx n) by(7 *)Un+ 8(Un*)Unll ll8(Un*)O(Vn*)VnUn e(Un*)nll < 3-(ad 0 (u n )) -0 (0= )11 < we obtain JJUn+l*O(Un+l) u*O(u)ll < 3 2 -n This shows that Un*0(Un) converges *-strongly to a unitary u M such that ad u -I00-I QED The relation between Int(M) Int(M), and Ct(M) has a strong bearing on the structure of M By Lemma 7.2, one always has p(Ct(M)) _c commutant of pC[ M)o consult the indicated references (all results are due to Connes)o Let N be a finite factor, the canonical trace on N Recall that N has pro__p_ert_y _ of Murray and von Neumann ([4]) if for each finite subset S. WRIGHT Xl,...,x n of N and xj]112 < g j i, n 7.4.THEOREM.([7], Corollary 3.8) Let N be a II factor with separable predual.Then Int(N) # Int N if and only if N has property F 7.5.THEOREM.([10], Theorem 2.1) Let N be a II factor acting stand- ardly on H Let K denote the compact operators on H C*(N,N') the C*- algebra generated by N and N' Then N has property F if and only if C*(N,N') K (0) Since N acts standardly on H C*(N,N') is irreducible.The next theorem gives a complete characterization of Int N for II factors.7.6.THEOREM ([i0], Theorem 3.1) Let N be a II factor acting standardly on H Then Int N if and only if e extends to an automorphism Aut (C* (N,N')) such that IN N' Identity on N' 7.7.THEOREM.([10], Corollary 4.4)
operators on H then C*(N,N') U K is either (0) or C*(N,N') Since I 6 C*(N,N') and H is infinite-dimensional, C*(N,N') K # C*(N,want to show that o F Int (N (R) N)To do this, let M be a standardly acting semidiscrete II factorai) bill (semidiscreteness again) tuple u (Ul,... u n) of unitaries defines a unitary representation LEMMAo Let ul,o.o,Un be unitaries in N For each > 0 there is a finite-dimensional factor Q and unitary operators Vl,..Vn Q not.Then for each finite subset o. of unitaries of N and 6 > 0 there is a state o.,6 on N such that (ii) Io.,6 (xj) -(xj) >_ for some j E {I,..., n } If we partially order the set of (o.,6)'s by setting (o.i,61) _< (o.2,62) if 62 then {o., } is a net and so by weak *-compactness of o.1 c o-2 and 61 6 the state space of N {o,6} weak *-accumulates at a state of N By (i), is unitarily invariant, and so But since o.,6 + re(weak*), we may find o.,6 such that lo,5 (xj) (xj)l < e, j I n contradicting (ii)This verifies the claim.Now use Proposition 5.1 to find a finite rank projection E # 0 such that for i I,..., n j I,..., m (iii) ll[x i, E]{IHS < e {{E{{HS

3 (
xEj ,Ej )HS IIEII2S -< (I(x(E-Ej), E)HS + I(xEj,E-Ej)HS "IIEII2S liE EjlIHsIIEII2S (IIEIIHS + IIEjlIHS) by(9.2) and induction on the length of the word m that flu(m) v'(m)llHS -< (length of m)-IIEIIH S (9.3)By (9.2), E commutes with all v so that (v'(m))E v(m) for each word m 3 Thus by(9.3) and the Schwartz inequality for the Hilbert-Schmidt norm, I(u(m)E,E)Hs (v(m)E,E)HsI < slIEII and the definition of this is what we seek.q QEDWe are now ready to construct an embedding of N into RThe construction depends on the following extensiorr-lemma of Pearcy and Ringrose ([15], Lemma I), which we record for the convenience of the reader.

IIxj
R) R) be the isomorphism of Lemma 9.4.As we saw in Section 8, the flip automorphism o F Aut(N (R) N) is in Int(N (R) N) so there is a unitary v N (R) N such that the trace and hence the L2-norm.Thus by (9.7) lls(xj (R) I) (v)S(l (R) xj)(v)*ll 2 in R such that (I (R) z.represents (I (R) xj) Then by PROOF OF THEOREM 8.1 COMPLETED.
This verifies condition C for N QED II.CLASSIFICATION OF INJECTIVE FACTORS OF TYPE II(R), lllk, k [0,I). yon Woods classification scheme was the first significant advance in the classification of factors beyond the initial results of Murray and von Neumann, and it was to have a great influence on the young Alain Connes.The second major development which Connes would put to good use came from Japan.A problem of interest at the time concerned the commutant of a tensor are von Neumann algebras, does 3. INJECTIVE VON NEUMANN ALGEBRAS AND THE HYPERTRACE.A von Neumann algebra N acting on a Hilbert space H is said to be in- Thus, by the previous claim, together with (iii) is what we are seeking.E.j u.Eu.*3 3 then ..IIEj-EIIHS,.< 611EIIHS and, by [i0], Lemma 1.4, there are unitary operators w.3 B(H) with w.E.w.*3 3 3 E and llwj.IIIHS.. < 3611EIIHS....
(In whatfollows,-WOT denotes closure in the weak operator topology)o 9.2.LEMMA.Let Mo be avon Neumann algebra, i a faithful, normal,