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Now if H itself is equivalent to M 0 , the function d(M)can (and must) be defined by: d(0) 0; for all M t 0, d(M) bfor some fixed b a . In this case fZ is said to be of type HI a .If H is a separable Hilbert space, every factor of type III mustbe of type III \ ; Murray and von Neumann wrote III oo f rthis type III *If M0 can actually be expressed as the union of a familyI of a orthogonal elements each equivalent to Mo (that is, ifthe cardinality a is attained) but H is not equivalent to MGthen d(M) can (and must) be defined by: d(0) 0; d(M) a ifM is equivalent to MQî d(M) d(M: MQ) if M is not equivalentto M 0 . In tMs case, let b d(H: MQ). Then d(M) takes onas values:0 and every cardinal K satisfying a X b. In thiscase the factor ii is said to be of type HI/ a j fo).Finally, suppose if possible that H is not equivalent toM0 and that a is not attained (necessarily a K0 and a hasthe property that it cannot be the sum of a set of cardinals2Z / r T ao w n each a «c a and cardinality of J c a)* Thenthere will be an element M\ in such that d(Mj : M0) aand yet Mj is not equivalent to M 0 . But for any function d(M)with the properties listed at the end of § 2, it is easy to seethat d(M0) would equal d(M ). Thus if a is not attained andH LËL n t equivalent to MQ there can be no such function d(M) If d(H: MQ) b (necessarily b a Xô ), iZ is said to be oftype HI/ , -px, the sloped mark indicating that a is not attained.It is not yet clear whether this type HI(a, fo\ actually occurs.7. Examples of factors. As stated at the end of § 4,for each finite or infinite a(a 1 , 2 , . . . , X0 , . . . ) and foreach Hilbert space H of dimension divisible by a, there existsa factor of type I a on H. Two factors of type I a with thesame a are isomorphic under a suitable isomorphism of theirH-spaces if their H spaces have the same dimension; in anycase the factors are isomorphic as rings with a -operation(a factor is always defined with respect to a particular Hilbertspace H but two factors can be *-ring isomorphic even thoughthey are defined on Hilbert spaces of different dimension).The construction of factors of types II and III is a moredifficult matter. Von Neumann and Murray first constructedexamples of factors Hi a n d H-X. o n a separable Hilbert spaceby using measure-theoretic methods [7] . In a later paper[8] von Neumann defined tensor productsTÏ H with aninfinite number of factor spaces Ha and used an infinite tensor278https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Press

product TT Ha with each HQ a 4 - dimensional Euclideanspace to construct a wide variety of factors.Itturned out that among these tensor product factors, were (i)factors of type 11 on spaces H of arbitrary dimension ) 0 ,(ii) factors of type II a with fixed but arbitrarily given a X 0 ,on spaces H of arbitrary dimension a, and (iii) factors oftype III a for fixed but arbitrarily given a on spaces H ofarbitrary dimension a. 15a)The factors constructed by von Neumann with the help ofinfinite tensor products are not difficult to describe, but theverification of their types, as given by von Neumann, is involved.In fact, von Neumann actually made the verification only for theIII factor on a separable Hilbert space by identifying the factoron a tensor product space with a certain III measure-theoreticfactor constructed in a previous Murray-von Neumann paper(the II /case then follows easily).Later, von Neumann [91 established the existence of afactor of type III ç on a separable Hilbert space, again bymeasure-theoretic methods, and asserted that some of thefactors previously constructed by him on infinite tensor productspaces could be identified with some of these (measure-theoretic)factors III vThe next two sections are devoted to a discussion of thesetensor product factors, avoiding the more involved measuretheoretic apparatus. Infinite tensor product spaces. Suppose H is afixed Hilbert space for each * in a set of indices J (assumeeach H has dimension 2), and suppose a fixed unit vector rf has been selected in each H Now let TT' H consist of all finite formal sums v Z i { TT f x ), where foreach * in J and for each i, f l is in the corresponding H and, with at most a finite number of exceptions, f l fi* .If w Z j ( TT g a C J) define (v w) to be Z i j TT* (f I g , J)(note that Z i j is a finite sum, and for each i, j , the product"FT* (f * * goc J) is a finite product since, with a finite numberof exceptions, (f i \g J) ( I f « ) 1). Note that thevector g T7§ * is in TT' H * .In TT Hoc , for each complex number c identify11with ZTi TlQiiJ)' where for each i, (f )' cf*cvfor one of the *and (f )' ** for all other * ; also identify v and w if279https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Press

(v-w lv-w) 0, With these identifications TT H becomes alinear space with inner product; its completion is called thetensor productTJ H . This tensor product depends notonly on the H but also on the particular choice of the CvHowever, different choices of the # give tensor productspaces which are isomorphic, all having the same dimension ")(von Neumann called this space an incomplete direct product;he wroteTT H to denote a much larger space which dependsonly on the H a , not on a choice of p and he showed that hislarger space decomposed in a unique way into orthogonal partseach of which is an incomplete direct product or tensor productas defined above).Each bounded linear operator T on a single factor spaceHo(0 determines a bounded linear operator T on the spaceH TT Ha in the following way. If v TR f with f a 4 OLfor all but a finite number of oc , let Tv TT S)f wheref f rt if c 4 x0 and f 0 Tf o . Now T is uniquelydefined by linearity for each v inTT&H (the correspondingTv is also inTT'oH* ) and by continuity T is defineduniquely on all of H.Let "G denote the set of all bounded linear operators onH and for fixed oc let 6 * denote the set of all bounded linearoperators on H . If 7 is a subset of 13 let 3 denotethe set of all T as T varies over . If for each * ,3 is a subset of 13 letTT denote the set of those,operators from 13 which can be expressed as JET i ( TT T )with ZZ a finite sum and eachTT T 1 containing only a finite1number of factors and each T inJ ; and let TT 2 J denote the strong closure l 6 ) of T7' 2 ,It is not difficult to prove the following.(i) IfTiandT2 are in different13 then T1T2 T2T1.(ii) For each fixed * , ( 15 ) 13 von Neumann algebra.the, i.e.15u is a(iii) The only von Neumann algebra which contains all is 13 .(iv) For fixed X , ? * is a von Neumann algebra (respectively, a factor) on H if and only if 3 is a von Neumannalgebra (respectively a factor) on H , If , 7 arefactors, they are of the same type.280https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Press

(v) If for each oc , 7* is a von Neumann algebra on HthenTT 3 is a von Neumann algebra on H and it is thesmallest one which contains all 7*(vi) If for each * ,is a factor on H. is a factor on H then "H" (vi) immediately presents a challenging problem, as yetnot completely solved, namely to determine the type of thefactorTT J , given the factor 9 in each H * VonNeumann emphasized that (vi) was also important because itopened up the possibility of constructing complicated factors onH by starting from known factors on the individual H . Infact von Neumann constructed factors of the form TT0 3bcof types I I } , II x o and HE e respectively by using spacesH , each an Euclidean space of dimension 4, and in each H a factor of type l .We shall discuss these examples of von Neumann in thenext section.9. F a c t o r s with a t r a c e function. Let "f? be any ringof o p e r a t o r s on a Hilbert space H such that "*? contains theidentity operator on H and P2 contains A* if A is in "ft .Then a function B with complex numbers (finite1.) as values, iscalled a finite t r a c e on "f if(i)(ii)B (A) is defined for each A in H0(A B ) ,9(A) 0(B) for all A, B in 1? ,(iii)a (A*A) is r e a l and 0 for every A 0 in "ft(iv)d (1) 1,(v)0(AB) 0 (BA) for all A, B in "R,.Suppose now that 1Z is known to be a factor and that itp o s s e s s e s a t r a c e . Then it must be of type II or I n for somefinite n, and if P j denotes the orthogonal projection operatoron a closed linear subspace M then the function 0 (Pj ) canbe taken as d(M). To see t h i s , recall that M « N means thatthere exists an operator T in -ft which maps M isometricallyonto N. Let W P N T P M . Then W*W P M , WW* P N so9 ( P M ) 6 (W*W) Q (WW*) a ( P N ) ; i . e . M s N implies281https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Press

t h a t 0 ( P M ) 9 ( P N ) * F r o m t h i s fact a n d ( i i ) , ( i i i ) , a n d (iv) itf o l l o w s e a s i l y t h a t d(M) k 0 ( P ) for s o m e finite p o s i t i v ec o n s t a n t k . T h i s i m p l i e s t h a t H i s finite so t h a t iZ i sn e c e s s a r i l y of t y p e I I i o r I n for s o m e finite n .Now m a k e t h e following d e f i n i t i o n .D E F I N I T I O N 6 . L e t "R be a r i n g of o p e r a t o r s on as p a c e H s u c h t h a t i c o n t a i n s t h e i d e n t i t y o p e r a t o r on H and-f? c o n t a i n s A * if A i s in iZ * T h e n a unit v e c t o r g inH i s c a l l e d a t r a c e - v e c t o r for -R if (ABg g) (BAg g) fora l l A , B in H .It i s e a s y t o s e e t h a t if iZ p o s s e s s e s a t r a c e - v e c t o r gt h e n (Ag g) i s a finite t r a c e o n R; h e n c e if "ft i s a f a c t o r itm u s t b e of t y p e I n o r I I I .F i n a l l y , s u p p o s e t h a t for e a c h * , -R i s a f a c t o r onH and t h a t &* i s a t r a c e - v e c t o r in H for -f2 (the q a r e t h e v e c t o r s i n H i n v o l v e d in t h e c o n s t r u c t i o n of TT H ).L e t g be t h e v e c t o r TT inTT H . T h e n it i s e a s yto p r o v e t h a t ( A B g g ) (BAg g) for a l l A , B i n TT' -R and h e n c e f o r a l l A , B i n -R TT(g -R T h i s s h o w s t h a t iZ m u s t t h e n be a f a c t o r of t y p e I n o rI I ! It i s e a s y t o p r o v e t h a t -f2 i s of t y p e I n if a n d only ifa l l tZ oc a r e of t y p e I a f o r finite a s u c h t h a t"oc3- isfinite ( n e c e s s a r i l y n; t h e n "R i s of t y p e I j for a l l but af i n i t e n u m b e r of oc ) . In a l l o t h e r c a s e s , i 3 i s n e c e s s a r i l yof tyPJL 1 * T h i s i n d i c a t e s a m e t h o d of c o n s t r u c t i n g II]. f a c t o r son a s p a c e H of a r b i t r a r y d i m e n s i o n a Xo N a m e l y , f i r s tc h o o s e a s e t J of i n d i c e s o so t h a t J h a s c a r d i n a l i t y a .T h e n c h o o s e f o r e a c h c s o m e s p a c e H of finite d i m e n s i o n a l i t ya 2 and c h o o s e a unit v e c t o r4 in H ; TT& H w i l lt h e n h a v e d i m e n s i o n a . N e x t , c h o o s e if p o s s i b l e in e a c h H a f a c t o r "R of s o m e t y p e I nw i t h n 1 so t h a t isa t r a c e - v e c t o r in H for -R and so t h a t t h e set J ' of o with n 1 has cardinality a' withK 0 a ' rs a . T h e n-R TT g w i l l be a f a c t o r of t y p e II on t h e a - d i m e n s i o n a l s p a c e TT S H * a s d e s i r e d .T o c o m p l e t e t h e s e l e c t i o n ofp r o c e e d a s f o l l o w s : L e t a be an , m a n d l e t H be i t s e l f awith E an E u c l i d e a n s p a c e withH , and 7? ,product n x m witht e n s o r product H EC F0 a complete -5 Published online by Cambridge University Press

basis h , . «, . , l and F an Euclidean space with a complete orthonormal basis k j , . . . , k r n . Let "R be thering of bounded linear o p e r a t o r s on H which depend only onE and let kc 1/V c U h i k x ) (h 2 k 2 ) .-. (h n k n J.-It is easily verified that is a t r a c e - v e c t o r in Hocfor HZ and so 71" - is a III factor ,on T7 Hoc Thesimplest case of this situation is when all E * and F are2-dimensional Euclidean spaces and J has cardinality I*0 .Note that in the general construction the II i factor",on T7 Hc depends on the cardinality of J ' ,To construct a 11 factor on a space H of dimension a(assuming a b X 0 ) proceed as follows. Let "iZ be a Hifactor on a space Hi of dimension b , let "8 2 D e * n e r i n g all bounded linear operators on a separable space H 2 , and let f o De t a e 1 factor consisting of multiplication operatorscl on a space H3 of dimension a. Then i2 1 -0 2 3is a factor of type H on the space H Hi S H 2 B H3 ofdimension a.The construction of factors of type III is easy to describeusing tensor products» Again let J be an infinite set ofindices & and for each * let H be a tensor product E F where E , F have orthonormal complete b a s e s h i , h 2 andfl» f2 respectively. Let be the ring of bounded linearoperators on H which depend only on E * .But now let j - c i ( h i fi) c 2 (h 2 f2) with 0 ci c 2and c c 2 1. The fact that c i c 2 prevents g TT from being a t r a c e - v e c t o r for i? TT0 -fZ on TT Hc ,and von Neumann indicated that this iZ is in fact a factor oftype III. However, the verification of this fact and constructionof factors HI/ a \ forX 0 s a b will not be given in thepresent a r t i c l e .10- Projection g e o m e t r i e s . For every von Neumannalgebra iZ on a space H the projection geometry is akind of generalization of complex-projective geometry (in thecase 12 is a factor of type I n , is actually a projectivegeometry). It has already been observed that sd is a complete,orthocomplemented lattice with a special congruence relation.It is not difficult to show that is irreducible as a latticeif and only if is a factor.283https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Press

Now is not in general a distributive l a t t i c e , that is(*) M n (N u Q) (M n N) u (MAQ)is not t r u e , in general. In fact, if "fi is a factor, the only Mfor which (*) holds for all N, Q a r e : M 0 and M H.Thus, even the projective geometry lattices a r e not distributive*But if sC is a projective geometry, does satisfy ar e s t r i c t e d form of (*); namely (*) with the r e s t r i c t i o n M N(this r e s t r i c t e d relation (*) is called the modular law). It t u r n sout that for a general factor -# , ; satisfies the modular lawif and only if iZ is of type I n with a finite n or of type II j .Von Neumann observed that if iZ is of type I n or II then is not only modular, but it satisfies the continuityconditions:(**) M x o M 2 c . . . c M c . . . implies LKM rx N j U M l o Nfor all N,M x M 2 . . M D . . . implies fKM uN) f î M u Nfor all N ,(Kaplansky l a t e r proved in [3] , see also [1] , that everyorthocomplemented complete modular lattice n e c e s s a r i l ysatisfies (**) ).Von Neumann considered the U projection g e o m e t r i e sto be a natural and very important generalization of c l a s s i c a lprojective geometry and he succeeded in characterizing themby lattice theoretic axioms (this manuscript has never beenpublished but an a b s t r a c t will be included in the forthcomingCollected Works of J . von Neumann). However, these p r o j e c tion geometries of 11 factors a r e closely associated with thecomplex number system and von Neumann found it convenientto axiomatize a wider class of l a t t i c e - g e o m e t r i e s which includedthe I n and I I \ projection geometries but no other projectiong e o m e t r i e s . This wider c l a s s of geometries he called continuousg e o m e t r i e s . His axioms for a continuous geometry L, were:L should be a complete lattice which is m o d u l a r , satisfies (**)and is complemented (for a lattice to be complemented m e a n s :for each a in L there exists at least one a' such thata w a' 1 and a A a r 0). Of c o u r s e , if a lattice is o r t h o c o m plemented it is 34-5 Published online by Cambridge University Press

A detailed account of continuous geometries is given invon Neumann's forthcoming Princeton lectures [13] Thefirst main theorem of von Neumann for continuous geometrywas: if the continuous geometry is irreducible then thereexists a unique dimension function d(a) defined for all a inL with the p r o p e r t i e s : 0 d(a) * 1 for each a in L; d(0) 0;d(l) 1; d(a o b) d(a) d(b) if a n b 0.REFERENCES1. I. Amemiya and I. Halperin, Complemented modular l a t t i c e s ,Canad. J . Math. 11 (1959), 481-520.2. J . Dixmier, L e s a l g è b r e s d1 o p é r a t e u r s dans l'espacehilbertien (algèbres de von Neumann), G a u t h i e r - V i l l a r s ,P a r i s , 1957.3. Irving Kaplansky, Any orthocomplemented complete modularlattice is a continuous geometry, Ann. of Math. 61 (1955),524-541.4.J . von Neumann, Allgemeine Eigenwerttheorie h e r m i t e s c h e rFunktionaloperatoren, Math. Ann. 102 (1929), 49-131.5. J . von Neumann, Zur Algebra der Funktionaloperatoren undThéorie der normalen Operatoren, Math. Ann.102(1929), 370-427.6.J. von Neumann, On a certain topology for rings of o p e r a t o r s ,Ann. of Math. 37(1936), 111-115.7. F . J . M u r r a y and J . von Neumann, On rings of o p e r a t o r s ,Ann. of Math. 37(1936), 116-229.8. J . von Neumann, On infinite direct products, CompositioMath. 6 (1938), 1-77.9.J. von Neumann, On rings of o p e r a t o r s , III, Ann, of Math.41 (1940), 94-161.10. F . J . M u r r a y and J . von Neumann, On rings of o p e r a t o r s ,IV, Ann. of Math. 44 (1943), 716-808.285https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Press

J , v o n N e u m a n n , On s o m e a l g e b r a i c a l p r o p e r t i e s of o p e r a t o rr i n g s , A n n . of M a t h . 4 4 ( 1 9 4 3 ) , 7 0 9 - 7 1 5 .J . von N e u m a n n , On r i n g s of o p e r a t o r s .A n n . of M a t h . (1949), 4 0 1 - 4 8 5 .Reduction theory,J . von N e u m a n n , 5 N o t e s in P r o c . N a t . A c a d . S c i . U . S . A . ,1 9 3 6 - 3 7 , and C o n t i n u o u s G e o m e t r y , m i m e o g r a p h e d l e c t u r en o t e s , The I n s t i t u t e for A d v a n c e d Study, P r i n c e t o n , 1 9 3 5 - 3 7 ,p u b l i s h e d by P r i n c e t o n U n i v e r s i t y P r e s s , I 9 6 0 .L. H. L o o m i s , T h e l a t t i c e t h e o r e t i c b a c k g r o u n d of thed i m e n s i o n t h e o r y , M e m . A m e r . M a t h . Soc. (1955).S. M a e d a , D i m e n s i o n functions on c e r t a i n g e n e r a l l a

von Neumann, partly in collaboration with F.J. Murray, and which grew out of von Neumann1 s analysis of linear operators in Hilbert space. */ 2. Continuous geometries were discovered by von Neumann and most of our present knowledge of these geometries is due to him. The first continuous geometries which he found were