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INTRODUCTION TO VON NEUMANNALGEBRAS AND CONTINUOUS GEOMETRYI s r a e l Halperin(received June 22, I960)1. "What is a von Neumann algebra? What is a factor (i)of type I, (ii) of type II, (iii) of type III? What is a projectiongeometry? And finally, what is a continuous geometry?The questions recall some of the most brilliant m a t h e m a tical work of the past 30 y e a r s , work which was done by Johnvon Neumann, partly in collaboration with F . J . M u r r a y , andwhich grew out of von Neumann 1 s analysis of linear o p e r a t o r sin Hilbert space. */2. Continuous geometries were discovered by von Neumannand most of our present knowledge of these geometries is dueto h i m . The first continuous geometries which he found wereprojection geometries of certain rings of operators in a separableHilbert s p a c e d (see definition 3 below for the definition of aprojection geometry). Roughly speaking, continuous geometrieswhich a r e projection geometries a r e a generalization of complex-projective geometry somewhat in the way that Hilbertspace is a generalization of finite dimensional Euclidean space.But to be m o r e p r e c i s e , we need some definitions.DEFINITION 1. Suppose U i s a ring of bounded linearoperators on a Hilbert space H. Then 1Z is called a vonNeumann algebra if(i) 1? contains the multiplication o p e r a t o r sfor every complex number c.See p . 286 for footnotes.Canad. Math. Bull, v o l . 3 , no. 3, September I960273https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Presscla' on H

(ii) ilcontains the adjoint operator T* for each T in-ft .(iii) "ft contains every T 0 in the strong closure of "7? .Von Neumann defined, for any family J of bounded linearo p e r a t o r s on H, the commutant 7 ' to consist of those boundedlinear o p e r a t o r s B on H such that BA AB and BA* A*Bfor all A in 3- He showed that J'' is always a von Neumannalgebra and he proved that ? is a von Neumann algebra if andonly if ( ? ' )' 7 .DEFINITION 2. A von Neumann algebra ft is called afactor if the only o p e r a t o r s in -ft which a r e also in ft' a r ethe multiplication o p e r a t o r s cl on H. 'Von Neumann showed that a von Neumann algebra is afactor if and only if it is irreducible as a ring, )DEFINITION 3. Suppose -ft is a ring of o p e r a t o r s on aHilbert space H. Then the projection geometry of ft ,denoted ( *# ), is defined to be the system0 , M, N, . . . . , H of those non-empty closed linear subspacesof H which have projection (i. e. , orthogonal projection)o p e r a t o r s in -ft . ' is to be considered as an orderedsystem, o r d e r e d by the relation of inclusion N d M. 'If -ft is a von Neumann algebra then its projectiongeometry is a lattice, in fact a complete lattice ) which isorthocomplemented. ' / F u r t h e r m o r e , on this lattice , therecan be defined in a natural way, a relation of equivalence, asfollows: M, N a r e called equivalent (written M S N) if thereexists in iZ an operator T which maps M i s o m e t r i c a l l yonto N . ' Von Neumann and M u r r a y showed:(i) If M, N in jL a r e such that some T in "ft mapsM in a ( 1 , 1) way into a set dense in N then M S N. *'(ii) The relation s is unrestrictedly additive for orthogonal families, that i s , if M N for each * and the M a r e mutually orthogonal and the N a r e mutually orthogonal,then U M S U NU .i(iii) If M, N in « a r e such that M N\ for someNl c N and N Mj for some Mj c M, then M N.274https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Press

(iv) If i ? i s a f a c t o r t h e n for e v e r y p a i r M , N i n e i t h e r M % N j for s o m e N j c N o r N M j for s o m eMi c M.F i n a l l y if 12 i s a f a c t o r , t h e r e e x i s t s a function d(M),with r e a l finite o r infinite n o n - n e g a t i v e v a l u e s , defined for thee l e m e n t s M in , s u c h that: d(0) 0; d(M) d(N) ifM N i for s o m e Nx c N; d(M) d(N) if and only if M % N;d( U M ) Z. d(Mx ) if the M u a r e m u t u a l l y o r t h o g o n a l . lz)3. In a n a l y s i n g the s t r u c t u r e of a p r o j e c t i o n g e o m e t r ytwo definitions a r e u s e f u l .D E F I N I T I O N 4 . An e l e m e n t M / 0 in i s c a l l e d a na t o m if for e v e r y N in , the r e l a t i o n s 0 N c M i m p l yt h a t N c o i n c i d e s with M .DEFINITION 5 . An e l e m e n t M in & i s c a l l e d finiteif N M , N a M i m p l y t h a t N c o i n c i d e s with M . If M i snot finite it i s c a l l e d i n f i n i t e .The f a c t o r s(i)i 1?a r e c l a s s i f i e d in t y p e s a s follows:i s s a i d to be of type I if contains an a t o m .(ii) Hi s s a i d to be of type II if /d o e s not c o n t a i nany a t o m , but « d o e s c o n t a i n s o m e n o n - z e r o finite e l e m e n t .(iii) "#i s s a i d to be of type III ifany n o n - z e r o finite e l e m e n t , d o e s not c o n t a i n4« F a c t o r s of type I . If , h a s a t o m s , t h e n any twoa t o m s a r e n e c e s s a r i l y e q u i v a l e n t and a n y n o n - z e r o M in &can be e x p r e s s e d a s the union of an o r t h o g o n a l f a m i l y of a t o m s :M ( M ; oc e I) w i t h the M o r t h o g o n a l a t o m s . It i snot difficult to p r o v e t h a t t h e c a r d i n a l i t y of I i s uniquely d e t e r m i n e d by M; d(M) c a n be defined to be the c a r d i n a l i t y of I (thisa m o u n t s to " n o r m a l i z i n g " t h e function d(M) by the a d d i t i o n a lc o n d i t i o n t h a t d ( a t o m ) should be 1). It i s e a s y to p r o v e t h a tt h i s function d(M) s a t i s f i e s t h e c o n d i t i o n s l i s t e d a t the end of§ 2 . If a d(H), w h e r e a i s a finite i n t e g e r o r infinite, wes a y f2 i s of t y p e I a .In t h e c a s e t h a t H is a s e p a r a b l e H i l b e r t s p a c e , a m u s tbe one of 1, 2 , . . . , n , . . . o r X 0. M u r r a y and von N e u m a n nw r o t e IQQ to d e n o t e I yj.275https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Press

A factor JE has finite linear dimension as a vector space '(this must be the case whenever H has finite dimension) ifand only if iZ is of type I n for some finite n. In this case , considered as an ordered system, can be identified withthe ordered system of all linear subsets of a complex-projectivegeometry, the atoms of jL being identified with the points ofthe projective geometry (the normalized function d(M) is thenthe projective geometry dimension i n c r e a s e d by 1). If H hasfinite dimension then d(M) differs from the dimension of M,considered a s a Hilbert space (see footnote 2), by a finitemultiplicative factor; but if H has infinite dimension then allnon-zero M in JL have infinite dimension as Hilbert s p a c e s .The case I a occurs if 1Z is the ring of all boundedlinear o p e r a t o r s on an H of dimension a. M u r r a y and vonNeumann obtained the important result that every iZ of typeI a is "essentially' 1 such a ring of all bounded linear o p e r a t o r son an H of dimension a. To be p r e c i s e , they defined thetensor product Hj S H ' of two Hilbert spaces H , U and they showed: a factor fî on a space H is of type I a ifand only if H can be expressed as a tensor product Hj & H »with H of dimension a, in such a way that "1Z is identifiedwith the ring of those bounded linear o p e r a t o r s on H whichdepend only on the factor H\. 15)5« F a c t o r s of type II. If has no atoms then everyM in H can be expressed as the union of two orthogonalequivalent elements; in other w o r d s , each M can be decomposedorthogonally into two equivalent p a r t s , each of which can becalled a half-M element.If now has no atoms but F is a n o n - z e r o finite element,the function d(M) can be constructed in the following way,which amounts to "normalizing" d(M) by the additional condition that d(F) 1.Let F j F and let Fl denote a half-F element. Byinduction, let F w n 1 denote a h a l f - l / 2 n element foreach n 1, 2, * . . ,Then an element M in is finite if and only if it canbe e x p r e s s e d a s a finite or countable union of orthogonalelementsM 7 . t F. Y eF, . *-* i li - n ll/2n276https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Press

rwherer i s finite, r 0, 1 , 2 ,i. . . , each of F i , . . . , F„iis*equivalent to F\,and e a c h F nto F 2 n « the value of r Z n 1i s either 0 or equivalent( l / 2 n ) ' where (l/2n)' 0or l / 2 n according a s ( F n ) i swill be uniquely d e t e r m i n e d by M,d(M) if M is finite.0 or i s equivalent to F I I DChoose this value to beIf M i s infinite, then M can be e x p r e s s e d as the unionof a countable or non-countable family I of orthogonal e l e m e n t se a c h e q u i v a l e n t to F\, the cardinality of I being uniquelyd e t e r m i n e d by M. Choose d(M) in this c a s e to be the cardinalityof I.The function d(M) s o defined for all M in i w i l l , inf a c t , be defined uniquely and w i l l satisfy the conditions g i v e nat the end of % 2 . T h i s construction of the function d o e sdepend on the particular choice of finite e l e m e n t F a s n o r m a lizing e l e m e n t , but a different choice for F will m e r e l y changethe function d(M) by multiplying it by a finite n o n - z e r o p o s i t i v econstant; in p a r t i c u l a r , if M i s infinite the value of d(M) i sindependent of the choice of F .If H i t s e l f i s finite it i s convenient to c h o o s e H to be F .Then the v a l u e s of d(M) are p r e c i s e l y all x in the i n t e r v a l0 x « 1. In this c a s e we s a y iZ i s of type 11 .If H i s infinite, and d(H) a then the v a l u e s of d(M)are p r e c i s e l y all r e a l x 0 together with all infinite cardinals . a. In this c a s e we say iZ i s of type I I a . If H is as e p a r a b l e Hilbert space then a factor of type II must be ll\ orII . M u r r a y and von Neumann wrote IIco for II 6. F a c t o r s of type HI. Suppose now that all n o n - z e r oe l e m e n t s in are infinite. It i s not difficult to prove thatthere e x i s t s in , a n o n - z e r o e l e m e n t M 0 such that0 f N c M 0 i m p l i e s that N M 0 . Then e v e r y n o n - z e r o e l e m e n tM in can be e x p r e s s e d a s the union of a family I of o r t h o gonal e l e m e n t s each equivalent to this M Q : M 2L *é I M ,M M 0 for all oc . If M is 0 and i s not equivalent toM 0 then it i s not difficult to prove that the cardinality of Ii s uniquely d e t e r m i n e d . Define d(M: MQ) to be the cardinalityof I. If M M Q the cardinality of I i s not uniquely d e t e r m i n e d ;define a to be the supremum of all c a r d i n a l i t i e s of such I( n e c e s s a r i l y a X 0 ).277https://doi.org/10.4153/CMB-1960-034-5 Published online by Cambridge University Press

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  . In a later paper 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  , see also  , 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  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