Von Neumann-On Rings of Operators III

Annals of Mathematics

On Rings of Operators. IIIAuthor(s): J. v. NeumannReviewed work(s):Source: The Annals of Mathematics, Second Series, Vol. 41, No. 1 (Jan., 1940), pp. 94-161Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1968823 .Accessed: 03/12/2011 09:30

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ANNALS OF MATHEMATICS Vol. 41, No. 1, January, 1940

ON RINGS OF OPERATORS. III

BY J. V. NEUMANN

(Received June 14, 1939)

INTRODUCTION

1. In two earlier papers ((1), (2)), F. J. Murray and the author investigated certain operator rings, called factors. (Cf. (1), p. 138, Definition 3.1.2. For an introduction into the theory of operator rings cf. (5) particularly pp. 372-376, 388-398.) The motives which led to those investigations are described in (1), pp. 116-123.

The main principle of classification for factors, which was found in (1), is based on the ranges of their relative dimension functions. (Cf. ibid., p. 165, Definition 8.2.1; p. 168, Theorem VII; and p. 172, Theorem VIII.) Thus all factors were found to belong to classes called (I.) (n = 1, 2, ..), (I.), (III), (IIJ), (IIIJ). It was shown that factors in each one of these classes actually do exist-with the exception of (III.), which had to be left undecided. (Cf. (1), p. 208, Theorem XII.)

The main result of this paper is that factors of class (III.) also exist. (Cf. Theorem IX, and Examples (a), (Il) in ?4.4.) Thus the above classification of factors is fully justified.

2. This result is obtained by the simultaneous use of two devices: The notions of norm and normedness, which form the subject of Chapter I; and the process At EIE2. which is the analogue of the process of taking the diagonal part of a finite matrix, and is discussed in Chapter II.

Both notions have an interest of their own, and we think that they deserve to be studied for their own sake. In this paper their analysis is restricted to the amount which is necessary for our immediate purposes, although we permitted ourselves a certain leeway in some cases where a little more generality than absolutely necessary seemed to make things clearer and easier. But we propose to discuss both of them independently and more fully at some other occasion.

Chapter III gives some explicit constructions of factors, generalizing those of (1), pp. 192-204. And then, with the help of the above-mentioned tools of Chapters I, II, it is shown in Chapter IV, to which classes the factors of Chapter III belong. At this stage examples of factors of class (III.) are also obtained.

3. The main questions concerning factors were stated in (1) as Problems 1-11. Of these, Problems 1, 2, 5, and parts of 3, 4, 6, 7, 8, 9, 10 were answered in (1); Problem 11 was answered in (2). Our present result will settle the remainder of Problems 3, 4. Important parts of 6, 7, 8, 9, 10 remain unsettled. Especially

94

ON RINGS OF OPERATORS, III 95

the following question is still open, and it is in our opinion of a quite decisive character: Are all factors of class (JJ1) isomorphic to each other? (Part of 6.) We think that the answer to this question will greatly affect the applicability of the theory to factors, especially to quantum mechanics.

F. J. Murray and the author believe that the answer is negative, and that further invariants-probably of an algebraical and group theoretical nature exist.

Certain partial results have been obtained, and they will be discussed elsewhere. But the main question (cf. above) is still unanswered.

4. The work (7) of the author also led to factors. The parts of the ring C" in the various incomplete direct products jJn=1,2,--. (,)(nJ) 0 ')(n,2)) ((7), p. 71, Lemma 7.3.1) were found to be in certain cases factors of classes (I.), (II,), and (IL,). (Cf. ibid., pp. 71-77, in particular Lemmas 7.4.1, 7.5.1.) It is not difficult to show that the above-mentioned parts are always (that is, in every 1I&n=1,2,-.. (,)(n,i) 0 ')(n,2)) factors.

Application of our present results, namely of our Theorem IX, shows that these factors are of class (III.) in certain cases.

More precisely: It was shown in (7), p. 71, Lemma 7.3.1, that the part of eCl in 0n=1,2,--- (,)(n,) 0 ')(n,2)) depended essentially (that is, up to isomorphisms) only upon a certain sequence al, a2, ... of constants > 0, < 1. It was also shown ibid., pp. 71-77, that its class was (I.) for a, = a2 = . . . = 1, (II,) for al = a2 =0, and stated that it was (II.) for =3= * = 0, 2 a4 = * = 1. Now we can show this: The class is (III.), if for some 5 > 0 we have for infinitely many n = 1, 2, ... * * X an < 1 -5.

The factors obtained by the above process have various instructive aspects, and will be discussed (together with the above-stated results) elsewhere.

The surmises stated at the end of (7) have to be modified in accordance with the results stated above.

5. For an account of modern operator theory in general, the reader is referred to (3) or (4). For the other topics touched, a general orientation may be obtained from (5), (1). A familiarity with the methods and results of these papers will be assumed.

Our notations are the same as in the papers enumerated above, for a detailed description cf. (1), pp. 126-127.

A detailed table of contents and a bibliography follow.

CONTENTS INTRODUCTION

1. Statement of the main result .................................. 94 2. Norm, normedness, A IE1E21. ... .. 94 3. The Problems 1-11 of (1) ............ ......................... 94

96 J. V. NEUMANN

4. Relation with (7) ............................................ 95 5. Literature and notations ............................... 95

CONTENTS......................................................... 95

BIBLIOGRAPHY..................................................... 96

CHAPTER I. THE RELATIVE TRACE IN AN ARBITRARY FACTOR .97 1.1. Generalities. ............................................... 97 1.2. Definition and properties of Ten (A). (A of finite rank)-

Theorem I ..97 1.3. Definition and properties of [[A]]. (A of finite rank)-Theorem II. 101 1.4. Fundamental sequences, normed A's, [[A]] for normed A's-

Theorem III .105 1.5. Properties of normed A's, of [[A]] for normed A's. Projections-

Theorems IV-V .112 1.6. Meaning of normedness and norm in each class (In)-(III1). 115

CHAPTER II. MAXIMUM ABELIAN RINGS AND THE OPERATION A' I . E21 118 2.1. Construction and properties of AI E'IE21 .-Theorem VI. 118 2.2. Criterion of OTR being of class (III.) .122

CHAPTER III. CONSTRUCTION OF THE RINGS .124 3.1. Generalities ................................................ 124 3.2. S, F, A, 11 = S=SI8,j. The "differentiation-theorem"-Theorem

VII .124 3.3. Relation of a group S to S, Ft. Being free, ergodic, measurable.. 131 3.4. Ad, S = tr. . 134 3.5. Constructions in js...................................... 135

3.6. Constructions in O 'R, R.'. 139 3.7. First discussion of OR, OTh'-Theorem VIII .145

CHAPTER IV. DETERMINATION OF THE CLASSES OF 'Do, 1t.. . 149 4.1. Generalities. ............................................... 149 4.2. g measurable: Cases (I), (II) .150 4.3. g non-measurable: Case (III)-Theorem IX .153 4.4. Examples of Case (III). Not coupled factors.158

BIBLIOGRAPHY

(1) F. J. MURRAY AND J. V. NEUMANN: "On rings of operators," Annals of Mathematics, vol. 37 (1936), pp. 116-229.

(2) F. J. MURRAY AND J. V. NEUMANN: "On rings of operators, II," Transac- tions Amer. Math. Soc., vol. 41 (1937), pp. 208-248.

(3) M. H. STONE: Linear transformations in Hilbert space, Amer. Math. Soc. Colloquium Publications, vol. XV (1932).

(4) J. V. NEUMANN: "Allgemeine Eigenwerttheorie Hermitescher Funktional- operatoren," Math. Annalen, vol. 102 (1929), pp. 49-131.


(5) J. V. NEUMANN: "Zur Algebra der Funktionaloperatoren," Math. Annalen, vol. 102 (1929), pp. 370-427.

(6) J. v. NEUMANN: "t~ber Funktionen von Funktionaloperatoren," Annals of Math., vol. 32 (1931), pp. 191-226.

(7) J. V. NEUMANN: "On infinite direct products," Compositio Mathematica, vol. 6 (1938), pp. 1-77.

(0) F. HAUSDORFF: Mengenlehre. (IJJrd Edition.) W. de Gruyter, Berlin (1935).

CHAPTER I. THE RELATIVE TRACE IN AN ARBITRARY FACTOR

1.1. Let D be a Hilbert space, and 'DR a factor in A. The notation of the relative trace Tqr(A) was defined for all Hermitean A e O1R in (1), pp. 212-213, Definition 15.1.1, and for all A E OR without restriction in (2), p. 218, Definition 2.2.1, provided that ODR belongs to one of the finite classes: (In) (n = 1, 2, * * ), (III). We shall now extend this definition to all factors ODR but (when ODR is infinite) not all A e OR. We shall see, in particular, that when 'DR is purely infinite, i.e. of class (III,), our definition covers A = 0 only. Hence the factors 'OR for which we really achieve something, are those in the not purely infinite classes: (I.), (II.c). (For (I.) in particular, we get the usual notion of the trace and the E. Schmidt class of operators in ODR cf. the discussion (B) in ?1.6 below. Nevertheless, we shall use this extended notion of the relative trace in this paper for a factor OR of unknown class, with the purpose of showing that ODR is just of class (IIIJ).)

1.2. Let OR be an arbitrary factor, D(9) its relative dimension function, in any normalization. ((1), p. 165, Definition 8.2.1, and p. 168, Theorem VII.) For any A e ORl the closed, linear set [Range A] can be formed, and is X ODR. (This notion is defined in (1), p. 141, Definition 4.2.1. If a proof of the statement is wanted, cf. (1), pp. 151-152, Lemmas 6.1.1, 6.2.1.)

DEFINITION 1.2.1. For every A e M we define a "rank" as follows:

Rank (A) = D([Range A]).

(The reader will easily verify that if OR = @ = the ring of all bounded operators in Sb-which is of class (I.) and if D(9J) is chosen in the standard normalization- (1), p. 173, Lemma 8.6.1 -then this is the matrix-rank in its usual sense.)

Together with D(9)), Rank (A) can assume finite or infinite numerical values. When the former is the case, we say that A has finite rank.

LEMMA 1.2.1. Rank (A) has the following properties: (i) Rank (A) > 0 it is = 0 if and only if A = 0. (ii) Rank (A) = Rank (A*). (iii) Rank (aA) = Rank (A) for every complex number a # 0. (iv) Rank (A + B) < Rank (A) + Rank (B). (v) Rank (AB) ? Rank (A), Rank (B). (vi) Rank (E) = D(E) for every projection E e DR.

98 J. V. NEUMANN

PROOF: Ad (i): Rank (A) = D([Range A]) is clearly _ 0, and = 0 if and only if

[Range A] = (0), i.e. Range A = (0), which means A = 0. Ad (ii): Coincides with (1), p. 152, Lemma 6.2.1. Ad (iii): Since A is linear, already Range (aA) = Range (A) if a # 0. Ad (iv): Clearly Range (A + B) C [Range A, Range B], hence [Range

(A + B)] C [[Range A], [Range B]], whence (1), p. 210, Lemma 14.2.3, gives the desired result.

Ad (v): Clearly Range (AB) C Range A, hence [Range (AB)] C [Range A], and so Rank (AB) < Rank (A). Combining this with (ii), we obtain: Rank (AB) = Rank ((AB)*) = Rank (B*A*) ? Rank (B*) =

Rank (B). This completes the proof.

Ad (vi): If E is the projection of the closed linear set 9W, then Range E = 9N, hence [Range E] = 9W.

LEMMA 1.2.2. The finiteness of rank has the following properties: (i) 0 has finite rank. (ii) A has a finite rank if and only if A * has a finite rank. (iii) A has a finite rank if and only if aA has a finite rank, a being any fixed

complex number 5 0. (iv) A + B has a finite rank, if the same is true for both A and B. (v) AB has a finite rank, if the same is true for either A or B. (vi) E has a finite rank if and only if it is finite in the sense of (1), p. 155, Defini-

tion 7.1.1, E being any fixed projection e DR. PROOF: These (i)-(vi) are immediate consequences of the corresponding

(i)-(vi) in Lemma 1.2.1.- Throughout what follows we will make use of the results of these Lemmas

1.2.1, 1.2.2, without referring to them explicitly. DEFINITION 1.2.2. A closed, linear set 9) q it9-or its projection E e Si ((1),

p. 141, Lemma 4.2.1)-"contains" an A e S1t if

[Range A], [Range A*] C P.

LEMMA 1.2.3. E contains A if and only if

EA = AE = A.

PROOF: [Range A] C 9N is equivalent to Range A C 9I, that is, to Af e 9) for every f e I, that is to EAf = Af. In other words: To EA = A.

Similarly [Range A *] C 9 means EA* = A *, that is (applying * to both sides) AE = A.

Hence both together amount to EA = AE = A. LEMMA 1.2.4. A (e i) is of finite rank, if and only if it is contained in a

finite 9) (7 Sit). PROOF: Sufficiency: [Range A] C 91, 9) is finite, hence [Range A] is finite,

that is, Rank (A) = D([Range A]) is a finite number.


NTecessity: A has finite rank, helnce A* has also finite rank, so [Range A], [Range A*] are finite, and with them 9T = [[Range A], [Range A*]] ((1), p. 160, Lemma 7.3.5). This 9 meets all requirements.--

Given a finite 9Y? (, Olt)-and its projection E (E Olt)-then those A e OR which are contained in 9) are by Lemma 1.2.3 precisely the elements which the correspondence X >? X(w) (discussed in (1), pp. 186-187, Definition 11.3.1, Lemmas 11.3.2-11.3.4) maps on the factor Olt(n) in 9) (instead of ! So we see: If A runs over all elements of 9Rt which are contained in 9)?, then A (p runs over all elements of 91t(p) .

The factor (w) necessarily belongs to one of the finite classes: (I.) (n = 1, 2, ... ), (II1) ((1), p. 188, Lemma 11.3.7, or p. 189, Lemma 11.4.3). Hence there exists a unique relative trace in 9,1l(p) defined for all A p e 6R(a) TOR(p)(A(p)). (For the definition of Tl'9t(p)(Ap) cf. the beginning of ?1.1, above.) We now define:

DEFINITION 1.2.3. If A e O91t and of finite rank, then define for every finite, closed, linear set 9? l O91t which contains A (such 9) exist by Lemma 1.2.4)

Ty)R(p)(A) = D(Z) Tu)tp) (A (p)).

(This TqR(p)(A) depends also on the normalization of D(9)), but we do not need to indicate this explicitly.)

LEMMA 1.2.5. The TqR(p)(A) of Definition 1.2.3 depends on A only, and not on 9) (as long as A, 9) are related as described there).

PROOF: In other words: If 9), 9 are both finite, closed linear sets, both -q ODR and both containing A, then we claim that

(1) TqR(a)(A) =Tg(%)(A)*

Consider first the case where C . Consider the operators Au e OR(X) . Form the operator An,% in 9, which is reduced by 9 and which coincides with Ace in 9) and with 0 in 9- 9. We claim that Awlp eOR(%) . Indeed: Ape xOR(p) implies A = A(w) for an A eDR. As E = Pp eOR, we have (EAE)(TI) = A(w), EA E eOR, so we may replace A by EAE, which coincides with 0 in t - 9). That is: We may assume that A itself coincides with 0 in S& - 9). Hence Awls = A(%) EOR(91).

Observe further: (i)* I(w) = E(w) is the unit in 9)?, hence for Y)R(p) . And I(D)I% = E(%) . (ii)* (a-Au),% = a-An,. (iii)* (An + Bn)1% = An,% + Bull. (iv)* If A is definite, then Awls is also definite. (v)* (A*)1w = A* (vi)* (ApBu)1% = Ap1%BpJ%

Now put for every An e DR(an)

(2) t(A u) = D (SR) T ( (A m |) . D(9I)?) %

100 J. V. NEUMANN

Comparing (2) and (i)*-(vi)* with (i)-(vi) in "Property III" in (2), pp. 218-219, we see immediately that t(Ao) fulfills (ii)-(vi), including (iv)', mentioned there (for 9)R() ,91 in place of A, @D). (vi)' is also fulfilled, since it is a consequence of (vi). (Put U-'A, U for A, B.) Let us finally check (i). Con- sidering (2) and (i)*, it becomes

1 - D(9) TM)(E(%)), that is D(91)

(3) TSR(%)(E,?)) = D(%))

Now we know from "Property III," (*), in (2), pp. 218-219, together with (1), p. 219 (end of the first paragraph), that we have for every idempotent

F = Pp, 3 C 91, TS(91)(F(1)) = D'(F(%)) = Y(f,

where D'(S) is the relative dimension function of R() in its standard normalization-that is, with D'(w) = 1. By (1), p. 188, Lemma 11.3.7, D(w) is also a relative dimension function of t(R), but in order to make its normalization the standard one, we must divide it by D(w). So D'(W) = D(3/D(W), and consequently TSRj(v)(F(g)) = D(fl/D(W). Now putting w = 91, F = E gives (3). Thus (i) is also established.

We have thus shown that t(Au) fulfills all conditions (i)-(vi)' in "Property III" in (2), pp. 218-219. Hence by "Property IV" in (2), p. 219, we have TOR (p) (Am) = t(A ). Or, if we substitute this in (2):

(4) D(9))TS(v)(Au) = D(9)TS(%)(AvI4),

Consider now our original A. Since it is contained in 9), so EA = AE = A by Lemma 1.2.3, therefore it coincides with 0 in & - 91. Therefore we have for Am = A(w) also Awls = A( %. By this substitution, however, (4) become

D(9)-T1(), (A(u)) = D(9)TR(,) (Ag()(),

which, by Definition 1.2.3, is precisely (1). This completes the proof for the case where 9 C 91. Consider next the case where W1I, 9 are arbitrary. Then form $ = [9S, 9].

Along with 9)1, 9 this $ will also be D On, and it will also be finite ((1), p. 160, Lemma 7.3.5). We have 9J, 91 C $3, and $ contains A since 9), 9 do. So we have (1) for 9)1, $ and for 9, $ (in place of 9)1, 9), and hence for 9)1, 91 too.

Thus the proof is completed. DEFINITION 1.2.4. Considering Lemma 1.2.5 we can drop the 9) from the

TM (A) of Definition 1.2.3. We shall therefore write for every A e SR of finite rank (in the sense of Definition 1.2.3):

TMR(A) = TI'(a)(A).


We obtain now with ease: THEOREM I. T'f)R(A) has the properties (ii)-(vi) in "Property III" in (2),

pp. 218-219). That is: (ii) ToR(aA) = aToR(A), a a complex number. (iii) TqR(A + B) = TqR(A) + TqR(B). (iv) TDR(A) _ 0 if A is definite. (iv)' TDR(A) > 0 if A is definite and 5 0. (v) TgR(A*) = TR(A). (vi) TqR(AB) = TiR(BA) if either A or B has a finite rank, and both are e 911. (vi)' T9R(U-'A U) = TqR(A) if U (e 'R) is unitary. We have also: (vii) TOT{(E) = D(E) if E (e 'DR) is a finite projection. (Throughout (i)-(vii) remember Lemma 1.2.2.)

PROOF: Ad (ii), (iv), (iv)', (v): They follow immediately from the corresponding statements in "Property III" in (2), pp. 218-219. We must only observe for (ii), (v), that 91 contains aA, A* along with A. And for (iv)', that if 9Z contains A, then A(w) = 0 implies A = 0.

Ad (iii) and (vi), when A, B have both finite rank: In order to deduce them from the corresponding statements of "Property III," as above, we only need a finite XN 'DR which contains both A, B. Let 9M, %1 9'D be finite and contain A, B respectively. Then v = [SI, %] meets all requirements. (For the finiteness by (1), p. 160, Lemma 7.3.5.)

Ad (vi): Since (vi) is symmetric in A, B, we may assume that A has a finite rank. So has BA, choose a finite v1 q 'DR which contains both A, BA (cf. above). So we have for E = Pp by Lemma 1.2.3, AE = A, EBA = BA. Consequently

AB = AE.B = A-EB,

BA = EBA = EB.A.

Now A has a finite rank, and EB has also a finite rank along with E. So our previous result gives T6,(A . EB) = TqR(EB -A), and consequently TqR(AB) - TR(BA).

Ad (vi)': Put U'1A, U for A, B in (vi). U'1A has a finite rank along with A. Ad (vii): Put E = Pf, WI is finite along with E, it is tq Olt, and obviously

contains E. E(p) = I(w) is the unit in 91Y, hence for 911(p). So we obtain from Definition 1.2.3, considering (i) in "Property III," as above:

TqR(E) = D() )T9R(()(E(w)) = D(9)

1.3. Our new relative trace TqR(A) was only defined for the A E 'DR with a finite rank. We now wish to extend its definition to a wider domain of operators A e 9O, with the help of a suitable limiting process.

102 J. V. NEUMANN

We first introduce a norm in the space of all A E t1l, of finite rank, to be denoted by [[A]], which is analogous to the notion of (2), p. 241, Definition 4.3.1, and Lemma 4.3.2 (especially the latter).

DEFINITION 1.3.1. If A E On has a finite rank, then we define the "norm" of A, a number [[A]] > 0, by

[[A]] = (T9R(A*A))' = (T- (AA*)).

(Remember Theorem I, (iv), (vi). A*A, AA* are obviously definite.) THEOREM II. [[A]] has the characteristic properties of a norm in a linear

space. That is: (i) Always [[A]] _ 0, and = 0 if and only if A = 0. (ii) [[aA]] = I a I [[A]] for every complex a. (iii) [[A + B]] < [[A]] + [[B]]. [[A]] also possesses the following further properties: (iv) [[A*]] = [[A]].

(v) [[AB]] { III A 111 [[B]], where only B need be of finite rank, (v <A] B i Il[[A]], where only A need be of finite rank.

(vi) l T9i1(AB) l < [[A]]. [[B]], where A, B must both be of finite rank. (vii) [[E]] = (D(E)) , where E is a finite projection.

PROOF: We prove these statements in a somewhat changed order: Ad (i): Immediately by Theorem I, (iv), (iv)' if we recall that A*A = 0 implies A = 0.' Ad (ii): Immediately by Theorem I, (ii). Ad (iv): Immediately by the form of Definition 1.3.1. Ad (vii): Immediately by Theorem I, (vii), since E*E = E' = E. Ad (v), first inequality: Put I A III = a, then a21 - A *A is definite,2 hence B*(a2I - A*A)B is definite.3 Now

B*(a2I - A*A)B = a2B*B - B*A*AB = a2B*B - (AB)*(AB),

and so this expression is definite too. Therefore Theorem I, (iv), gives, together with (ii), (iii), eod.:

0 < T(PI(a2B*B - (AB)*(AB))

= a R~f(B*B) - TOR((AB)*(AB)) = a2[[B]]2 - [[AB]]2,

[[AB]] ? a[[B]] = I I A I II[[B]].

Ad (v), second inequality: Combine the above result with (iv):

[[AB]] = [[(AB)*]] = [[B*A *]] < ? I I B* I I I [[A *]] = 41 !l B III .[[A]] I 11 Af 112 = (Af, Af) = (A*Af,f) = 0, 11 Af 11 = 0, henceA = 0. 2 ((a2l - A*A)f,f) = a2(f,f) -(A*Af,f) = a2(f,f) -(Af, Af) = a2 1I f 112 -11 Af !12 > (.

3B*CB is always definite along with C:

(B*CBf, f) = (CBf, Bf) _ 0. 4 11! B* III = III B I I, cf. (5), p. 384, footnote 36.


Ad (vi): Considering (iv), we might as well prove

(5) I TqjR(A*B) I < [[A]] [[B]].

In what follows we shall make use continuously of Theorem I, (ii), (iii), (v), and of (ii) above:

0 ? [[A - B]]2 = T -t((A-B)*(A - B))

= Tqj(A*A + B*B - A*B- B*A)

= Tqjj (A*A) + Tc01t(B*B) -T jt(A*B) -Tqjt(B*A)

= [[A]]2 + [[B]]2 - Tcijt(A*B) -Tt(A*B)

= [[A]]2 + [[B]]2 - 29 T~qj(A*B),

T1'qNf(A*B) < '[[A]]2 + 4[[Bfl2. 1

Replace A, B by aA, - B, where a > 0; then this becomes a 2

TT()1t(A*B) < a

[[All + 1

[[B]]2. - 2 ~~2a2 The greatest lower bound of the right side for all a > 0 is [[A]]. [[B]], hence

Now replace B by e"aB, a real, then this becomes

T(e'c'TqR(A*B)) < [[A]] [[B]].

The maximum of the left side for all real a is I Tcpjt(A*B) 1, hence (5) results. Ad (iii): Use Theorem I, (iii), and (iv), (vi) above:

[[A + B]]2 = T6)R((A + B)*(A + B)) = T61jt(A*A + B*B + A*B + B*A)

= T)pjt(A*A) + T')fl(B*B) + T6)jt(A*B) + TQJR(B*A)

< [[A]]2 + [[B]]2 + 2[[A]] [[B]] = ([[A]] + [[B]])2,

[[A + B]] < [[A]] + [[B]].

Throughout what follows we shall make use of the results of thcse Theorems I and II without referring to them explicitly. Theorem II, (i)-(iii), give us also the right to use [[A]] as a norm and [[A - B]] as a distance, without any further comment.

In III A III and [[A]] we have two different numerical evaluations for the size of A. It is therefore desirable to evaluate each one in terms of the other. This is only feasible to a very limited extent, and is expressed by the Lemmas 1.3.1, 1.3.2, which follow. (Cf., however, B.3 in (B) ini ?1.6, when 'ORJ{ is in class (I.).)

104 J. V. NEUMANN

LEMMA 1.3.1. [[A]] _ (Rank (A))' |||A I|I l. PROOF: Put 9) = [Range A], E = Po. Then A e DR implies X7 IDR,

E e 9OT (cf. Definition 1.2.2), and D(E) = D(93Q) = D([Range A]) = Rank (A), [[E]] (D(E))' = (Rank (A))'. Every Af belongs to 9)1 so EAf = Af, that is EA = A. Hence

[[A]] = [[EA]] _ [[E]]-III A II = (Rank (A))'.*II A jj.

LEMMA 1.3.2. If [[A]] < be with 6, e > 0, then there exists an E e 9OT with D(E) < e2, so that 11 A(I-E) Ii < d.

PROOF: A*A is Hermitean and bounded, so it has a spectral form

(6) (A*Af, g) = f Xd (E(X)f, g.

(This is an application of the well-known spectral theorem for bounded operators, cf. e.g. (5), pp. 389-390, footnote 42, and p. 418, or (1), p. 212, Definition 15.1.1. The Stieltjes integral converges, cf. eod.) E (X), - 00 < X < + 00

is a resolution of unity (cf. eod.). Put f = g in (6), since (A*Af, f) =

(Af, Af) = 11 Af 1122 (E(X)f, f) = I IE(X)f 112, we then obtain

(7) I Af 112 = Xd(IIE(X)f112). at

For Xo < 0 put f = E(Xo)g. Then the right side in (7) becomes

= JLXOX d(lj E(X)g12) < xfd(I IE(X)g 112) = X 11 E(o)g 12.

The left side of (7) is _ 0, hence Xo < 0 gives I I E(Xo)g 112 < 0, SO I JE(xo)g 1 = 0 E(X\o)g = 0. That is:

(8) X < 0 implies E(X) 0.

Now (7), (8) give for f = E(52)g: 52 rz2

II AE(62)g 112 = fXd(2 E(X)g 112) _ 62 J d(j1 E(X)g 112)

= 62 IIE(62)g 1<62< 5jg 112,

IAE(62)g l< a jgfI,

that is

(9) IIIAE(52) III <5.

On the other hand we have by (7), (8)


({ A*A - 6(I -E(62)) }f, f) = (A*Af, f) - a 11 (I- E(6))f 12

= f Xd(j IE(X)f 12) -_2fd(jI E(X)f 1 2)

>.f Xd(I E(X)f 112) -622 d(I IE(X)f 112)

00

= J (X _ j2) d(I IE(X)f 112) _ 0. 2

So A*A - 62(I -E(62)) is definite, hence

[[A]]2 - Tc'JR(A*A) ? 62T Tj V (I- E(62)) = 62D(I - E(62)).

But [[A]]2 ? 62C22 consequently

(10) D(I - E(6)) _ 62

So (9), (10) state that E = I - E(62) meets all our requirements.

1.4. DEFINITION 1.4.1. A sequence of operators A1 2 A22 ... is called "regular" if it possesses the following properties: (i) All A. e O. (ii) All An have finite ranks. (But their ranks need not be bounded.) (iii) The numbers I A1 11, III A2 111, *. . are bounded.

DEFINITION 1.4.2. A regular sequence A1 2 A2, .** is "fundamental" if

lim [[Am - A]] = 0. m n -boo

DEFINITION 1.4.3. Two regular sequences A1, A2, ... and B1, B2 , ... are "equivalent" if

lim [[An- Bn]] = O. n --,c

In all our considerations about operator-convergence which will follow, we shall use strong convergence. This is the definition (cf. (5), pp. 381-382):

DEFINITION 1.4.4. A sequence A 1, A2, . . is "strongly convergent" to the limit A, in symbols

lim str An = A) n -boo

if these conditions are fulfilled: (i) The numbers I I A1 1 K A2 111,... are bounded. (ii) For every f e &

lim 1Anf-Af 1=O m, n -

106 J. v. NEUMANN

We shall use only the most obvious properties of strong convergence. (Cf. (5), pp. 382-384. In Definition 1.4.4 above, (i) is a consequence of (ii)-cf. (5), p. 382, footnote 35 but we do not need this fact.)

We proceed now to establish connections between fundamental sequences and equivalence on one hand, and strong convergence on the other.

LEMMA 1.4.1. Let Al, A2, be a regular sequence with

lim [[An]] = 0. n -400

Then it possesses a subsequence Ai , Ai2, * i(in 0o0) with

lim str Ai,, = 0. n -en

PROOF: For every m = 1, 2, ... there exists an no = no(m), such that n _ no implies [[An]] < 1/4m. Choose ii < i2 < ... with in > no(m). Then we have

[[Ai]] <

Apply Lemma 1.3.2 with a = e = 1/2m, then an Em e O)R obtains, with

(11) D(Em) <?m

(12) Aim(I - Em) III < Wr

Put Em = Pn9Jm 91m = [9m,) Tn+l, * ...], Fm = P Then we have also

Fm 6 OR, and (1), p. 168, Lemma 8.3.2 gives 00

D(91.) < E D(TWn)y n=m

00 00

D(Fm) ? < n 1 3 1 n=m n-m 3.4m-1

(13) D(Fm) < 1 =3.4m-1

Also, by construction of the 91m 97m D 91m+1 Fm _ Fm+l that is

(14) T1 D T2 D , F, >! F2 _> *

Hence by (4), p. 77, Theorem 19, we have lim strm-oo Fm = F P911.912. and by (1), p. 169, Lemma 8.3.4,

DW911%2.. 1= lim D(9%,), D1(F) = lim D (Fj). m-.o0 m-o0e

This is = 0 by (13), so F = 0, that is

(15) lim str Fm = 0. n -00


(12) states 11 A itm(I - Em)f II I _ " mf I1, hence for f e - -JRm

(16) || Aijf 1j < 2m Ilf l.

By construction of 91m 9J'm C %.m X hence S& - 9m D 4- , m, and so (16) holds for all f e S) - 1m . Assume finally f E ID - SM, , m > p. Then (14) gives 9M, D Sm so - Z,, C & - XZm, and so (16) holds again.

So if f e - SMX then (16) holds for all m > p. That is: Then

(17) lim 11 Aif =0. m--+x

As (17) holds in every % - M, and considering (15), we see that the domain of validity of (17) is an everywhere dense set in S.5

But the domain of validity of (17) is a closed set, because the Al, A2, are uniformly continuous by (iii) in Definition 1.4.1. Hence it comprises all A. Thus (ii) in Definition 1.4.4 is established for the sequence Ail, Ai2 . (i) eod. follows from (iii) in Definition 1.4.1.

LEMMA 1.4.2. Let Al , A2, * be afundamental sequence. Then lim strom An exists.

PROOF: Since (i) in Definition 1.4.4 follows from (iii) in Definition 1.4.1, we must only prove the validity of (ii) in Definition 1.4.4 for a suitable A. That is: For every f e & the existence of a g E ii (which then is to be Af) with lim~ boo1 A nf-g - - 0. And since & is complete, this is certainly the case if

(18) lim 1I Anf-Anf II = 0 ,n, n --,c

Therefore assume the opposite: That (18) is not true for a suitable f E

So we have anfo e I and a 6 > 0, such that

(19) 11 A fO -A jdo f I > 3 for n 1, 2, * * , where inin -X OC ?

From the fact that A1, A2, X * * is a fundamental sequence, we conclude immediately that Ail - Aj2 , Ai2 - A i2 . is a regular sequence with linO [[A in-A inA] = 0. Therefore we may apply Lemma 1.4.1 to A i- A j,3 Ai2 - A i2, . .. (in place of Al, A2, X.. ). This sequence possesses a subsequence Aikl - Aikl , A ik,2 - A k2 ... (kn -* ), with lim strom (Aikn -A ik.) = 0. Thus in particular limn- II (Aikn - Aikn)fO II = 0, that is

(20) lim Aiknfo - Aknfo I = 0 n -oo

(19) and (20) contradict each other, and this completes the proof. LEMMA 1.4.3. Let Al, A2, ... be a fundamental sequence with lim strom

An = 0. Then it possesses a subsequence Ai,, Ai2, ... (in ct) with

lim [[Ain]f = 0. n -*

6 Consider an aribtrary f e 't. Then fp = (I - Fp)f e - S9p, and limp--. II f - II =

limp.. II Fpf I = 0 by (15).

108 J. V. NEUMANN

PROOF: Since A1, A2, is a fundamental sequence, we have limm,n-* [[Am - A.]] = 0. Hence there exists for every p = 1, 2, . an n0 = no(p) such that m, n > n0 imply [[Am - AJ]] - 1/4'. Choose i1 < i2 < ... with ip no(p). Then we have

[[Ai - Ai,,+]] < - =4v

Apply Lemma 1.3.2 with 5 = E = 1/2', then an E,, ER obtains, with

(21) D(Ev) < 4p,

(22) E (As,- Aj)(I- E) I < -1.

Let us now proceed as in the proof of Lemma 1.4.1. Put E,, = Pop, 91,p = [, p+l, I ...], Fp = P%,,. Then we have again Fm e t, and as there

*000

(F() ) _D (E(,) <

D(Fp < T

(Q -< 4 = 3 .4q-1

(23) D(Fv) < 1 -3.42-1-

By the construction of the 1p, 1, D %1,,?, so 91 D 912 D * , and

(24) - I2 C C - 2 C .

Consider now an f e .-1q. Then we have for every p _ q also f e -91p by (24), hence (I - Ep)f = f. So (22) gives for such an f

-1 A fAip+jf 11- 2P 1l f I.I

If r > s 2 q, then the above inequality may be applied to p = s, s + 1,..*, r - 1.

Adding gives

As*f - Aif || = |2 Az Ap-As f|

r_1 r-1 () Af - Aivf - A 21 I< IL

(25) Ii Aiwj-Air+lf 11 <- 2'- lf .1


Now lim strnx A, = 0 implies lim,_,. II AJf II = 0, and a fortiori 1imr r Airf A I = 0. So if s > q is fixed, and we let r -+00 with r > s then (25) gives:

(26) II AiJf I C 2<L1 jjf ifs > q.

This is true for every f E -91q that is for every f = (I - Fq)g. Substi- tuting this into (26) gives:

||Ai -F 11<--Fq)g j11 1 I - II.

That is:

(27) I Ais(I - Fq) flj < 1

Having obtained (27), we can now proceed to a general evaluation of the An. Since the numbers III A1 1 1 11I A2 1, ... are bounded, by (iii) in Definition

1.4.1, there exists a fixed number a with

(28) IllAnlII a forn=1,2,---.

Hence (23), (28) give:

[[AisFq]] ? II Ai. I * [[Fq]] = I|| Ai. *I (D(Fq))' a (3.4`-1)I

(29) [[Ai8Fq]] < a_______

Consider now any finite projection G E 'R. Then (27), (29) yield:

[[GAiJ I ' [[GAi8FJII + [[GAiJ(I - Fq)]I ? 11G 11 [[AiJFaFq]] + A iA.,(I- F) 111 [[GI]

< 1 * )a + (D(G))' * 1 (3.4q-1)i 8-

a (D (G))'

(30) [[GAl ll < a + (D (G))' ( (3. 4-1)J 28-1

Put q = [Range Aiq], Gq = Pk8 . Again ?$q j DR, G e OR. And always A qfe Eq, GAitQf = Aiqf, that is GgAij = Ai,. So (I - Gq)Aiq = 0, and consequently

(I - Gq)(Ai. - Aiq) = (I - Gq)JA

110 J. V. NEUMANN

Hence, by using our original construction of the il. i2, ,

[[(I- Gq)Ajis] = [[(I - Gq)(Ais- Ai q)]]

? GI - Gq Ih-[[A i- AI] ?1-= 4q 4q~

(31) [[(I -Gq)Aij]] < ifs ? q.

Now combine (30) with G = G. and (31), remembering that D(G9) = D(3) =

Rank (Aiq). So we obtain:

(32) [[Aj + 1 a + (Rank (AX)) if s > q_ +q (3.4q-1) ~ 281 q

1 a Let finally an e > 0 be given. Choose a q = q(e) so that - + <- - -2 4q 3.q-1).! 2

and then an sO = SO(e, q) so, that so > q and (Rank (Aiq) - 2 Then (32)

gives immediately for all s ? so that [[A ij] < e. In other words:

lim [[AiJ]] = 0. s3 --00

Thus the proof is cotlIpleted.-- Before we go on, we observe this: Both Lemnmata 1.4.1, 1.4.3 could have been

easily strengthened by replacing the subsequence A Ai, by the sequence Al, A2, itself. But they suffice as they stand for our immediate purposes, and our ultimate results (Theorem III) yield those strengthened forms immediately.

THEOREM III. (i) For every fundaturital sequwnewc AI , A2, lim str A A, exists, and belongs to ORh. (ii) For two fundamental sequences Al, A2, and B1, B2, the equation

lim str An -- fin str Bn n -too n -boo

holds if and only if the two sequences arc equivalent. (iii) For every fundamental sequence Al, A2, lime SO [[AJ] (numerical!) exists, and it depends on lim stro0 A-, only (and not on the sequence Al , A2, itself!).

PROOF: We prove these statements in a somewhat changed order: Ad (i): Existence of lim strn SO An: Identical with Lemma 1.4.2. Ad (i): lim strnS.O An Ec-TO: Clear, since Al, A2, . OeRi, and en being a ring, is closed in the strong topology. Ad (ii): Sufficiency: Let Al, A2, ... and B1,B2, be two equivalent fundamental sequences. Then Al, B1, A2, B2, ... is also a fundamental sequence; denote it by Cl, C2, -* e . By (i) above lim strn.-0 Cn exists. Hence lim strn_ SOC2n1 -lim strn., Con , that is lin strn., A", = lim strm00 Bn -

ON RING(S OF 0P'U1ArTO1S, III i11

Ad (iii): Existence of limn, [[An]]: Since Al , A2, is a fundamental sequence, we have limmn_ [[Am - An]] = 0. Now I [[Am]] - [[An]] I [[Am - An]],6 hence limmQno ([[Am]] - [[An]]) = 0. Therefore the (numerical) limit lim" N [[An]] exists. Ad (ii): Necessity: We may replace AI, A2, . and B1, B2 X** by Al -B1, A2- B2 ! and 0, 0, . That is: We may assume, that B1 = B2 = ** = 0? So Al, A,2 , . is fundamental, and we assume that lim strn S An = 0. Then Lemma 1.4.3 excludes that lim,-,O [[An]] should exist and at the same time be $0. But we proved just now that part of (iii) which states that limns [[An]]

does exist. Hence limnO [[An]] = 0. In other words: Al , A2 X . and0, 0, .* are equivalent. Ad (iii): lim strn_.0 An determines limn-, [[An]]: We have two fundamental sequences Al, A2, X * -and B1, B2, - with lim strn SO An = lim strod Bn, and we want to prove that lim O [[An]] = limn, [[BnI]. We proved just now the necessity of the criterion of (ii), hence A1 , A2 , and B1 , B2 , - * are at any rate equivalent. That is: limbe, [[A - Bn]] 0 . But I [[An]] - [[Ba]] I < [[A,, - Bn]] (cf.6 above), hence limnr ([[An]] - [[Ba]]) = 0. Consequently limn_O: [[An]] = limn.o [[B.]].

We can now define: DEFINITION 1.4.5. The fact that A1, A2, -is a fundamental sequence

and that limn strn, An = A, will be denoted by

(A1, A2, A.

DEFINITION 1.4.6. We shall say that an operator A is "normed" if a sequence Al, A2, .. with (A1, A2, ...) - A exists. By (iii) in Theorem III, limn-, [[An]] exists for every such sequence, and it depends on A only (not on the sequence Al, A2, * ** itself!). We shall define the "norm" of A, the number [[A]], as this limit: [[A]] = limn,. [[Af]].

REMARKS: (i) Every A of finite rank is normed, since we then have (A, A, * *) - A. This makes clear also that in this case-i.e., when the old [[A]] is also defined-the new [[A]] coincides with the old one. (ii) By (i) in Theorem III every normed A is E 0911. (iii) By (i), (ii) in Theorem III the relation

(Al) A2 , **) A

establishes a one-to-one correspondence between all equivalence-classes of mutually equivalent fundamental sequences Al, A2, -on one hand, and all normed A on the other.-

Throughout what follows we shall make use of the results of Theorem III. Definitions 1.4.5, 1.4.6, and of the above Remarks, without referring to them explicitly.

6Always I [[A]J - [[B]] I < [[A - B]]. Proof: [[A]] = [[B +(A - B)]] < [[B]] + [[A -B II, [A]] - [VB <-I IA -B II. Interchanging A, B gives [[B] - [[A]] ? [IA - B] . Hence (t[tAI - [L[BI) < [[A - B]], that is I [A]] - [B]] I ? [[A - B].

112 J. V. NEUMANN

1.5. LEMMA 1.5.1. (i) (A1, A2, ) A implies (aA1, aA2, ) ~ aA for every complex a. (ii) (Al, A2, * -) PA, (B1, B2, ) Bimply (A? + B,, A2? B2, )

A +B (iii) (AI, A2, ) MA implies (A1B,A2B, ) AB and (BA1, BA2, * )

BA for every B e OR. (iv) (Al, A2, ..) A is equivalent to (A*,A', *) . .. A

PROOF: Ad (i): aA1, aA2, is fundamental along with A1, A2, because [[aAm - aAn]] = [[a(Am - An)]] = I a [[Am - An]]. And lim strom An = A implies lim strn,. aAn = aA. Ad (ii): Al + B1, A2 + B2, *.. is fundamental along with Al, A2, and B1, B2, ... because [[(Am + Bm) - (An + B.)]] = [[(Am -A.) + (Bm-

Bn)]] < [[Am - An]] + [[Bm - Bn]]. And lim strn,0 An = A, lim strn - in, Bn = B imply lim strn---. (An + B.) = A + B. Ad (iii): A1B, A2B, ... and BA1, BA2, - are fundamental along with Al, A2, because

[[AmB - AnB]] [[(Am - An)B]] < III B II -[[Am -A,

[[BAm - BAn]] = [[B(Am - An)]] < ? I B III[[Am - An]].

And lim strom An = A implies lim str.o AnB = AB, lim strom BAn = BA. Ad (iv): Since ** is identity, it suffices to prove the forward implication. Now Al', A2, X... is fundamental along with Al, A2, * because [[A* - A]] = [[(Am - An)*]] = [[Am - An]]. Hence a B with lim strn Ad An = B exists, and we know that we have lim str,0 A,, = A.' Hence we have for every f e i limn0 11 A4 - Af 11 = 0, lim r I A*nf - Bf II= 0. So for allf, g e t limn 0*, (Anf, g) = (Af, g), limb (A*f, g) = (Bf, g). Replacing f, g in the first relation by g, f and application of - gives limn_, (A*f, g) = (A*f, g). Hence (A*f, g) = (Bf, g) for all f, g E Sg. Thus A* B, and consequently lim stro A* = A*. This completes the proof.-

It is now easy to prove the fundamental properties of the class of the normed A and of their norm [[A]].

THEOREM IV. [[A]] (for normed A) has the characteristic properties of a norm in a linear space. That is: (i) Always [[A]] > 0, and = O if and only if A = 0. (ii) aA is normed along with A, and [[aA]] = i a [[A]] for every complex a. (iii) A + B is normed along with A, B and [[A + B]] < [[A]] + [[B]]. The following further statements are true: (iv) A* is normed if and only if A is normed, and [[A*]] = [[A]]. (v) AB is normed if either A or B is normed (but both e 'DR), and

< III A IIIj-[[B]] if B is normed,

[[ IB] B I-II[[A]] if A is normed.

7r Mstr.,.(A*) = (lim strn..-An)* is not a generally valid equation in the sense that the existence of either side implies that one of the other side also, (Cf. (5), p. 382.)


(vi) If Al, A2, is a regular sequence and A is normed, then (A1, A2 ) .,A is equivalent to limn--, [[A - A]] = 0.

PROOF: Ad (i): Choose an (Al, A2 , ) ) A. Then [[A]]= limn-. [[An]]. Since all [[An]] t 0, so [[A]] ? 0. And [[A]] = 0 means limn_.O [[An]]-= 0 that is, the equivalence of Al, A2, . * andO, 0O... But as (Al, A2 *I ) - ) A, (0, 0, *) -) - 0, therefore this means A = 0. Ad (ii), (iii), (iv), (v): Immediately by (ii), (iii), (iv), (iii) in Theorem III, respectively. (In the second part of (v) interchange A and B.) Ad (vi): Sufficiency: Assume (Al, A2, ) -)- A. Clearly (-An, -An, X) - -An Xhence by (ii) in Lemmal1.5.1. (Al-A.,XA2 -A,, X -) -A-An .

Therefore limm 0, [[Am - And] = [[A - An]]. Now Al, A2, ... must be fundamental, so limm ,nO [[Am-AnA]] =O. Since limm.-o [[Am-An]] exists,we have a fortiori lim,. (lim. [[Am - An]]) = 0, limn,. [[A - An]] = 0, that is limn_-- [[A, - A]] = 0.

Necessity: Assume limn_-- [[An - A]] = 0. By (iii) above, [[Am -A]] = [[(Am - A) - (An - A)]] ? [[Am - A]] + [[An - A]]. Hence the above assumption implies limnn_,o [[Am - An]] = 0. So Al, A2, being regular, is even fundamental. Therefore a normed A' with (Al, A2 X ) *-* A' exists. Hence by our above result limbo [[An - A'l]] = 0. Now again by (iii) above [[A - A']] = [[(An - A') - (An - A)]] ? [[An - A']] + [[An - A]]. As

limn-,. [[An- A']] = limber [[An -A]] = 0, this proves [[A -A']] 0. There- fore by (i) above, A - A' = 0,A = A'. Hence (A1, A2, -**) - A.

LEMMA 1.5.2. If (Al, A2, ) o ) A, (B1, B2, ) ') B, then limbo

Tu(AnB n) exists, and it depends on A, B only (not on the sequences A1, A2, and B1, B2, . . . themselves!).

PROOF: It suffices to prove this for limed iT Ti(A nB) and lime DO T6X(AnB) separately. And since the latter obtains from the former by replacing A and the An by iA and the iAn , so we need to consider limn-. S TY) (A nB*) only.

Now

([[An + Bn]]2 - [[A ]]2 -[[B,]]2)

= 21 (T'jR[[A,, + Bn)(A,, + Bn)*) - TV (AnA*) - TV(BnB*))

= 2 (T)R(AnB*) + Tgt(BnA*))

=-2 (OrC(A2'B*) + T9R(BnA*)) = TT6

(33) T T",j(AB*) =2([[An + B ,j12 _ [[An]]2 _ [[B n]]2).

We have (A1, A2, - A,(B1, B2, .**) B, hence by (iii) in Lemma 1.5.1, also (A1 + B1, A2 + B2, ...) - A + B. So lim,,o [[An]] = [[A]], lim,, [[Bn]] = [[B]], lim..w [[An + Bn]] = [[A + B]]. Consequently (33) gives

lim TTR(AnB*) = "([[A + B]]2 - [[A]]2 _ [[B]]2). n --oo

114 J. V. NEUMANN

This completes the proof. DEFINITION 1.5.1. Let A, B be normed. By Lemma 1.5.2 lin1,O T6p(AnB*)

exists for any two sequences. A , A2, * and B1, B2, with (Al, A2, * -*) ' A, (B1, B2, *I ) ' B, and it depends on A, B only (not on the sequences A1, A2, . * .

and B1, B2, * themselves!). We shall define the "inner product" of A, B, the number ?A, B,>> as this limit: <<A, B>> = limn---. T6R(AnB*).-

Throughout what follows we shall make use of the results of Theorem IV and Definition 1.5.1, without referring to them explicitly.

THEOREM V. (i) <<A, B>? is linear in A, conjugate-linear in B, and of Hermitean symmetry in A, B. That is

<<aA, B>> = a<<A, B>>,

<<A + A", B?> = <<A, B>? + <<A, B>>

<<A, aB>? = a<<A, B>?,

<<A, B' + B">? = <<A, B'>? + <<A, B?>>,

<<A, B? = <<?B A>>.

(ii) <<A, A> = [[A]]2.

(iii) <<AD, B> = <<A, BD*>?,

<<DA, B>> = <<A, D*B>>,

where A, B are normed and D e AD. PROOF: All these statements are immediate by Definition 1.5.1, and the

properties of Ti1(X) for X of finite rank. (Remember, in particular, that

TiR(XY) = Tjj(YX) if either X or Y has a finite rank, and both are e We are now in a position to prove: LEMMA 1.5.3. A projection E is normed if and only if it is finite. PROOF: Sufficiency: If E is finite, then it is of finite rank, hence it is normed. Necessity: Assume that E is normed. We want to prove that it is finite,

that is: If F e 'DR is a projection < E, with F - E ( *.. 'DR) ((1), p. 151, Defini- tion 6.1.1), then F = E ((1), p. 155, Definition 7.1.1). So we assume that F has the above properties (excepting of course F = E).

F ' E( ... OR) means F = U* U, E = UUU* for a partially isometric U e- ((1), p. 151, Definition 6.1.1), that is a U with UU*U = U, U*UU* = U* ((1), p. 142, Lemma 4.3.2). Then U = UU*U = EU and F = U*U are both normed, along with E. F < E means FE = F.


Now we obtain, with the help of (i)-(iii) in Theorem V:

K<E, F?2 = K<E, F*F>> <<FE, F>> = <<F, F>> = [[F]]2,

<<F, E>> <<E, F>> = [[F]]2 = [[F]]2,

[[E - F]]2 = <<E - F, E -F>>

= <<E, E>> + <<F, F>> -<<E, F>> -<<F E? = [[E]]2 - [[F]]2,

(34) [[E - F]]2 _ [[E]]2 _ [[F]]2.

Also:

[[E]]2 = <<E, E>> = <<UU*, UU*>> = <<UU*U, U>> = <<U, U>>,

[[F]]2 = <<F F>> = <<U*U, U*U>> = <<U U*U, U>> = <<U U>>,

(35) [[E]12= [[F]]2.

Combining (34), (35) gives [[E - F]]2 = 0, [[E - F]] = 0, and consequently E = F.

Thus the proof is completed.- After this we could determine the normed A completely by investigating the

spectral form of A*A and the projections of its resolution of unity E(X), -c < X < + cc. We shall not do this here, however, because for our further purposes (especially for Lemma 2.2.1) Lemma 1.5.3 is sufficient.

1.6. We want to say a few words about the meaning of our abov e notions of normedness and norm in factors O)R of various classes. The contents of this section, however, are not needed for the understanding of our subsequent deductions.

(A) 6"?)t is of class (Jn) (n = 1, 2, ) or (I1i): UR is in this case of a finite class, hence every 9W - O. is finite, every A E ' has finite rank. Tlit(A) is defined for all A E UR and coincides with the relative trace TUK(A). (We can choose ST = & in Definitions 1.2.3, 1.2.4, and D(T)) in the standard normalization.) Hence every A is normed, and [[A]], <<A, B>> mean the same thing as in (2), p. 241, Definition 4.3.1, and Lemma 4.3.2.

(B) OU is of class (I): We might as well take OR as the ring ?s3 of all bounded operators in St. (Cf. (1), p. 173, Lemma 8.6.1.) Let s(p, Se2, ** be a complete, normalized, orthogonal set in '5 and represent every A e OR = U@ by its matrix:

A {aii}, i, j = 1, 2, ...

ai, = (A~pi Ip,)

116 J. V. NEUMANN

Choose D(9YV) in the standard normalizations; then it is the common notion of dimension. (Cf. loc. cit. above.) Hence Rank (A) has its usual meaning too. We show now:

B.1. (i) If Rank (A) is finite, then aii is absolutely convergent, and TSB(A) = 1 aii (ii) If Rank (A) is finite, then EZ',jo I aij 12 is absolutely convergent, and [[A]]2 = f, j-1 I ai, j2.

PROOF: Ad (i): If Rank (A) is finite, then Rank (A) = D ([Range A]) = k = 0 1, 2, .., hence [Range A] - [sI', . ,'I, 'k] where 4s,.. , 4fk are a normalized orthogonal set. Put El = P[,p, I = 1, * , k, then P[RangeAl = E1 + + Ek, so A = PIRange A]A = E1A + *; + EkA. Each EA has at most rank 1 so it suffices to prove (i) when the rank is at most 1. In this case clearly A =aij, aii = ai~j. If all ai = 0 or all As = 0, then all aii = 0 in which case our statement is trivially true. Hence we may assume that some ai $ 0 and some As $ 0. As every D.0 as; I2 and every fX=1 I ai; I2 must be finite,8 this implies that both Et.1 I ai 2 and Zi= hi are finite. So we have:

to co

0 < I Iai 12, 1 l12 < +0.

This proves in the first place that E12 a- 1 = aiI3 is absolutely convergent (by Schwarz's well-known inequality). Now put B = {bi C = { ci}, where

b ai for j=1! f i for i= 1 bll= O otherwise ' ciij= 0 otherwise

Then BC = A, CB = {dii, where

e ai ii for i,j 1 dii{ i- = 0 otherwise

So CB = (17==1 aiji)E, , where E1 = P[,1] . B, C are clearly of finite rank (= 1). Now we bave (remember D(E1) = 1, hence Tgp(E1) = 1):

TFL(A) = ~1Tg(BC) = Tgi(CB) = (E ai,) TD1(E1) =2 aE3i.

Thus the proof is completed. Ad (ii): Immediately by (i), since A*A {git}, git = Sk akiaki , hence

9ii = Ek=l I aki 2 i2O=l gii = Ek,i=i I aki 2 2

B.2. A is normed if and only if EXy~l I ai, 2 is finite. Again [[A]]2 = DO 2. Sm1j I aii 12.

8 I aij 12 = a. , I (Awij, pj) 12 = f1 Awpi 12, so I aii 12 is finite. The same

results for EI._ asi 12 by considering A* l7a i in place of A = ai .


PROOF: Necessity: Assume that A is normed. Put E. = P(,.*,sj. Then E, E I = 1, hence EIAEn is normed, and [[EAAEJ]] _ [[A]]. Now EnAEn

has even finite rank, along with En, and A - {ais} implies ERAE- {aI~7n}, where

a(n) f= aoi for , j= 1, n..,n %I= otherwise

Hence by (ii) in B.1:

[[EA E2 = E an) 2 = la 12, i,j-1 id-I1

and consequently ESi-l i aij 12 < [[A]]2 for every n = 1, 2, This proves the finiteness of Esi-= I as; 12.

Sufficiency, and value of [(A]]2 Assume that E!ij aI 2 is finite. Form E. as above, put An = ERAE E. Each An has finite rank, and III A =

III EAE, I < I II A II1. Hence the sequence Al, A2, . . is regular. It is also fundamental: If m > n then Am, - { a

' - a ̂) , and

( a {-~ =aii for ij 1,..., but not i j =1, n -2 0 otherwise

and consequently (using (ii) in B.1) 00 n1 f

[[Am =nc42 -A ]])2 =Iai 12 =I aii 12 i,j-1 i~j=1 i,j=1

So for m ->n m ~~~n

[[Am -An]]2 I Mai 12 I lai 12 ij=l ij=1

and so the finiteness of Whirl I aii I2 implies limmn--.. [[Am- An]] = 0. Finally lim strn-<oo En = I, hence lim strn-.. An = -im str >00 EnAEn = A.

Thus we have,

(Al YA2^ A. Therefore A is normed, and (using (ii) in B.1)

X ~ ~ ~~n oo [[A]]2 = ihn [[An]]2 = lim EI a~n) 12 = lim E I ai =

2 I aii 12. n >ea n--*a ^,jl n -o i~jl i1 ~

Thus the proof is completed.- B.2 shows that in this case the class of all normed A coincides with the well-

known class of E. Schmidt. We finally observe that in this case the condition limm,n_,, [[Am - An]] = 0

of Definition 1.4.2 (which implies the existence of limn_-O [[An]] hence the boundedness of the sequence [[Al]], [[A2]], *.. ) implies (iii) in Definition 1.4.1. So in the present case (In) condition (iii) in Definition 1.4.1 is superfluous- which is not true in the case (II), as will be seen in C.1 below.

118 J. V. NEUMANN

Indeed we have: B.3 I11 A I _ [[A]] if A is normed. PROOF: That is: I Af ? _ [[A]] IIf II for every fES . We may obviously

assume IIf I = 1, and then we can choose the complete, normalized orthogonal set spl, 02, . so that spl = f. So we want to prove II Alpl 1 ? [[A]] that is, Il Asp 112 ? [[A]]2. This, however, is obvious.9

(C) 911T' is of class (II): In this case our notions are genuinely new. We only wish to observe that the opposite of B.3 is true, which justifies our formula- tion of Definitions 1.4.1, 1.4.2. (Cf. the above discussion before B.3.) Indeed we have:

C.1. Even for an A of finite rank I A III/[[A]] can be arbitrarily great. PROOF: Given any C > 0, choose a projection E E 'a with D(E) > 0, < 1/C2.

Then E is finite, hence of finite rank. E = 1, [[E]] = (D(E))* ? 1/C, and consequently 1 1 E fI/[[E]] > C.

(D) OR is of class (III): If A is of finite rank, then [Range A] is finite, hence (since ORy is of class (III.)!) [Range A] = 0, A = 0. Hence for every regular, and a fortiori for every fundamental sequence Al, A2, necessarily A, = A2 = ... = 0. Thus the only normed A is A = 0.

In this case, therefore, all the notions we have defined are vacuous.- In spite of this, just these notions will help us to prove that the examples of

factors which we shall construct in ??4.3-4.4, belong in certain cases to class (III.). (Cf. in particular ?4.3, Theorem IX.) The procedure will be essentially the reverse of (C) above: We shall succeed in showing that in those factors 9V? every finite projection E must be 0, the proof being mainly based on the properties of the notion of normedness. From this we then conclude that such an O1lt must be of class (III.). Any direct attempt to prove such a thing will show that of all possibilities alternative to (III) the case (II.) is the hardest one to exclude. It is therefore very appropriate that our notions of normedness and norm are in this case genuinely new.

We also want to point out, that Theorems IV, V show, that the normed operators in OR11 form a (possibly) incomplete Hilbert space. This space is actually complete in the following cases: (I.): Obvious. (I<<,): Clear by (B) above. (III): Clear, since it contains 0 only, cf. (D) above. In the cases (II1), (II) it is not complete, but can be completed to a space Q(9)R), by adjoining to 9O certain (but not all) unbounded operators 7 911. This process was carried out in detail for (III) in (2), pp. 236-242. In this connection C.1 above is significant. We will not go here any further into this subject.

CHAPTER II. MAXIMUM ABELIAN RINGS AND THE OPERATIONS A IlIE 21

2.1. DEFINITION 2.1.1. For any bounded operator A and any projection E we define

A E = EAE + (I - E)A(I - E).

I l Api 112 = 12 by footnote 8, [[A]], = E'. al I aii 12 by B. 2.


Consequently A - AIE = (I- E)AE + EA(I - E).

LEMMA 2.1.1. (i) If A, E e OR then AlE e )1. (ii) If A, E e OR and A is of finite rank, then A IE is also of finite rank.

PROOF: Both statements are obvious. LEMMA 2.1.2. A E commutes with E. PROOF: Results by an obvious computation. LEMMA 2.1.3. If E1, X , En commute with each other, and if (ir, , in,,)

is a permutation of (1, , n), then

AlE 1 . IEn = A I Erl1 1 I... r

PROOF: It is obviously sufficient to consider the case where (iri , rn,,) is just a transposition of two neighbors, say of m, m + 1 (m = 1,.. , n- 1). Then we must prove BIEmIEm+ - BIEmllEm put B = AIR, E,,i, and apply lEm+21 .En to the result. Writing A, E, F for B, Em, Emil we see: We have to show

Al EI = AIFIE

if E, F commute. This, however, results by an obvious computation. COROLLARY. If El, . , E,, commute with each other, then A E1 I

... I En commutes with all of them.

PROOF: Consider an i - 1, ..., n. Let (lri, TO, n) be a permutation of (1, ... , n) with in = i. Then AIEl' .JEn - A IET,1 ... Ern by Lemma 2.1.3, and this commutes with En = Es by Lemma 2.1.2.

LEMMA 2.1.4. I A"E" ,,En _I < II A PROOF: We shall prove III AIElI. En-IEm l < f AJE1... fEl. From this

the desired result obtains by putting m = 1, . , n. Write in the above inequality A, E for AlE,! .Em1i Em. Then we obtain II A E IIl < _II A In other words: We want to prove

(36) 1 (A'Ef, g) I < IIIA IIIIfII Ig I .

Now clearly

(AlEf, g) = (EAEf, g) + ((I -E)A(I - E)f, g)

= (AEf, Eg) + (A(I - E)f, (I - E)g).

Hence

(37) 1 (A'Ef, g) I < Ill A (11 Ef 11 || Eg 11 + 11 (I - E)f 11 11 (I - E)g II).

By Schwarz's well-known inequality

jj EfI1 Eg 11 + 11 (I - E)f 11 1I (I - E)g II

< [(Il Ef 2 --+ 11 (I - E)f 112)(11 Eg 112 + 11 (I - E)g 112)]'

120 J. V. NEUMANN

But

fl Ef 12 + 11 (I - E)f 112 = (Ef, Ef) + ((I - E)f, (I - E)f)

= (Eff) + ((I - E)ff) = (ff) = 11ft11 similarly

Eg 112 + fl (I-E)g - _ |g2 Therefore

11 Ef 11 f 11 Eg I + 11 (I -E)f 11 11 (I- E)g ?l Ilf I11 g 11. And this carries (37) over into (36), and thereby completes the proof.

LEMMA 2.1.5. Assume that A, E1, E2, * O * RThA, A of finite rank, the E1, E2. ... commute with each other. Then

[[AEl..1 l -Em AIE1il* IEnh]2 = [[Ail ...

m]] - [[A 1I'En]]2

if m < n, m, n = 1, 2, PROOF: All Al El2 **..* are of finite rank by Lemma 2.1.1, (ii), so all quantities

which occur in the above equation are defined. We want to prove

(38) [[X + y]]2 = [[X]]2 + [[y]]2

for X = A11...IEn Y = AIEll Em - AIEll.. n

Now

[[X + y]]2 = <<X + Y, X + Y>>

= <<X, X>> + <<Y. Y>> + <<X, Y>> + <<Y. X>>

= [[X]]2 + [[Y]]2 + 2(<<X, Y>>,

hence (38) is certainly a consequence of

(39) <<X, Y>> = 0.

We shall prove (39) for the above X, Y. And since n-I

AIE1I {Em _ A IE1I I.B = E (A IElII...EP - A 1B A+

p=m

so we may prove (39) for X = Al1l I{n Y - A-El1'-..Ep _ A-El'-..|Ep|Ep+

with p < n, p + 1 < n. Let (Xi, * , wrn) be a permutation of (1, * , n) with 7r. = p + 1.

Then Lemma 2.1.3 gives Al 1I . n = AI Erl I.Iern-lee} - AIEril .IEwrnlILEp+l

Write B C, E for AIEr"I...IEwn-l, AlEl' 1IEP, EP+l, then we see: We need to prove (39) for X = BIE, Y = C - CIE only. By Definition 2.1.1, BIE = EBE + (I - E)B(I - E), C -Cl = (I- E)CE + EC(I - E). Hence we need to prove (39) only for the following combinations of X, Y:

(40) fiX = EBE or (I- E)B(I -E) (Y= (I - E)CE or EC(I - E).


Now we have for the various alternatives of (40)

(41) fEither FX =X, FY = 0 for F = E or I-E, for XF =X, YF = 0 for F = F or I-E.

Owing to (41) we now have:

Either <<X, Y>> = <<FX, Y>> = <<X, FY>> = 0, or <<X, Y>> = <<XF, Y>> = <<X, YF>> = 0.

Hence (39) holds in any case. Thus the proof is completed. LEMMA2.1.6. Let A, E1, E2, be as in Lemma 2.1.5. Put An= AE". En

for n = 1, 2, *. . Then Al, A2, - is a fundamental sequence. PROOF: Al, A2, - is a regular sequence: We must verify the conditions

(i)-(iii) in Definition 1.4.1. Ad (i): Obvious. Ad (ii): Immediate by repeated application of Lemma 2.1.1, (ii). Ad (iii): Immediate by Lemma 2.1.4.

Al, A2, * is a fundamental sequence: We must verify further the condition of Definition 1.4.2, that is

(42) lim [[Am - An]]2 = 0. mn-oo

Now Lemma 2.1.5 gives [[Am]]2 > [[A .]]2 for m < n,

[[A1]]2 > [[A2]]2 2 ... > 0.

Therefore limn_-., [[An]]2 exists. Hence

(43) lim i [[Am]]2 _ [[An]]2 .= 0 m,n-+00

But

(44) [[Am - A ]]2 = I [[Am]]2- [[A n]]2 I. Indeed: By symmetry in m, n we need only to prove (44) for m _ n, and then it follows from Lemma 2.1.5.

Now (43), (44) give together (42). DEFINITION 2.1.2. Let A, El, E2, * be as in Lemma 2.1.5. Then there

exists by Lemma 2.1.6 a unique A, so that

(Al, A2, *** ) - A, where An = AIElll.

We shall write

A = AIE1IE21.

THEOREM VI. Let A, El, E2, X** be as in Lemma 2.1.5. Then we have: (i) AIEl1E21I". is normed. (ii) AIB1I B21 - - commutes with E1, E2, .

122 J. V. NEUMANN

PROOF: Ad (i): Obvious by Definition 2.1.2. Ad (ii): Consider a fixed i = 1, 2, . For n > i A n = A I EIII En commutes with Ei by the Corollary to Lemma 2.1.3. But A'ElIE2l = lim str,._o An by Definition 2.1.2, hence AIElIE21 commutes also with Es.

2.2. DEFINITION 2.2.1. A ring S C OR1 will be said to be "purely infinite" if every projection E e 2 is either = 0 or infinite (with respect to OR, cf. (1), p. 155, Definition 7.1.1).-

Observe that the following terms: finite, infinite (cf. loc. cit. above) of finite rank (Definition 1.2.1) normed (Definition 1.4.6) are always to be understood with respect to 'OR. Being purely infinite, however, is defined in Definition 2.2.1 above for any ring 2 c OR. But it has, of course, for V = OR the old meaning of ODR being purely infinite ((1), p. 172, Theorem VIII): That ORl belongs to class (IIIJ).

We prove now: LEMMA 2.2.1. A ring V C Oft is purely infinite, if and only if the only normed

A eisA = 0. PROOF: Sufficiency: Assume that the only normed A E 2 is A = 0. Apply

this to projections E e V'. They must be = 0 or infinite. Hence ?, is purely infinite.

Necessity: Assume that A' is purely infinite. Consider a normed A e '. Then A *A is also normed and also e 2 along with A.

We now proceed as in the proof of Lemma 1.3.2, and obtain the spectral form of A*A, in particular the equations

(6) (A*Af, g) = f X d(E(X)f, g),

00

(7) 11 Af 112 = fXd(lIl E(X)f 12), 00

(8) X < 0 implies E(X) = 0

where E(X), - c < X < + a, is a resolution of unity. (Cf. loc. cit. above.) Consider now a Xo > 0. Form the bounded function

for X?>Xo a(X) iX-?X = 0 otherwise

We can form the operator B = a(A*A) which is characterized by

00

(45) (Bf, g) =La(X) d(E(X)fl g) 00


((6), pp. 202-205), and which belongs to , along with A*A ((6), ). 213, Theorem 6--only the easy, suifficiency par-t of this theorem is nieeded). Now B.A*A =

fl(A*A), where

A(X) = Xa(X) =- otherwise

((6), pp. 205-206, property e) on p. 205); hence

(B .A*Af, g) = I 3 ,(X) d(E(X)f, g) = ((I -E(Xo))f, g)

((6), pp. 202-205), and so

B A*A = I - E(Xo).

B, A*A belong both to If and A*A is normed; hence B.A*A = I - E(Xo) belongs also to !' and is also normed. So we have shown:

(46) X > 0 implies E(X) = I.

(8), (46) and (7) give together I Af 112 = 0, Af = 0 for all f e So. Hence A = 0.

This completes the proof. LEMMA 2.2.2. Consider an Abelian ring dT C OR. There exists a sequence of

projections El, E2, * * * e( Uwith (X = iR(E1, E2, ..). (Cf. (5), 401, lower part of the page.) Since LT is Abelian, these E1, E2, I .. commute with each other. So we can form A I El I E21 ... for every A of finite rank (Definition 2.1.2). ? We consider Cf and E1, E2, - * as given.

We make the following assumptions: (i) The ring R.X Cf'(C (R) is purely infinite. (ii) For every finite projection F(e OR)

F }EIE21 | Es|= 0 implies F - 0.

Then OR is of class (IIIO). PROOF: Let F(E OR) be a finite projection. Then we can form FIE1IE2*.

F EIIE21 .. is normed ((i) in Theorem VI), and so in particular e OR. Further-

more F El I E2 ... commutes with El, E2, *- . ((ii) in Theorem VI), hence with all of C = tR(E1, E2, * .. ) and so it is ECf'. Thus FIE1lE21 -OR ICT'.. Now F ElIE21... is normed and OR (C' is purely infinite by (i), hence Lemma 2.2.1 gives F' ElE21 - 0. And (ii) permits us therefore to conclude that F = 0.

In other words: A projection F e OR is either = 0 or infinite. Hence OR is of class (III.).-

The above Lemma 2.2.2 is the device by which we are going to establish

10 We could show that AIlE1!21... depends only on A and Cf (and not on the sequence El, E2, ... itself). We omit the proof, because this fact is immaterial for our further purposes.

124 J. V. NEUMANN

that some of our examples 6X are of class (IIJo). (Cf. Lemma 4.3.5.) We shall apply it to rings (f which are -= It Lt, so-called maximum Abelian rings in OR1s. The general theory of such rings is of some interest, but we do not propose to go into it in this connection.

CHAPTER III. CONSTRUCTION OF THE RINGS 'DR, 'DR' 3.1. This chapter will be devoted to the construction and investigation of a

family of examples of factors 'DR in appropriate Hilbert spaces A5. Some of these 'D will be shown to be of class (JIII). More precisely: We shall succeed in determining the class of each 911 of this family, and we shall see that all classes (In)-(IIIo) occur among them.

The examples which we shall consider are a generalization of those of (1), pp. 192-209, which belonged to the classes (In)-(IIc) We desire, however, to treat the intervening notion of (generalized) Lebesgue measure in a slightly different way than was done loc. cit. above. Furthermore, this part of our discussion will have to be made on a broader basis than was necessary there. For these reasons we begin with a discussion of the properties of (generalized) Lebesgue measure and integration in an arbitrary space S.

3.2. DEFINITION 3.2.1. Let S be a fixed set. A system r of subsets of S will be called a "Borel-system" if it has the following properties:

(i) 0 (the empty set) e r. (ii) M eF r implies S - M e r. (iii) M1I M2, -... r imply m. + M2 +

LEMMA 3.2.1. For a Borel system r in S we have further:

(i) se r. (ii) M,Ne r imply M - N e r. (iii) M1, M2 ... e rimply M1-M2-. e r.

PROOF: We prove these statements in a somewhat changed order:

Ad (i): By (i), (ii) in Definition 3.2.1, since S = S- 0. Ad (iii): By (ii), (iii) in Definition 3.2.1, since

MIM2 - -=S-((S-M ) + (S M2) + )

Ad (ii): By (ii) Definition 3.2.1 and (iii) above, since M - N = M. (S - N).

DEFINITION 3.2.2. Let A be an arbitrary system of subsets of S. Then there exists a minimum Borel-system r zD A in S," which will be denoted by r(A).

11 The intersection of all Borel-systems r s A in S.


DEFINITION 3.2.3. A Borel-system r in S will be called "separable," if there exists a system A of subsets of S with the following properties:

(i) r = r(A). (ii) If x, y e S and if x e M is equivalent to y e M for all M e A, then x = y. (iii) A is finite or enumerably infinite.-

We can now define our notion of (generalized) Lebesgue measure. (To be called r-measure, cf. below.)

DEFINITION 3.2.4. Let a separable Borel-system r be given in S. A function ,u(M) will be called a "r-measure" if it possesses the following properties:

(i) ,Z(M) is defined for all M e r and only those. (ii) The values of /.(M) are real numbers > 0 < + 00.

(iii) There exists a sequence T('), T (2)* e r with T(') + T(2) + S.. = 8,

such that all ,4(T(t)) < + so.

(iv) If M1, M2, e r and i $ j implies Mai M = 0 then

A (Ml + M2 + * X )-= A(Ml) + A (M2) + *

We shall always consider measures with /h(S) $ 0. DEFINITION 3.2.5. A complex valued function f(x), defined for all x e S, will

be called "r-measurable," if the sets

(x; 9Rf(x) > a), (x; 3f(x) > a) are e r for every real a.-

It is not necessary to develop the properties of the class of all r-measurable functions in detail since they are the same and obtained in precisely the same manner as those of the class of all Lebesgue-measurable functions of one real variable. (Cf. any standard exposition of that subject. For a systematic discussion of such function classes as ours, cf. e.g. (0), pp. 232-247. Our r-measurable function's f(x) are to be characterized in the terminology loc. cit. as follows: Tf(x), Caf(x) are both functions of class (r, *) or, which is equivalent in this case, of class (r, r).) It may suffice to observe that the following functions in particular are r-measurable:

(i) The characteristic function OM(x {-0 oth erw sM of any M e r. (ii) af(x), f(x) along with f(x), a being any complex constant. (iii) f(x) + g(x), f(x)g(x), Max (f(x), g(x)), Min (f(x), g(x)) along withf(x), g(x). (iv) f(x) = limn-, fn(x) (if this limit exists for all x e S) along with fi(x),

f2(x), * e

For the r-measurable functions f(x) we now define the notion of r, ju-integra- bility (with respect to the r-measure ,!) and the numerical value of the r, ,U- integral

(47) f f(x) dox (M e r)

126 J. V. NEUMANN

in the usual way. (Cf. any standard exp)osition of Lebesgue-integration. The observations of (1), p. 194, top of that page, apply to the present situation.) There is again no need to enumerate the well-known details. Let us mention this, however: For a general, complex valued, f(x) we restrict, as usual, (47)

to the case where f(x) is even F, y-summable over M that is, where I f(x) I dx

exists and is finite. In such situations, however, where f(x) is restricted to real and _ 0 functions, we shall consider (47) as meaningful for all F-measurable f(x), expressly permitting in this case the value + o also for the integral. This convention too agrees with the general usage.

Throughout the rest of this section, as well as in ? ?3.3-3.7 which follow, we assume that S, F y are given, and also that r is a separable Borel-system. Other 1'-measures v, Pt, v" and groups Ai, which will be considered, may vary.

We construct next a functional Hilbert space over S. DEFINITION 3.2.6. Let !9s = tsar,, be the set of all (complex) valued F-measur-

able functions f(x) (x e S) with a finite f i f(x) 12 do x. Consider f, g as identical

if f(x) g(x), except for an x-set of .-measure 0. Define

(fY, q) = (,fS,g) r;. = ff(x) g(x) d x.

LEMMA 3.2.2. The space ?~s fulfills the postulates A, B, C, E of (4), pp. 64-66. (This includes the existence and finiteness of the integral which defines (f, g),s for f, g e !)s.) Thus I~s is a finite-dimensional Euclidean or a Hilbert space.

PROOF: Existence and finiteness of f f(x) g(x) dx: Literally as in (4),

p. 109. Ad A, B, E: Literally as in (4), pp. 109, 111. Ad C: Same as in (4), pp. 109-111, except that the system of neighborhoods in which occurs there (Q is a metric space which plays the role of our S), is to be replaced by the following sets in S (all e r): T(t)T, i, j = 1, 2, ..., where the T(l) 2)7 ..

2 are the sets from (iii) in Definition 3.2.4, and the T1 , T2, are all sets which obtain by any (finite) number of applications of the operations M - N M + N, M . N to the elements of a finite or enumerably infinite basis A of F in the sense of Definitions 3.2.2, 3.2.3.-

It may be well to recall at this stage the examples a) and b) of (1), p. 193. (The sets T(l), T, given there serve both for (iii) in Definition 3.2.4, and to construct A, F by A = (T(1), T ..2 - ), = F(A). F turns out to be in the example a) the system of all sub-sets of S and in the example b) the system of all relative Borel sets in S in the usual topological sense.) The example a), with a finite set S = (xl, ... , x,) shows in particular that i;s may actually be a finite-dimensional Euclidean space.

We conclude this section by proving the. "differentiation-theorem" of Lebesgue-Nikodym (Theorem VII below). It would be possible to carry out

ON Ri1NGS OF OPEl{ATrORS, III 127

most of what we want also without this theorem, but such a procedure would necessitate qualifications and restrictions in the hypotheses of the lemmas and theorems which follow.

LEMMA 3.2.3. Consider two F-measures y, v in S. Then we have:

(i) If 4(M) = 0 implies v(M) = 0 then there exists a r-measurable function K(X) > 0, such that for all r-measurable (complex) functions f(x) and all sets M e r

fM(x) d, x = If(X)K(X) dx.

(This is meant to include that if either side is defined and finite, then both are.) (ii) The K(X) of (i) is unique, up to x-sets of 4-measure 0. This is even the case if (i) is restricted to f(x) -1, M e F. (iii) The equivalence of ,4(M) = 0 to v(M) = 0 is equivalent to having always K(X) > 0 in (i), except for an x-set of 4-measure 0.

PROOF: Ad (i): Consider the T('), T(), of (iii) in Definition 3.2.4 first for 4: T") T and then for v: T ( . Let T T be the ' ' i, j = 1, 2, ** *, arranged as a sequence. They will then do for the

F'-measure defined by

(48) p(M) = ,(M) + V(M) (M e r)

in the sense of (iii) in Definition 3.2.4. The same is true for T('), T(2). if we define TW = T - (T?1 + + T 'l), i = 1, 2, . And these have the further property that i $ j implies To) T() = 0.

Let us form

(49) L(f) = fAx) dx for any f c As, rp. o00

(Observe: v in the integral, but p in Isr p!) Since

I f(x) 12dx _ lf(x) 12dx = f If(x) l dx = IIf 112

and

1 dx = (l'(o))

are both finite, the integral in (49) is (by Schwarz's well-known inequality) existing and finite, and even

I Li(f) I = Lf(x) d x _ [L I f(x) 12 d, x vl 12 d x]

(0 [(TL())) I | C| f | (C r [ p

(50) i Li(-") I < CiX ||I f I|s IS, p (Ci = [V(T"o))1*).

128 J. V. NEUMANN

Li(f) is by (49) a conjugate linear functional in As, r, and by (50) a continuous one. Hence a lemma of F. Riesz applies to it (cf. e.g. (4), p. 94, footnote 52): There exists a Kit E Br , such that Li(f) (K , f) that is

fT() f(x) d, = ff(X) Ki(x) dx = ff(x) K, (X) dK + J ff ) K,(X) dx.

Consider now a r-measurable f(x) > 0 which is = 0 for x 4 Too) and bounded in S. Then f e Isxr,, (because p(T(o)) is finite!), and the above equation may be written

(51) f,, f(X)(1 - K,(x)) dx = f(X) K'(x) dzx.

Since it does not matter in (51), what f(x) does for x 4 T(o we can drop the requirement of f(x) = 0 for x 4 T(o).

Taking the real part of both sides of (51) replaces (since f(x) is real) Ki(X)

by RK'(X). Hence we can assume that K'(X) is real. Now put

{= for x e M f(x) {O ( _ otherwise

first with M = M' = T(o) (x; K'(X) < 0), and then with M = = T(). (x; K'(X) > 1). In both cases f(x) meets the requirements of (51). The left side of (51) becomes > 0 and _ 0 respectively, and so we obtain

f Ki(X) d x > 0, KW(X) do x ? 0.

But K'(X) < 0 for all x e M' and K'(X) > 0 for all x e M"'; hence necessarily ,u(M'),= u(M") = 0. This also implies v(M') = v(M') = 0. Hence we can change K'(x) arbitrarily in M' + M' without affecting the validity of (51). We redefine K4(X) = 0 for x e M + Mti. So we have

(52) 0 _ K'(X) < 1 (X e T~o)

So the integrands on both sides of (51) are ? 0. Therefore if f(x) is merely r-measurable and > 0, we can put

fn (x) = f(x) for f(x) ? n {- 0 otherwise

replace f(x) by fn(x) in (51), and apply limnO: Then (51) for the original f(x) results. In other words: (51) holds for all r-measurable f(x) > 0. (Of course then both sides of (51) may be infinite.)

Now put Ki(x) Then (52) gives 1 -K5(X)

(53) Ki~(x) > 0 (X e 1()


We can replace f(x) in (51) by f(X) ; then (51) becomes: 1 -Ki(X)

(54) Lf x) dv x = f(x) Ki(X) dx,

again for all F-measurable f(x) _ 0. Now define K(X) - Ki(X) for x ET") i = 1, 2, . Then (53) gives

(55) K (X) 0 (X ES).

And adding (54) over all i = 1, 2, gives

(56) f (x) dV x = ff(x) K(x) d x.

Consider now any F-measurable (complex) f(x). Apply (56) to i f(x) I (which

is _ 0): It shows that if either of I f(x) I dox, f f(x) K(x)dx is finite,

then both are. That is (f(x) is complex!): If either of ff(x)dvx, ff(x)K(x)dcx

is defined and finite, then both are. Combining this with the validity of (56) for F-measurable f(x) ? 0 shows: (56) holds for all r-measurable (complex) f(x), this being meant to include, that if either side of (56) is defined and finite, then both are.

f= f(x) for x e M~1 Now define f'(x) ,M Er. Then (56) becomes (in the

~= 0 otherwise same sense as above)

(57) fAf(x) dvx = ff(x) K(X) dox.

And (55), (57) complete the proof of (i). Ad (ii): Assume that K'(X), K"(X) both meet the requirements of (i), with the

restriction made in (ii). Consider T('), T(2). of (iii) in Definition 3.2.4 for ', and put M/,n = T(t). (X; K (X) < K"(X) < n). Consider the function f(x) 1

and M = M$,n . Then we have v(M',i) = K'(X) d, x = K'(X) d, x.

Since v(M$,,,) ' v(T(")) is finite, we can infer f (K"(X) - K'(X))dgX = 0.

But K"(X) - K'(X) > 0 for x e , n hence we have (M ' n)=O. Put M'= Zin= Mz i = (X; K'(X) < K "(X)), then this gives y (M') = 0. By symmetry in K'(X), K"(X) we have similarly for M" = (x; K'(X) > K'(X)) II(M") = 0. And so for M = M' + M" = (x; K'(X) id K'(X)) again ,i(M) = 0. That is: K'(X)

K"(X), except for an x-set of u-mneasure 0 (namely: M). Ad (iii): Sufficiency: Assume K(x) > 0, except for an x-set of 4-measure 0.

v(M) = 0 implies f K(x)dcx = 0, and since K(X) > 0 for xe M (except for an

130 J. Ir. NEUMANN

x-set of /-measure 0), so ,i(M) = 0. Hence p(M) = 0 and v(M) = 0 are equivalent.

Necessity: Assume that p(M) = 0 is equivalent to (4M) = 0. Put

f(x) _OaM( {-= 1 for xE ill f(X) OMW = 0 otherwise

M = (x; K(X) =0 ). Then we obtain v(M) = K(x)d,,x 0, hence M(M) = 0

(by the above assumption). So we have K(X) > 0, except for an x-set of ,u- measure 0 (namely: AI).

DEFINITION 3.2.7. We denote the unique (up to r-sets of u-rneasure 0!) /\(x)

of (i), (ii) in Lemma 3.2.3 by dp (x).

THEOREM VII. Let j1 be a fixed r-measure in S. T hen the relation

d,

establishes a one-to-one correspondence between all F-measurable functions K(X) >_ 0

(up to x-sets of j,-measure 0!) on one hand, and all r-measures v, for which i(M) = 0 implies v(M) = 0, on the other. The inverse relation is

v(M'1) = f K(X) dx.

PROOF:. Given v(M), as described above, the existence and uniqueness of this K(X) was already stated in (i), (ii) in Lemma 3.2.3 and in Definition 3.2.7. So we must only consider the ease where K(X), as described above, is given.

Consider the ~ ~T, .. of (iii) in Definition 3.2.4, and let T( T, be the T(t). (x; K(X) < n), i, n = 1, 2, arranged as a sequence. They will then do for the F-measure defined by

(58) v(M) = f K(X) dox (AI eP)

in the sense of (iii) in Definition 3.2.4. It is also clear by (58) that ,4(M) = 0

implies v(M) = 0. Hence we can form d (x).

Put f(x) 1, then we have v(M) f dx= f| (x) ddgx. So (58) gives

fd(x)d x= fM(xxd~x (ME r). dm

Hence by (ii) in Lemma 3.2.3 d, (x) -K(X), except for an x-set of M-measure 0.

This, together with (58), completes the proof.


d COROLLARY. (i) W (x) 1.

(ii) d, (x) dp (x) _ dP (x) if pu(M) = 0 implies v(M) = 0, and v(M) = 0 implies d., dy d...

p(M) = 0. PROOF: (i), (ii) are both immediate on the basis of the characterization of the

d, (x) in (i), (ii) in Lemma 3.2.3.

3.3. Let S, rF, u be as in the preceding section. We now describe a certain relationship between S and a group 9.

DEFINITION 3.3.1. 9 is an S-group if it has the following properties: (i) 9 is a "group," that is, a "composition" aco, an "inverse" aol and a "uinit" 1 are defined in 9 with these properties:

(a)-y = a(g3y), aa-1 = a-1a = 1 al = la a.

(So 9 is "associative," but not necessarily "commutative.") (ii) Every a e 9 defines a certain one-to-one mapping of S on itself:

x -xa (x,xa ES).

These mappings "represent" 9, that is

(xa)( = x(a3) (x E S, a, 3 e9).

(iii) 9 is finite or enumerably infinite. One sees immediately (by (ii) above) that xl x and that x >i xa1 is the

inverse mapping to x >? Xa. Wre shall, as usual, denote by Ma the set (xa; x E M) (M, Ma C S). DEFINITION 3.3.2. An S-group 9 is an "S, r-group" if for every a e 9 M E I'

implies Ma e r. Hence if pi is a r-measure in S, then the ia, defined by

Ala(M) - i(Ma) is also a r-measure in S.

DEFINITION 3.3.3. An S r-group 9 is an "S, r, /i-group" (tz a r-measure in S), if for every a e 9 ,A(M) = 0 implies 1.(Ma) = 0.

LEMMA 3.3.1. Let 9 be an S, F, i-group. Then we have: (i) 4(M) = 0 and ic(M) = 0 (that is: 1u(Ma) = 0) are equivalent. Hence we

can form den (x).

(ii) Each one of the following formulae holds, except for x-sets of si-measure 0:

d, (x) > 0,

d,' (x) o (x)-1,

d,, (xa) dn (x) _ (x). da dp da

132 J. V. NEUMANN

(iii) (ta (x) is uniquely characterized, up to an x-set of 4-measure 0, by this property

f f(xa) do (x) d,,x = f f(x) do .

(f(x) is any F-measurable function, M e r, everything in the same sense as in (i) in Lemma 3.2.3).

PROOF: We prove these statements in a somewhat changed order. Ad (i): pu(M) = 0 implies pu(Ma) = 0, and Ap(Ma) = 0 implies pu(Ma. as-) = 0,

p(M) = 0. Hence p(M) = 0 is equivalent to p(Ma) = 0, that is to A,,(M) = 0. Ad (ii), first formula: Immediate by (i) above and (iii) in Lemma 3.2.3.

Ad (iii): According to Definition 3.2.7, do- (x) is characterized, up to an x-set of

p-measure 0, by

(59) ff(x) d.= fx)x(x)d,

(f(x) L'-measurable, M E r). Now obviously f f(x)dx = f f(xa-')dox.

So (59) becomes

(60) f| f(xa') d,,x= f(x) f (x) dx. Ma M di,

Replacement of M, f(x) by Mcai', f(xa) in (60) gives the desired formula. Ad (ii), secondformula: Coincides with (i) in the Corollary after Theorem VII.

Ad (iii), third formula: Two applications of (iii) above show, that d-# (xa) L` (x) do do

satisfies the characterization of do (x) by (iii) above.

DEFINITION 3.3.4. An S., F,-group 9 is "free" if it has this property: If a e q is = 1 then xa = x holds only for an x-set of p-measure 0. DEFINITION 3.3.5. An S, r, pA-group W is "ergodic" if it has this property: If M e r is such that for every a e CP the difference of the sets M, Ma (i.e. the set

(M + Ma) - (M . Ma)) has pl-measure 0, then either ,1(M) = 0 or (t - M) = 0. Cf. the comment in (1), p. 195, bottom of the page, concerning this notion of

ergodicity. LEMMA 3.3.2. An S, r, pu-group Vi is ergodic if and only if it has the following

property: If f(x) is a F-measurable function such that for every a e 9 f(xa) f(x) (except

for x-sets ofp-measure 0), then f(x) a (constant!, except for an x-set of pi-measure 0). PROOF: Sufficiency: If f(x) runs only over characteristic functions

f(x) OM() { 0 otherwise

then this coincides with Definition 3.3.5.


Necessity: Since we may consider ?f(x), lf(x) instead of f(x) we may amsulm(e fAx) that f(x) is real. And since we may consider 1 +f(x) instead of f(x), we may

even assume that

(61) -1 <f(x) < 1.

For any real a the set Ma = (x; f(x) < a) has clearly the property used ill Definition 3.3.5; hence either ,4(M.) = 0 or ( - Ma) = 0. Let 2 be the set of all a with A (Ma) = 0. By (61)-1 e , 14 ,andasfor a b MaC Mb, so in this case b e ; implies a e Z. Hence z possesses a least upper boulnd, say

ao and ao e X, ao + - 2 for n = 1, 2,...

Hence the sets (x; f(x) < ao) = E'--1 Mao-1/n and (x; f(x) > ao) = - Mao+iin) have /h-measure 0, and so has their sum, the set (x f(x) $ ao).

That is: f(x) ao , except for an x-set of M-measure 0.- Our present notion of a free, ergodic, S, I, a-group corresponds to what was

called an ergodic m-group (for S, s) in (1), page 195, Definition 12.1.5, but there the stronger requirement ,4(M) = js(Ma) replaced our present requirement that; 4(M) = 0 imply 4(Ma) = 0.

This situation motivates the following definition: DEFINITION 3.3.6. An S, F, u-group 9 is "measurable" if therc exists a 1'-

neasure v with the following properties: (i) tt(M) = 0 is equivalent to v(M) = 0. (ii) Always v(M) = v(Ma).

We conclude this section by proving two properties of the above notions. LEMMA 3.3.3. The properties of the F-measure v given in Definition 3.3.6, arc

equivalent to the following properties of its K(X) =d (x) (remember Theorem VII!),

always except for x-sets of Mi-measure 0:

(i) K(X) > 0.

(ii) (K(x) d, (x) K(Xa) dT

PROOF: (i) is equivalent to (i) in Definition 3.3.6: Coincides with (iii) in Lemma 3.2.3.

(ii) is equivalent to (ii) in Definition 3.3.6: (x) K-(X) hence clearly kv (x)-

K(xa). So (ii) in the Corollary to Theorem VII gives (by evaluating (x) ill

two ways):

(62) K(X) (X) K(xa) Lda (.

134 J. V. NEUMANN

Now (ii) in Definition 3.3.6 means v = va, that is, by (i) in the Corollary to d K (X) 'Tlheorem VII `(x) = 1. Hence (62) becomes the desired equation

*10

LEMMA 3.3.4. Let li be ant ergodic S, r, ,-group. Then thc I'-measurc v of I)cfinition 3.3.6 is uniquely determined, up to a constant factor a > 0.

1'IOOF: Let v', v" be two such I'-measures. Form their K(X) = d" (x), K(x) =

(.r), iin accordance with Lemma 3.3.3. Then (ii) in Lemma 3.3.3 gives dZ K (X)/K'(Xa') = K'(X)/K (xa ), K (X)/K (X) = K'(xa1)/K"(xa'). Put f(x) =

K (X)/K"(x) and replace a by a-1, then we have: f(x) = f(xa). All this, of course, holds with the exception of x-sets of ,u-measure 0. Now Lemma 3.3.2 gives f(.r) a, K'(X) aK"(X) (ill the same sense) with a constant a. Hence v'(M) =

(IV"(M), and clearly a > 0.

3.4. DFIsrrioN 3.4.1. (i) Let 1' be the set of all (complex-valued) functions f(a) (a E .ts) with a finite LaE I f(a) 12. (Remember that Sj is finite or enumer-

ably infinite!) Define

(f, g) = 2 f(a)g(a). a f C,

(ii) Let $' - : be the set of all (complex-valued) functions Rf(x, a) (x E S, a E VJ) which are r-measurable in x for every a e ci and with a finite

Ea f K) Tf(x, a) I dyx.

Consider ', ',Xas identical if 'F(x, a) YX(x, a) except for an x-set of g-measure O for every a E (L. Define

= ({f-,x))a J {f(x, a) X(x, a) d, x. s s r y ~~~~a f J

LEMMA 3.4.1. (i) The spaces Do, D3 fulfill the postulates A, B, C, E of (4), pp. 64-66. (This includes the existence and finiteness of the sums and integrals which define (f, g)3, (f, XM)d.) Thus 'D, D are finite dimensional Euclidean or Ililbert spaces. (ii) Using the complete, normalized, orthogonal set of all functions

{ =1 for a=ao soao (a)= 0 otherwise |' E

ON RIN(,S OF OPERATORS, III 135

in $S and setting up the correspondence

tf(x, a) < "f(X, &?), k-f(Xj, 2), - >

(al, (7X2. II *** .-titerationrn of VJ, so that the abovc a( rUns ove a, 72 ) i Is 1S

itso'norphic 'vith S ? s in the sensc of (1), p. 135, Definitiorn 2.4.1 and Lernnma 2.4.1.

In what follows we! uset the inore symmetric notation

.f (x,, a) -'< <f(x, a); a e 0 >.

PaROOF: We prove these statement's in a solnewhat chlanlged order. Ad (i): Character of S?3: 1ut f(&n) = In = 1, 2, ( , 2, an eutitner- ation of t3!), and f(a) - I 1, then our definitions coincide with the original definition of a finite dimensional uc(lidean or a Hilbert space ((1), ). 69, lower lpart of the page). Ad (ii): Immediate, by comparing our )efilnition 3.4.1 with (1), p. 135,

Definition 2.4.1 and Lemma 2.4.1. Ad (i): Character of y Considering what we showed about vSW above, an(l

about 's in Lemma 3.2.2, this follows from (ii) above, since 't is isomorphic to

We are now in a position to begin a systematic investigation of certain oper-

ators in 8s and S8.

3.5. Throughout what follows, i will be a fixed S, F, ,u-group. We assume from now on throughout ??3.5-3.7, that cj is free and ergodic, but not that it is measurable.

We begin by constructions in t, . LEMMA 3.5.1. Define the operators:

Fdjgal (i) UgAf(X) [!:a (x)] f(xa) forany a e6. (ii) L,,(.)f(x) _ p(x)f(x) for any (complex-valued) F-measurable, bounded function ip(x). These operators are bounded, Ua is even unitary.

PROOF: Ua is bounded and unitary: By (iii) in Lemma 3.3.1

f I Uaf(X) 12 dx = f f(xa) 12 da (x)d.x I = f(x) 12dx.

So f e !gs implies Uaf e .Ss and II Uaf I I = I f IIs. L,(2) is bounded: We have I Vo(x) I < C for all x c S and a suitable C > 0.

Hence

f L,,(x)f(x) |dx = f I p(x) 2 I f(x) I2dx < C2 I f(x) j d.x.

So f c '~s implies Lz,(,) f e ps I and 11 Lp,()f IIs < C I I f I1I.

136 J. V. NEUMANN

LEMMA 3.5.2. (i) U1 = 1. (ii) Ua-1 = U1 = Ua.

(iii) Ucg = UUjo.

PROOF: We prove these statements in a somewhat changed order: Ad (i), (iii): Immediate by (ii) in Lemma 3.3.1. Ad (ii): (i), (iii) above give Ua-i = Ui'. And since U. is unitary, so U =' U

LEMMA 3.5.3. Let 2 be the set of all operators L,(.) from (ii) in Lemma 3.5.1. Then 2 is a ring, and ? = t'.

PROOF: As ?' is necessarily a ring it suffices to prove S -2'. As LS(x)L,(z) = Lp(,^4(x, yso L,,(,) commutes with every L*(2). As L*(x) = L6x, so Lr,(x) also commutes with every L+(2). Thus L,,(,) e 2', that is, 2 C 2'. So we must only prove S' C 2.

Consider therefore an A e 2'. Then A commutes with every L,(2,, that is ALV(C)f = L,,(c)Af. That is

(63) A (sp(x)f(x)) (p(x)Af(x).

The assumptions are: so(x), f(x) r-measurable, ,o(x) bounded, f f(x) I2d#x finite. (63), as well as all subsequent relations in this proof, holds with the exception of an x-set of A-measure 0.

Consider the T(') T(2) ... of (iii) in Definition 3.2.4; we can replace them by T () ... where T(i) = T( - (T(1) + * + T(t-)), and then we have also that i 5 j implies T(i), T() - 0. Define

ei(x) OT(t)(X) forXe 0 = O otherwise0

Then f ei(x) 2 dx = i.(T(')) is finite, and so we can put f(x) c-(x) in (63).

We obtain:

(64) A(sp(x)e,(x)) (Ae,(x))jp(x).

As ei(x) is bounded, we can also put So(x) - ei(x) in (64), and so we obtain A (e,(x)) (Aei(x))ei(x) that is:

(65) Aei(x) 0 for z 4 T(i)

Now define p(x) = Aei(x) for x e T('), i = 1, 2, * . . Then (65) means

(66) Ae;(x)_=p(x)ej(x),

and consequently (64) becomes:

(67) A(jp(x)ej(x)) _p(x)(p(x)ej(x).

Summing (67) over i = 1, * , n, and writing f(x) for 50(X) E' ei(x), we obtain

(68) Af(x) p(x)f(x),

if f(x) is bounded and = 0 for x 4 T(1) + * + T for any n = 1, 2, ...

ON RINGS OF OPERATORS, III 1.37

A is bounded; so I Af 11s C.1I f IlI for a suitable C > 0. Put M, = (x; lp(x) i > C + 1, x e T(') + + T (n)) n = 1, 2, We may use

f(x) _ l(X) {_ 1 for xe M 0 therwise

in (68). Then we have:

A ~112 = I AO1n(x) 12dAx = I p(x) 12d, x

? f (C + 1)2dyx = (C + 1)2 (M,)

11 OM,, di = f |MA(x) I2dx = f cx = bi~~~~~~~n

So II A Om. I Is C II Om. I Is gives u(Mn) = 0. Consequently for M = E = (x; I p(x) I > C + 1) also A(M) = 0. Hence we have (with the usual exception)

(69) i p(x) I _ C + 1.

So p(x) is bounded, and we may form L,(,) . And (68) becomes:

(70) Af(x) Lp(z)f(x)

for all f(x) of (68). But these f(x) form an everywhere dense set in bs. In- deed, if any f e S~s is given, then define

e ( \f{=f(x) for If(x)I < n, x e T + * .+ Tn = O otherwise

Then all fn(x), n = 1, 2, * , satisfy the requirements stated after (68), and clearly IiMn-.1 IA - f 118 = 0

As A, L.(,) are both bounded, this means that (70) extends by continuity to all f e 1~s . Hence A = L,(x e S. So we have established 2' C ? too, thus completing the proof.

LEMMA 3.5.4. L,p(.) = Li(,)Ua holds if and only if either a = 1, w(x)-_(x) (except for an x-set of si-measure 0), or a $ 1, j(x) --(x) 0_ (except for an x-set of ,i-measure 0).

PROOF: Sufficiency: Obvious. Necessity: Assume L,0(,) = L#(x) Ua. That is

(71) p(xPfAX) -+(z) Wdy() f(xat),

whenever f f(X) 12 d,,x is finite, (71), as well as all subsequent relations in this

proof, holds with the exception of an x-set of M-measure 0.

138 J. V. NEUMANN

Consider the ""), (2),. of (iii) in Definition 3.2.4. We may put

f(X) OTM('+. +T(n)(X) f = 1 for XET + + = 0 otherwise

in (71), since m(T(') + + T(n)) is finite. Then (71) implies

(72) w(A) --(x) [ W(x)]

for those x for which x, xa e 7T" + * + A Ad suinning over n =1, 2, gives (72) for all x.

(72) settles the case a = 1, owing to (ii) in Lemma 3.3.1. Assume therefore a $ 1. Substituting #(x) from (72) into ( 71) gives cp(x)f(x) -(x)f(xa), that is

('73) fi (x) 12 f(x) - f(xa) 12 d.x = 0.

Now consider the system a of Definition 3.2.3. We have A = (Ml 1112, *2) by (iii) eod. We may put

f() ~~~~= for x eT A'i.M f(x) VT(%),i(x\ = 0 otherwise

i, j = 1, 2, , since jA(T7 .Mj) -A(T'(i)) is finite. Now clearly

f = O if neither or both x, xa E T() *.1j1 j\X -gxa I= 1 otherwise

and the latter x-set is

Kii = ((7T(i') A M) + (T('i Mj)a') - ((7(i) . MA) ) (T(') -.)

Hence (73) becomes

(74) f l(x)I2d',x = 0.

Consider now K = Fi Kij. If x = xa, then never x e Kij, and hence not x e K. If x $ xa then we know from (ii) in Definition 3.2.3 that for somle j- 1, 2, . one, but not the other of x, xa is e Mj. The one e M, is also e TGi for some i = 1, 2, ... , hence one, but not the other of x, xa is e 7'" . Mi. This means x e Ki,. Consequently x e K. So we see

(75) K = (x; x $ xa).

Since .3 is free, and as a $ 1, so we have ly Definition 3.3.4, x $ xa, except for an x-set of ;A-mneasure 0. That is

(76) ,A(S - K) = 0.


Summing over i, j = 1, 2, . converts (74) into fI $(x) I2dyx = 0, and so,

by (76) f i(x) '2 dx = 0. That is

(77) so(X) 0.

So we have for a * 1 by (72), (77), Sp(x) -i(x) 0, and the proof is completed.

LEMMA 3.5.5. Let ?? be as in Lemma 3.5.3, and let 6t( be the set of all U,, a e 9. Then t' 611' = (aI) (a any complex number).

PROOF: alI belongs obviously to 611', and as aI = La. , so aIe E. '6t', 2.t 611'D (aI). So we need only to prove 2. 1t' c (aI).

Assume therefore A E !t' ql, Then A E 611' so for every a E W UaA = AUa, UaAUA' - A. Also A E 8 A = L-() I, UaAU'f = Lo(,a) , hence Lp(xa) L,,(,,) By Lemma 3.5.4 this means (p(xa) = s(x), except for an x-set of ,u-measure 0. Since t3 is ergodic, this implies by Lemma 3.3.2 that <0(x) = a (constant!) except for an x-set of g-measure 0. Hence A La = al. Thus t' .6l c (aI), completing the proof.

3.6. We now pass to constructions in tS, . LEMMA 3.6.1. Define the operators

(i) Uot( f(x, a) d ? (x)a ) f(xao ,aao) LdA ~ for any ao E

(ii) Viao f(X, a) 1F(x, ao a)J

(iii) W'f(x, a) [dd (x)] 'T(xa', a').

(iv) Lx(I)<f(x, a) sp(x),F(x, a) for any bounded and F-measurable (V) M<,(I)<F(X, ax) _ sp(sxa')Lf(x, a) <p(x) which is defined for all x e S.

These operators are bounded, UaTo V, W are even unitary. PROOF: UaCo VaO, W are bounded and unitary: By (iii) in Lemma 3.3.1

Z f | r;'f(X, a) 12d Z f f (xao, aao) 12 dA? (x)

d.x a eG s ae fo)I2

= E | j'f(x, aa) 2dysx Z JIF(x; a) 2d x, af e (8 a e ( s

and

I i WTf(x, a) I2d x = Z f I (ra-'. a-1) 12 - (x) d,,x d

= ? |I 1 fS(srcQ.1) l-dpX Z ? | -f1#(X a) 12d a. Sr et 8 a f 0 S

140 J. V. NEUMANN

And obviously

Z f I 7ao2F0(x s)I2daIx I E f(X7a-1a)I'dpx I =f(x, a) 12d x. af S aea g

So Je tP implies Uco , {# . V F0f7 W fE e and 1Uao|= |ao t =

_ _

L,,(2), M,,() are bounded: We have j so(x) I < C for all x e S and a suitable C > 0. Hence

? L| 4y(c)ff(x, a) l2dx = Z f I(x) 12( a) 192dx ated aeg

< C2 Ce Y(a) 12dyx,

t et S

and

EI f I M"(,)(X, a) 12d = x f o(xa') 12 Y(x, a) I2d.x a eW 8a e3

?C- ? fi(xa)12dpx.

So 5Ye tO implies.L,,(c)f, MRW yFe 'f, and I i, 1 x MJcIFIF, ? C 11 I119. LEMMA 3.6.2. (i) U1 = I. (i)' V1 = I.

(ii) Ua l = UTao = Uao (ii)' Va 1 = 7ao = fa0

(iii) Uao Ugo = aoo . (iii) VVTao V = Va00o (iv) W = W-= W*

(V) WULaOW = V9aO

(vi) WL,,(X)W =l(v) PROOF: We prove these statements in a somewhat changed order: Ad (iv): We determine W2. One verifies immediately:

W2I(X, a) (x) d-- (xa-1) {F(x, a). By (ii) in Lemma 3.3.1 (with a-', Ld; di

a in place of a, ,) 7 (x) (x ) s a (x) 1, s0 W2Y(, ca) Y(x, a),

that is: W2 = I. Hence WT = -' and since T is unitary, so -= W?. Ad (v): Owing to (iv) we may as well prove WVVGOW = U a. Now

WVaOWJ(X, a) - (x) dWd0 (xa)] 9(xao , aao). By (iii) in Lemma 3.3.1

(with a-1, aao in place of a, )

ON RINGS OF OPEtUATORS, III 141

d (x) (Xa ') d"'o (x), so TWVaW(x, a) d cl d dm a

_[a (x)] $(xao, aao) - Ua0T(XI a),

that is: WVaoW = Ua0. Ad (vi): Owing to (iv) we may as well prove WL,(z) = MR0(I)W. This, how-

ever, is obviously true, since WLP(.)J(x, a), MR,(I)WiF(x, a) are both

Ad (i)', (iii)': Obvious. Ad (ii)': (i)', (iii)' above give Vf-l = V-. And since Vas is unitary, so

V-1-=V~*

Ad (i)-(iii): Considering (iv), (v) above, they result from (i)'-(iii)' above, respectively.

LEMMA 3.6.3. Let g be the set of all Uao and all L), and N the set of all Vao and all M,,W. (Cf. Lemma 3.6.1.) Form for each bounded operator A in S! the decomposition from (1), p. 136, Definition 2.4.2: A - <Aa,,g>, a, A et where every A ag -is a bounded operator in As. (Cf. the decomposition !09 =

0 s !from Lemma 3.4.1. As described there, we replace the indices ts = 1, 2, ... by indices a, 3 E9, as already the complete, normalized, orthogonal set

(Pao, ao e N, cf. loc. cit. above, in tSN has been indexed this way.) Then we have: (i) A e5' if and only if Aag = LXajg-1(xv).

(ii) A e g' if and only if Aa,0 = Lxa-1i()UV-Ia. Here x,(x) must be a (complex valued) F-measurable, bounded function of x for every y e A. We do not determine for which systems x,(x), y e (W, a bounded A satisfying (i) or (ii) actually exist. This, however, is true: (iii) The systems x,(x), y e 9, obtained from all A 5 4' by (i), and those obtained from all A - g' by (ii), are identical. (iv) An A, e 5' and an A2 e g,' which have the same x,(x), 7 e ', by (i) or (ii) respectively, obtain from each other by the involutory automorphism If 2? WY

of 19. PROOF: Remember the definition of the Aa,, ((1), p. 136, Definition 2.4.2,

using our present notations): A <f(x)epa' (a); a E = < Aa' f(); A ft>; that is (if we replace a', a by a, ( so that now x e S and 3 e 9 are the variables, and a e $3 is a parameter): Af(x)(pa((3) = Aa,(x). Then the definitions of Ua,,

Lso(p), W give: If A - <Aa,,,,>, a, ( e H, then

(78) UlAUao <UCJ2Aaa- 1l ,ao Uao>, a, (3 E t3,

(79) WA W a <Ulal,-0> c-b A e

(80) LIW)A ' <Lo(x)Aag>, a, e (3

(81) ALf (z) ~ <AaJ3L<p(z)>, a, (3 e h.

142 J. V. NEUMANN

We now proceed to prove the statements (i)-(iv) in a somewhat changed order:

Ad (i): A e g' means that A commutes with all L.(), L*(-) _ Ua0, Uy C . Since = L ,, -L = Ula- I it suffices to consider the Lp(x), UP(X). That is

(82) Lv(j)A = jLv(x)

(83) U-lAUao = A.

(82) means owing to (80), (81), that all Aa,, O ', that is, by Lemma 3.5.3, A ao 1e A. SO

(84) Aas = Lw.,a(x) y

cua,(X) a (complex-valued) F-measurable, bounded function of x. (83) means, owing to (78), that

(85) Aa,# = UalAaao1,ga61Ua0

Substituting (84) into (85) transforms it into Lwa,,(x) = Lwaai, ra5j(xa-1) that is

(86) Wcao(x) - 0 )

except for an x-set of .t-measure 0. If we put ao = 3, and write x,(x) for w7,j(x), then (86) specializes to

(87) Wajt(X) -0W-1),

except for an x-set of ,u-measure 0. Again (87) obviously implies (86).

So A e 5' is equivalent to (84), (87), that is, to

(88) Aca,g = LXa-01 tx0B)

This completes the proof of (i). Ad (ii), (iv): Since W is unitary, Em Willis an automorphism of St, since

W = W-, W2 = I (by (iv) in Lemma 3.6.2), this automorphism is involutory. This automorphism carries an operator A in into the operator WAW. Hence it carries Uao, L4(x) into VTa MS,(X) (by (iv)-(vi) in Lemma 3.6.2), and so 5 into g. Hence it carries g' into g'.

So the general A2 E g' obtains by forming A2 = WA1W' for the general A1E '. Or (by (iv) in Lemma 3.6.2) equivalently by forming WA1W. Now (88) gives the Aai of the general A1 e 5', hence (79) permits us to find the A ar, of the general A2 g'. We obtain in this way for the latter A aO (89) Aax = Us'Lx,-&1(xft)Ua.


But since Use = U0U# 1 a Lxa-l,(x#) Up = UgLxa.--fi(x) , (89) can also be written

(90) Aa,0 = Lx,-1 #(x)U--l a

This completes the proof of (ii). And we have also established all properties of J fi W'J claimed by (iv). Ad (iii): Immediate by (iv). REMARK: We have also seen (in the proof of (ii), (iv) above), that 1F iWf

carries S into g. And since it is involutory (cf. (iv) above), it also carries g into g.

DEFINITION 3.6.1. (i) We denote the set of all systems x7(x), 'y e X, which occur in (i) or in (ii) in Lemma 3.6.3 (both are the same, cf. (iii) eod.) by Xo . (ii) We denote the relation between an A e 5' or an A e g' and a system x,(x), ,y e 9, from Xo, as described in (i) or in (ii) in Lemma 3.6.3, respectively, in both cases by

A {xy(x);7 'l}.

LEMMA 3.6.4. A(S) - J' .k(1) = 5'. PROOF: Since the involutory automorphism J Hi W7F of St interchanges 5

and I (cf. (iv) in Lemma 3.6.3 and the remark after this lemma), and consequently also 9k(5), g' and A?(gS), g', it suffices to prove Rk(5) = g'.

Proof of AR(S) C g': One verifies immediately, for A = Uao

that Aa,` = U0 otherwise a X and for A = Lp() that

A =0 Lp,(x) for a = A abo= 0 otherwise * So we have A a,# = LX,-(x)UOla

wher x7() =1 for y = ao-oX7x = (p(x) fory = 1 w = otherwise -0 otherwise'

respectively. So A e g' by (ii) in Lemma 3.6.3. We have thus proved A C J'. Since g' is a ring, this implies AR(g) C g'. Proof of 9I(4) D g': Consider any A e g', B e g', then we have by Lemma

3.6.3, Aag = Lxa#-0l(x1-) I Bag = Lwal_0j(x)U0g1a . Put C = AR, D = BA, then (1), pp. 136-137, (iv) in Lemma 2.4.3, gives:

Ca,0 = E A.(,gBa,7 = E LxY0#l(x-1)LWa-lY(x) Uy-a

= EI LXyft-1(X#O1)Wa-ly(X) Ue-Y1a

= EI LXay-. 0- 1(x- 1).O-.I(x) U7

-Y 6 9

144 J. v. NEUMANN

(we replace(d the summatiotn index y by ay71), aind

D a, 1 =E B-y,; AM" = E ]JCO Ip1(l)Ug-,_ybx.,_(xzy-,) 'Y fe,

- en

= I L l(x) Lxa lI (z3- i) Uv3 I .Y

= E LXal(xi-1)a? (z) UO-i1- 'Y e 't

= ~ .,_1- X (:2rXg- 1 ) (z) Ue

(we replaced the summation index y by #3y). Remembering that the order of summation does not matter in the above d E (cf. loc. cit. above), we see from the above equation that Cr Ad = D,,,,o for all a, ,3 e Wi. That is: C = D, or in other words: A B = BA. Since 5' is a ring, and A E 5' arbitrary, this means that B E 5". And as B E J' was arbitrary, so we have g' C g".

As I e g (I = U'), so LR (.) = .4" by (5), p. 397, Theorem 7. Hence we have A' c A(S).

The proof is thereby completed. LEMMA 3.6.5. Put O)R = ,R(4) = a'. Then Ot' = @R((cf) - $1', and OR 9R'

are factors (in the sense of (1), p. 138, Definition 3.1.2). PROOF: Clearly 'DR' = R2(#)' = g', this and Lemma 3.6.4 prove all our equa-

tions. It remains for us to prove that p1, AR' are factors, that is that A e A -ORt' implies A = a4.

Consider an A E OR. O'. Then A E = ' g' and A E OR' = g', hence Lemma 3.6.3 gives that

(91) Acat, = Lxa-l(x-1)

and

(92) A a, = Lwe_ l(z)Up 1a

simultaneously. Comparing (91), (92), and using Lemma 3.5.4 gives: For Wla 5$ 1, that is a 5$ 3: x,,#-t(xF'1) = wa-i#(x) 0, except for an x-set

of M-measure 0. That is: In this case (91) gives Aaf = 0. For 13-a = 1, that is a = j3: xi(x0-3) =w(x), except for an x-set of M-measure

0. Comparing herein ,3 = 1 with the general 3 E9 gives x,(x#-3) =-- xl(x), with the same exceptions. Since P is ergodic, this gives, by Lemma 3.3.2 (with xi(), -1 in place of f(x), a): Xi(x) a, except for an x-set of r-measure 0. Hence (91) gives A,,,, = La = aL.

So we see:

A ,O= aI for a = al= 0 otherwise

that is: A = aL. Thus the proof is completed.


3.7. We derive now the basic rules of algebraical computation in the factors O1)R,, 0'..

LEMMA 3.7.1. Both in R11 and in .OnR' the following rules of comtputation hold: If A {xY(x); y e E, B {It(x); fy e (JI, I {ti.(x); fy e .} (all of them e O, or all of them e OR'!), then we have: (i) If C = aA, then 71.y(x) axy(x). (ii) If C = A*, then -qy(x) xy-i(xr ) (iii) If C = A + B, then -q,(x) xy(x) + tz(x). (iv) If C = AB, then th(x) 3 ae X,((X)ya-1(Xa1). The Eag converges "en mesure" (cf. the precise description of this notion at the end of the proof), irrespective of the order in which the a e 9 are gone through.

PROOF: Considering (iv) in Lemma 3.6.3, it suffices to consider the case where A, B, C e ODR. We prove the statements (i)--(iv) for this case in a somewhat changed order:

Ad (i), (iii): Immediate by (i), (iii) in (1), pp. 136-137, Lemma 2.4.3. Ad (ii): This follows from (ii), loc. cit. above, by the following consideration:

Cad = Ln,,_i(z#-I) Ca, = A;.a = LX#._l(xa-1) , ?71-1(x1-1) Xpa-i(xa-1), that

is t,( e X-)(SY Ad (iv): Apply (iv), loc. cit. above. We have C,# = L Cam =

Eys Ay,#Bcty = Eze LX^ -l(x4-)LLaYl(z1-l) = ?^ye Lxyp- (xp- l) tar-I(x-i) in

the sense of strong convergence, irrespective of the order of the y e W. So we have tta91(@fl)- = t Ly7zl(yB-x1)a-,l(Zy-1). Putting ,3=1 and writing a, -y in place of y, a this becomes:

(93) L,.,(X) = E Lxa(z)t.alr(xa-I)

in the sense of strong convergence, irrespective of the order of the a et. Consider now an M e r with a finite ,p(M), and form

{ = 1 for x eM EM(x) 0= otherwise'

Then OM E Cs , and so (93) implies: If al , a2, a** is an enumeration of X, then

n 2

lim L7y(,) GM - E Lxb-i(x)t a a 7')OM =0.

That is n 2

lim ] f8(X)Om (X) - E2 xai(x ) Oz'x 1)OM(x) dx = 0, n boo .Si=l

n 2

(94) lim () x(x) - a()t-y 1 (xat1) djx = 0 n -oo h ~

146 J. v. NEUMANN

So we have for every e > 0 and M e 1' with a finite ,A(M), owing to (94):

(95) lim A ((x;71(x) - Xa (x)tta'(x&a') > e) M =

That is: j X= xai(x)tzas1(x&a7') converges "en mesure" to q(x). As a&, a2, * was an arbitrary enumeration of I this proves (iv). (Of course, if $ is finite, the convergence considerations are unnecessary, the finite sum E

being simply equal to q,(x).)- We sum up the results obtained so far. THEOREM VIII. Let S be a fixed set, r a separable Borel-system in S (Defini-

tions 3.2.1, 3.2.3), A, a r-measure in S (Definition 3.2.4) and 9 an S, F, u-group (Definitions 3.3.1-3.3.3; hence 9 is, in particular, finite or enumerably infinite), which is free and ergodic (Definitions 3.3.4, 3.3.5). Form the unitary (finite dimensional Euclidean or Hilbert) spaces 1ds, 1Dq, 1Dq (Definitions 3.2.6, 3.4.1).

In St form the operators UT 2, VY , r W. 4 , MI ROW (Lemma 3.6.1). Let g be the set of all Ua0 , L,(x) and g the set of all V,, ,,, . Then we have:

a = A,(g) = (' is a ring, its ' is OR' = %(,IJ) ='.

1Y >? WTY is an involutory automorphism of SAt which interchanges 5, Ak(g), 5', S 'a1 and g, Jk(g), g', OR'. OR, 'DR' are factors. The general element A e 'DR or A e OR' has a representation A { x,(x); 'y e ' } (Definition 3.6.1), for which the rules of algebraical computation are known (Lemma 3.7.1).

PROOF: This is merely a restatement of the essential results of various Defini- tions and Lemmas in ??3.2-3.7. Besides those mentioned in the text, cf. also Lemmas 3.6.4, 3.6.5, (iv) in Lemma 3.6.3, and finally (i), (ii) in Lemma 3.6.3.-

The results and notions enumerated in Theorem VIII will be made use of throughout what follows, without being referred to explicitly.

We conclude this section by two lemmata on certain special operators in OR and 6n)k'. (All -relations in both of them are to be understood to hold with the exception of x-sets of Au-measure 0 only.)

LEMMA 3.7.2. Assume A em % or 9)R', A Ix,(x); -y e }. Then we have: (i) A is Hermitean if and only if x,(x)- X-l(X-Y-') for all -y e A. This implies

that xi(x) is real. (ii) If A is definite, then xl(x) > 0. (iii) If A is definite and xi(x) 0, then A = 0.

PROOF: Considering the isomorphism 'f W{F of O1, 'Dt' (cf. in particular (iv) in Lemma 3.6.3), it suffices to consider the case A e 91t'. Ad (i): A - {x_(x); - e }, A* txi-(xFy-'); - e}. A Hermitean means A = A*, that is x,(x) -x,-(x-y'). For ey = 1 this becomes xi(x) - Xl(x), so Xi(x) must be real.


Ad (ii): Consider an f e ts and define

7Fx )~= f(x) for a = 1 9(xy ca) 0 0 otherwise1

Then ef and as A is definite, (A0, f)O > O. Now one verifies easily that

(AR, i = fxi(x) \f(x) I2dox. So we see:

(96) fxi(x) If(x) 12d, x > 0 for every f e ts.

If M e P, /(M) finite, we can put

f~x) O~x) { 1 for x eM A~)-Xz) {-O= otherwise '

then f e '8,} and (96) becomes

(97) Xi(x) dyx ? 0.

Now xi(x) is real by (i) above. Consider the T"), * * of (iii) in Defini- tion 3.2.4, and put M = (x; xl(x) < 0) . T(t). Then 4(M) ? 4(T(t)), hence it is finite, so (97) holds. And as xi(x) < 0 for all x E M, so 4(M) = 0. Thus xi(x) > 0 for all x E T(') (exceptions as above!). Summing over i = 1, 2, gives xl(x) > 0, unrestrictedly.

Ad (iii): Consider an ao e W3, ao $ 1, and two f, g e 's. Define

r= f(X) for a 1 >F(x, a) ' = g(x) for a =ao

= 0 otherwise

Then f e Sl and, as A is definite, (AfSJ > O. Now one verifies easily, considering x0(x)-e that

(98) (At ~i = f xao(x)f(x)g(x) dx + f xa- (Xao )g(x)f(x) d, (x).

Now by (i), xao(X) =- xO-1(xa-1), so the second term in (98) is the complex conjugate of the first one. Hence (98) becomes

(99) (ASF, iF)t = 29 f Xa(X)f(x)g(x) dox.

By (99), (AM, Jtt > 0 states that

(100) f Xao(X)f(X)9(i) dyx > 0.

148 J. V. NEUMANN

If M e I', A(M) finite, we can put

Xf)= 1 for x E M = 0 otherwise

q(x) -sgn xi(x) for x e M12 {: 0 otherwise

thelnf, g E s and (100) becomes

(101) f I X(x) I d;,x ? 0.

Thus xa.(x) 0 for all x E M (exceptions as above!). Now with the 'Al', 17(2) . . of (iii) in Definition 3.2.4 we may put M = T(t) and sum over i =

1, 2, *.; this gives xa0(x) --0 unrestrictedly. So we have xa(x) 0 for a $ 1. For a = 1 it holds by assumption. Thus

A = O.- The statements which follow will be made for 91O only, but the automorphism

'ff? WS in SO (cf. in particular (iv) in Lemma 3.6.3) extends them immediately to DIV' too.

DEFINITION 3.7.1. (i) We denote the set of all Lp(x) by S.

(ii) For every M F r put E(M) = LxM(z) 7 where Om,(x) {_ o Mtherwis

LEMMA 3.7.3. (i) 2 is an Abelian ring c 'OR, we have

(ii) The projections in 2 coincide with the E(M). PROOF: Ad (i): It suffices to prove 2 = 6X A': Since OR .) ' is clearly a ring,

it proves that Si is also a ring, and implies 2 c Olt, and also 2 C A', that is, that 2 is Abelian.

Proof of S C OR -It': LTV(X) eE 5 c 9k(S) = 'DR, so 2 C OR. Every Lp(x) commutes with every Lp(x), and L4(x) = L, hence it is E2', so i' C 2'. Hence

2 c OR u '.

Proof of 2 D OR Ak': Consider an A E 'DR. A'. Since A E OR, so A {x,(x); -y E S3. Since A E A', so A commutes with every L,(x). Now L,,(x) {<p(x)b; 'y e}, where

= 1 forty = 1 = 0 otherwise|

[=a for a 54 0 12 For any complex a sgn a jaI for So always sgn a ?1 and

__0 for a=O a sgn a = I al.

ON RI.NGS OF OPERATORS, III 149

So L,() A {P(x)x,(x); y E q}, AL<(z) {p(x_ ')x,(a); y e }. E That is

(102) f (X)xW( -(XT))x (X).

Put

- O (X) -= 1 for x EM (P(X) OMW = 0 otherwise

for an M E F, then (102) becomes

(103) XY(x) 0 for x EM, X'yJ 4OM, or x'-y' EM, x OM.

Now let M run over all elements of the finite or enumerably infinite system A in Definition 3.2.3; owing to (ii) eod. (103) becomes, if we sum over all M EA:

(104) x7(x) 0

if x'zea - x that is, if x x'y. Now if y X 1, then since 9 is free, this means that (104) holds without any further qualifications. So x,(x) --xi(x)67 (a, as above), and hence A = LX1(X) (see above), and therefore A E Ad.

Thus we have shown that m)R . ?' c V?, and thereby completed the proof. Ad (ii): That an A E ? is a projection, means this: A = Lp(x) and A =A*

A = A= But A* = L(,(x) , AZ = L (X)2 so it means (p(x) x), o(x)- = That is: sp(x) -0, 1. But these sp(x) are precisely the

o(x)-= M(x) {=1 for xe MI

for all M E F, and so the A are the Lo mCx) = E(M) for all M E P.

CHAPTER IV. DETERMINATION OF THE CLASSES OF 9R1, n)R'

4.1. The essential distinction which has to be made when determining the classes of 91, Y)R'/ is whether the 8, F, it-group r3 is measurable or not. (Cf. Definition 3.3.6.)

Consider first the case when .) is measurable. This means that a F-measure v as described in Definition 3.3.6 exists, or, equivalently, that a function K(x) as described in Lemnma 3.3.3 exists.

In this case it is possible to replace At by v altogether: } is an S, F, v-group as well as an S, F, Au-group, and its being free and ergodic depends only on the

sets of measure 0-which are the same for At and for v. s and Si become

A r ̂ and S' if we replace f(x) and iF(x, a) by (K(x))!f(x) and (K(X))'{f(X, a) s, Ir, v

respectively. That is: This replacement compensates for the changes brought

about by substituting v for At in to and St. And finally (ii) in Lemma 3.3.3

shows that this replacement has also the right effect in the definitions of Uao Vaeo I WI LA(x) I A2x .

150 J. V. NEUMANN

In other words: If 9 is measurable at all, then we may even assume that ju

itself has the properties required for v, especially (ii) in Definition 3.3.6. That is:

(105) 4 (M) = 4 (Ma) = -A(M), di- (x) 1.

This means, however, that for a measurable (C our oR, or' coincide essentially with those of (1), pp. 192-204, and so the exhaustive discussion given loc. cit. disposes of our OR, OR' also in this case. Indeed, it is for the sake of the not measurable 9, to be discussed in ??4.3-4.4 below, that this work was under- taken. (They correspond to case (III.,,), cf. Theorem IX.)

We shall nevertheless, for the sake of completeness, discuss the measurable 9 also, and that without bothering about replacing Au by v. We shall, however, keep this discussion as short as possible, and not enter into questions which could also be answered now: Find criteria for normedness, expressions for TOR(A), [[A]], <<A, B?, etc.

4.2. We assume throughout this section that 9 is measurable, and make use of the (essentially unique) v, K(X) of Definition 3.3.6 and Lemma 3.3.3.

All our results, throughout ??4.2-4.4, will be stated and proved for OR only:

The automorphism He WF of Aft (cf. in particular (iv) in Lemma 3.6.6) extends

them immediately to OR' too. DEFINITION 4.2.1. If A e OR is definite, then we have for A {x (x); -Y e 9}

(by (ii) in Lemma 3.7.2) xi(x) _ 0. Hence we can certainly form the integral

J Xi(X)K(X) dX.

Its numerical value is _ 0, but it may be finite or infinite. We denote this value by T0(A).

LEMMA 4.2.1. We have for A, B, C e OR, A, B definite: (i) 0 <-T (A) _ + 00 .

(ii) T7(A) > OifA s 0. (iii) T0(A + B) = TV(A) + T(B). (iv) C { xy(x); or e 9}. C*C, CC* are both e OA and definite, and

1 (C*0) - TO(00*) E J | X (X) I2 K(X) dx.

(Since all terms in this , 9 are > 0 it is clear what it means.) PROOF: Ad (i): Immediate by (ii) in Lemma 3.7.2.

Ad (ii): Immediate by (ii), (iii) in Lemma 3.7.2 Adl (iii): Obvious


Ad (iv): 0*0, CI* e E6talong with C 'e )R, their definiteness is obvious. We have C {xI(x); oy e i}, C* Ixe--(xe-y'); y e }, hence CC* py(x); y E } with p7(x) = x xa(x)xay-i(x"-V). Hence for y- 1

(106) pW(x) i xa(X) 12. a e

These a . converge "en mesure" in the sense of (iv) in Lemma 3.7.1. But in the Ea e of (106) all terms are _ 0, hence it converges numerically (to finite or infinite values). Hence both limits agree, and (106) holds in the sense of numerical convergence.

Now To(CC*) = f pl(X)K(x) dox, hence (106) gives:

(107) To(CC*) X = I I xa(X) I2 K(X) dx. a ei (1

We replace C by C*, that is xa(x) by Xa-l(xa-'). Then (107) becomes, if we also replace the summation index at by a-:

(108) TO(C*C) = f I Xa(Xa) I2 K(x) dx. a ft 9

Now by (iii) in Lemma 3.3.1 and (ii) in Lemma 3.3.3 we have always L f(xa)

K(x) dx = f f(X) K(X) dox, hence the right sides of (107) and (108) are equal.

Therefore they combine to

(109) TO(C*C) = TO(CC*) = E xa(x) I2K(x) d, , a fe p

which completes our proof. LEMMA 4.2.2. (i) If E runs over all projections e OR, then T0(E) is always

defined, and it is a relative dimension function in the sense of (1), p. 165, Definition 8.2.1. We now assign to the relative dimension function DON (E) this normalization:

DR(E) = T0(E).

(ii) We have for the E(M) of (ii) in Definition 3.7.1., M e F:

D9 (.E(M)) = f K(X) dx = (M).

(iii) There exists an M e P with v(M) 7 0, oo.

152 J. V. NEUMANN

PROOF: We prove these statements in a somewhat changed order: Ad (ii): We have E(M) = LoM(x) where

-=1 for xe M\xJ X= 0 otherwise

Now this means (cf. also the proof of (i) in Lemma 3.7.3) E(M) = LoM(x) {OM(x)6; My e }, where

= 1 for y = 1 = 0 otherwise|

So by Definition 4.2.1.

(110) To(E(M)) Om OM(x>)(x) d}x c f (x) d}x = V (M).

As soon as we shall have proven (i) and thereby established Dqj(E(M)) = To(E(M)), (110) will go over into the formula we want to prove.

Ad (iii): Consider the V'), V') ... of (iii) in Definition 3.2.4. If all v(T(t)) = 0, then summation over all i = 1, 2, ... would give v(S) = 0, hence equivalently Au(S) = 0, which is not the case. Hence there exists an i = 1, 2, * with v(T(t)) 5 0. Since we have also v(T(t)) # a, this completes the proof.

Ad (i): Since every projection is definite, therefore for every projection E e O1R To(E) is defined (Definition 4.2.1) and > 0, < + X ((i) in Lemma 4.2.1). And To(E) # 0, oo does occur: Put E = E(M) for an M from (iii) above, and use the already established part of (ii) above, (110): T0(E(M)) = v(M) 5$ 0, o.

Therefore the character of T0(E) as a relative dimension function can be established by (1), p. 170, Lemma 8.3.5: It suffices to show that T0(E) fulfills the conditions (ii), (iii) in (1), p. 165, Definition 8.2.1.

Proof of (ii) eod.: We must show that E - F(... OR) implies T0(E) = T0(F). By (1), p. 151, Definition 6.1.1, and p. 141, Lemma 4.3.1, it implies E = U*U, F = UU*. Hence To(E) = To(F) by (iv) in Lemma 4.2.1.

Proof of (iii) eod.: We must show T0(E + F) = T7(E) + T0(P) for orthogonal E, F. Now this is immediate by (iii) in Lemma 4.2.1.

This completes the proof. COROLLARY: 'DR belongs to the classes (I) or (II). (( is assumed to be meas-

urable!) PROOF: (ii), (iii) above together show that a projection E e O1R with DOR(E)

s 0, x exists. So OR is not of class (III).- A precise determination of the class to which O1R belongs, when 9 is measurable,

could now be carried out. Owing to the above Corollary the possibilities are these: (In) (n = 1, 2, - .), (I.), (II,), (II.). But since this discussion would coincide in all details with that one of (1), pp. 204-208, we omit it. The reader is referred to that work, and especially to the complete enumeration of the results in (1), p. 208, Theorem XII. (The role of ,u there is played by our v.) That theorem shows in particular that all the cases enumerated above do actually arise when S, i, ,u are chosen suitably.


4.3. Later in this section we shall make the assumption that C3 is not measurable.

Without knowing what the class of OR is, we shall nevertheless use its relative dimension function Dgj(E). (Cf. (1), p. 165, Definition 8.2.1, and p. 168, Theorem VII-it does not matter now in which normalization.) We shall see ultimately, in Lemma 4.3.5, that the class is (III.), and hence the range of Dq(E) consists of 0, X only-but this need not concern us now.

We begin by investigating the Ds0pj(E(M)), M e r (cf. (ii) in Definition 3.7.1). LEMMA 4.3.1. (i) If M1, M2, * * r and i # j implies MiMi = 9, then

Dq(E(M1 + M2 + *.A))- D6Oy(E(Ml)) + DM(E(M2)) + -

(ii) If M er, then for all a e 9

D%,(E(M)) = D91(E(Ma)).

PROOF: Ad (i): One concludes immediately from (ii) in Definition 3.7.1 that E(M) is the projection of the following closed, linear set 9Y(M):

(111) E(M) = P9(M), W(M) = (F; 1F(x, a) = 0 if x 4 M).

Now it is clear that under the above assumptions on M1, M2, * * * U (M1 + M2 + * .) = [9N (M1), I9N(M2), * * *], and the 9N (M1), 9(M2), * * * are mutually orthogonal. Hence (1), p. 168, Lemma 8.3.2. gives D((E(M1 + M2 + ...)) =

D'lt(E(Ml)) + DJX-9(E(M2)) + * * * as desired. Ad (ii): Ua (cf. (i) in Lemma 3.6.1) is unitary and 0 4 C 9k(,) = $RT, and it is

clear from (111) that Ua maps 9)I(M) on 9W(Ma). Hence UQXE(M) is partially isometric and has the initial set 9)(M) and the final set 9)(Ma). So M(M) - 91 (Ma) (. .. O9) and hence Dq(E(M)) = DqX(E(Ma)) by (1), p. 151, Definition 6.1.1., and p. 165, (ii) in Definition 8.2.1.

LEMMA 4.3.2. There are only the two following alternatives: (a) For no M e r is DEyL(E(M)) $ 0, o. (b) There exists a sequence M1, M2, * * F, M1 + M2 + * -S, such that all DqR(B(Mi)) X ?, o

PROOF: Define a subsystem K of r as follows: M e K if and only if there exists a sequence M1 M2 *. * F with M1 + M2 + = M, such that all Djrc(E(Mi)) #0, Oo.

Observe:

(112) M()(2) ... e K imply MW') + MM2) + K.

Indeed: Form the sequence M"), M(t), ... for M(t) (i = 1, 2, * , cf. above), then M3i), i, j = 1, 2, * * , written as a simple sequence M1, M2, . * will do for M( ) + M(2) + ... . Next:

(113) M e K implies Ma e K for every a E i.

154 J. V. NEUMANN

Indeed: If Ml, M2, ** do for M, then Mia, M2a, ** will do for Ma considering (ii) in Lemma 4.3.1. And finally:

(114) If M EK, M C M', 1u(M' -M) = 0, then M' eK.

Indeed: Form the sequence Ml, M2, *** for M, then replace M1 by M1 + (M' - M) and leave M2, M3, *** unchanged: This sequence will do for M'.

Consider now the T"' T 21 ... of (iii) in Definition 3.2.4 for jU. Define:

(115) ai = Least upper bound of (,M(M. T"'); M e K).

For i, j = 1, 2, we can choose an Mi,j with

(116) Mi,i e K, Iu(Mi,j. T"')) > ai -

Write the M ,j, i, j = 1, 2, ... , as a simple sequence M() M * . Then by (112) By; l Mi,j = M(1) +M2) + ***e K, that is

00

(117) M 2 Mj,pEK. ij-1

We have ,(M.T(i)) _ A (Mij .T()) _ a,-i . Since this holds for all j = 1, 2,

and 1A (Mi T(")) does not depend on j, so it implies /u(M T()) _ a, . By

(115), (117) also ,u(M T-')) < a , so

(118) 4(M.T()) = aj.

Now consider an M eK with M D M. Then we have MeT~') D M-T'' and both sets are CTT"), so their u are finite. This, together with (115), (118), gives:

((M -M)* T(') = -T(i) (i)

= ju(M. T(') - u1(M T(i)) _ a, - a, = 0, that is:

(119) ,((M -M)T"i) = 0.

Summing over i = 1, 2, * and restating our hypothesis, we obtain:

(120) If M eK and M D M, then ,u(M-M) = 0.

(117), (113) give for every a e Ma eK. Hence by (112) M + Ma eK (put M()= M, M2 = M(3)* = * = Ma). So M = M + Ma fulfills the hypotheses

of (120), and therefore

(121) UA((M + Ma) - M) = 0.


Replacing a by a-' in (121) gives ,(M + Ma-1) - M) = 0, hence eu(((M + Ma-') - M)a) = 0, that is

(122) /((M + Ma) - Ma) = 0.

Summing (121), (122) gives:

(123) p((M + Ma) - (M.Ma)) = 0.

Since (123) holds for all a e X, and since P is ergodic, we must either have

(124a) g(M) = 0

or

(124b) y (S -M) = .

CASE (124a): Consider any M e r with DqR(E(M)) $ 0, co. Then we see, by replacing the M1, M2, .. of M by M, M1, M2 ..* , that M + M e K. M + M fulfills the hypotheses of (120), hence u((M + M) - M) = 0. This and (124a) give together ,(M + M) = 0, henceuv(M) = 0. Now (ii) in Defini- tion 3.7.1 shows E(M) = 0 and so D6pJE(M)) = 0, which contradicts our assumption.

So we have never DR(E(M)) $ 0, co, which is the alternative (a). CASE (124b): (117), (124b), and (114) give together S E K, which is the

alternative (b). LEMMA 4.3.3. If the alternative (b) of Lemma 4.3.2 holds, then q is measurable. PROOF: For every M e r define

v(M) = D(R(E(M)).

This v is a F-measure, because it fulfills the conditions (i)-(iv) of Definition 3.2.4. Indeed: Ad (i), (ii): Obvious. Ad (iii): Immediate by (b) in Lemma 4.3.2, if we put Ml , M2, for T('), T (2), ...

Ad (iv): Coincides with (i) in Lemma 4.3.1. We now prove the measurability of P by showing that this v also fulfills the

conditions (i), (ii) of Definition 3.3.6. Indeed: Ad (i): v(M) = 0 means D J(RE(M)) = 0, that is E(M) = 0, that is, by (ii) in

Definition 3.7.1, LOM(x) = 0. But this is clearly equivalent to 4 (M) = 0. Ad (ii): Coincides with (ii) in Lemma 4.3.1. This completes the proof.- We next derive a property of the operation A I El I E2 (cf. Definition 2.1.2). LEMMA 4.3.4. Since L C Y.R is an Abelian ring (by (i) in Lemma 3.7.3), so every

sequence of projections E1, E2, * ... * is commutative and hence for every A e q1T of finite rank A 1911lE21

... can be formed and is c.OR (by Definition 2.1.2). PutA = Ix,(x);,y E }, AIEIE2 { X(X);Ye . .

156 J. V. NEUMANN

Then X1(X)--X1 (x), except for an x-set of p-measure 0. PROOF: All x-relations in this proof are to be understood to hold with the

exception of x-sets of p-measure 0 only. We define

(125) AIEll ... lEn = {xe .(x);y for n = 1, 2, . I

and proceed in several successive steps: Proof of xI(x) =X"' (x): We have

(126) AIE1 = E1AE1 + (I - E1)A(I -E).

Now E1 e T, so by (ii) in Lemma 3.7.3, E1 = E(M1) for an M1 E r. That is, RI = LM, (x)I

O(m {- =1 for x eM1 = 0 otherwise

This means, as we know, E1 O M1(x) 6, ;-y c (},

6 = 1 for -y =1 { = O otherwise

This and A ,x (x); -y E I give, owing to (126):

A I[ OM,(X)OM,(X)y' ) + (1 - Om ,(x)) (1 - OM,(x-y )]xe(x); -y E }.

Hence by (125):

(127) xe' (x) =[OMJ(X)OM,(x7y) + (1 - OMJ(x)) (1 - OM1(xy-'))]X(x) .

Put -y = 1 in (127); then, since Om(x) = 0, 1, it goes over into X(1)(x) Xi(x). Proof of X(n)(X) X(n+l)(x) (for n = 1, 2, ): We proved Xi(x) X(1) W.

Replace A, E1 by A E1I I En+l, then this becomes (x) (x) Combining the two above results gives:

(128) Xi(X) x? (x) I(x)

Proof of XI(x)= x* (x): We have (by Definition 2.1.2)

lim str A'Ell .IEn = - F, ^21

nz - oc

That is: For every Y S'c- lim, oo A,' and-hence for all , e3C

(129) lim ((A ll1fl5 jXC)t - (AI ElE2I V f 9JC)) O.

n-oo


Choose anyf, g e As, and define

f= fAx) for a = 1 .f(x, a) = 0 otherwise'

r( a = g x) for a = 1 SC(x, a) -= 0 otherwise

then X, X e aid, and

(A (EI... EnT 5JC)q = ffl)(n)f(x)g(z) don, (A A, :C~t =s xi (x)f(x)g(x) dot.,

(AJElIE21...Y 5JC)g fx(x)f(x)g (x) dx.

Hence (129) becomes

lim f (x(8) -xl (x))f(x)g(x) d, x = 0, n~o or, considering (128),

(130) f (xi(x) - X(x))f(x)g(x)d,,x = 0.

If M e F, ,u(M) finite, we can put

AX { =1 for xeMl j~x){= 0 otherwise I

{fsgn (x(x)- *(x)) for x MI 0 ~~otherwise

(cf. footnote"), then f, g e As, and (130) becomes

(131) f - (x)- xl(x) J dx = 0.

Thus xi(x) = x1 (x) for all x e M (exceptions as above!). Now with the I"() 7i(2)y ... of (iii) in Definition 3.2.4 we may put M = T(), and sum over i = 1, 2,

this gives xl(x) =x (x) unrestrictedly. Thus the proof is completed.- We are now in the position to take the decisive step. LEMMA 4.3.5. If .3 is not measurable, then O)R is of class (11100). PROOF: Assume that 9 is not measurable. We shall show that SWy is of class

(11100) by applying Lemma 2.2.2. For the (d mentioned there we shall choose the ? of (i) in Definition 3.7.1. So we must only prove the hypotheses of Lemma 2.2.2 for (d = 2.

JT is an Abelian ring CMR: By (i) in Lemma 3.7.3.

158 J. V. NE UMANN

Ad (i): The ring R-P'(c'DR) is purely infinite: By Lemmata 4.3.2, 4.3.3, since 9 is not measurable, the alternative (a) of Lemma 4.3.2 holds. But this means, considering (ii) in Lemma 3.7.3 and Definition 2.2.1, that 2 is purely infinite.

Now by (i) in Lemma 3.7.3, ? = OR 1.', hence we have shown that )11 A2' is purely infinite.

Ad (ii): For every finite projection F(OR) FIE1Rh21 = 0 implies F - 0: Assume F as described. Put F {X (x); 'y e Od}, FlJElIE1 . { X *(X); -Y E }.

Then F'E1E2 0 = 0 implies xl (x) 0. (All x-relations are to be understood to hold with the exception of x-sets of p-measure 0 only.) By Lemma 4.3.4, Xi(x) xl (x), hence XI(x) 0. Now P being a projection, is definite. Hence xi(x) 0 implies, by (iii) in Lemma 3.7.2, that F = 0.

Thus the proof is completed.- We may sum up our results as follows: THEOREM IX. Let S, F, ,u be as in Theorem VIII, and form US, g and

R, ORt' as described there. Then we have: If g is measurable (cf. Definition 3.3.6), then the factors OR, 9k' belong to the classes (I) or (II). The precise determination of their classes is given in (1), p. 204, Lemma 13.1.1, and p. 206, Lemma 13.2.1. (Cf. the end of our ?4.2.) If 9 is not measurable, then the factors )T,'R' belong to the class (III) (that is, (IIIJ )).

PROOF: It suffices to consider 'DR only: The automorphism 'for WMf of t S

(cf. Theorem VIII) extends our results from )R to SR'. Our statements concerning a measurable i result from the Corollary to Lemma

4.2.2, and the discussion which follows, at the end of ?4.2. Those concerning a non-measurable i result from Lemma 4.3.5.

4.4. Examples of S, F, A, and q, as required by Theorems VIII, IX for any one of the classes (I)-(II)--that is, of free, ergodic, and measurable groups 3-were given in (1), pp. 206-208, Lemma 13.2.1, and the specific examples (a) - (-y). So there is only one more thing left for us to do: To give examples of S, r, Au and g as required by Theorems VIII, IX for the class (III)-that is: of free, ergodic, and non-measurable goups 9.

A certain class of examples was mentioned in the Introduction, ?4, but we shall not investigate it here. Our examples will be based on the following lemma:

LEMMA 4.4.1. Let S, F, At and g be as described in Theorem VIII. Denote the set of those a E X, for which always Mu(M) = A (Ma), by go. go is a subgroup of AJ and a free S, F, u-group along with q. It is also measurable, with A for v. (Cf. Definitions 3.3.3, 3.3.4, and 3.3.6.) Now we assume: (i) go is ergodic. (Cf. Definition 3.3.5.) (ii) g0 5 3.

Then 9 is not measurable. PROOF: The statements about go which precede (i), (ii) are obvious on the

basis of the definitions referred to.


Assume now (i), (ii), and, per absurdum, the measurability of C. Consider the v of Definition 3.3.6 for A. This v will also do for to. So A, v both do for qo. Now 0o is ergodic by (i), hence Lemma 3.3.4 applies for to in place of X, and gives g(M) = av(M) for a constant factor a > 0. Since we have for every a E C v(M) = v(Ma) by the definition of v, so this gives M(M) = M(Ma), that is a E C0 . Hence 0o = Ci, which contradicts (ii).

This completes the proof.- We can now give specific examples. EXAMPLE (a). Let S be the set of all real numbers, F the system of all Borel

sets in S (in the usual sense), and ,u the common Lebesgue measure (for Borel sets). Consider the following one-to-one mappings of S on itself:

(132) a,(p, a): X Z? px + a, p > 0, p, a- rational.

These ai(p, a-) form an obviously enumerably infinite group. Let C be this group One verifies immediately that 9 is an S, F, u-group, and also that it is free.

Clearly M(Ma) = pM(M) for a = al(p, a-), hence the set CWo of all a E q with M(Ma) = M(M) consists of all c = a,, (p, a-) with p = 1. So 9o # A.

Co is also an S, F, A-group, and ergodic by (1), p. 206, Lemma 13.2.1, and p. 208, example ((). Hence 9 is a fortiori ergodic.

So S, F, > and C fulfill the requirements of Theorems VIII, IX, and to fulfills the conditions of Lemma 4.4.1. Hence 9 is not measurable.

Observe that in this case M(S) = oo. EXAMPLE (f): Let S be the set of all complex numbers z with I z = 1, F

the system of all Borel sets in S (in the usual sense). Let ,u be the common Lebesgue measure to be applied after S has been mapped by the mapping z = e2rip on the set S. of all real numbersso 0, < 1 which can be (and in what follows will be) looked at as the set of all real numbers (mod 1).

Consider the following one-to-one mappings of S on itself:

(133) a2(O, z) Z: z ?z + U 01 Ju1l < 1.

These a2(0, u) form an obviously continuous group. Denote this group by ,.13 Consider the following elements of C3: The a = a2(O, u) with 0 = e2tP, p rational, u 0, and that one with 0 = 1, u = 2. They form an enumerably infinite set, and consequently they generate an enumerably infinite subgroup of c. Let q be this subgroup.

In combining two a = a2(0, u) of (133), their 0 simply multiply. Hence the 0 of all a = a2(0, U) e 3 are the products of the 0 of those a = a2(0, U) with which we started, that is: the 0 = e2rip p rational. Therefore we have:

(134) a = a2(O, 0) E 0 if and only if 0 = e2TiP, p rational.

One verifies immediately that g3 is an S, r, g-group, and also that it is free.

13 ( plays a well-known and important role in the theory of conformal mappings.

160 J. V. NEUMANN

Let us now determine to, a e Weomeans that i(Ma) = 8(M). Since a = a2(G, u) maps arcs on arcs, this requires that it conserve their lengths, hence, that it be a rotation or a reflection. That is:

a: z O Oz, or z TO-, 1= 1. z

(133) excludes the second alternative, and the first one means u = 0 for a =

a2(0, u). So go is characterized by u = 0, and consequently by (134). So W $ go , since a = a2(l, a) is e9 but not eg0 .

Now map S by z = e2 i, on the V-set S, (cf. above). Then the mappings of 9o go over from z >? e2Tt"z (in (134)) into V, ( p + p, p rational. This proves that go is ergodic (in S1, and hence also in S), by (1), p. 206, Lemma 13.2.1, and p. 208, example (f3). Hence 9 is a fortiori ergodic.

So S, r, la and 9 fulfill all requirements of Theorems VIII, IX, and 9o fulfills the conditions of Lemma 4.4.1. Hence 9 is not measurable.

Observe that in this case ,u(S) = 1. Besides establishing the existence of factors 'OR of class (III), the above

examples can also be used to determine all existing types of factorizations 9RT, 912. To this end we need the lemma which follows.

LEMMA 4.4.2. Consider the S, F, A, 9, 9o and 9T, 9/' of Lemma 4.4.1. Every A e 9R' is A {= I(x); -y e 9) ; consider those for which x,(x)- 0 whenever oy 4 90. Denote their set by 91. Then we have: (i) 9.is a ring. (ii) 91'. 9.' = (aI) (a runs over all complex numbers). (iii) 9tR, 91 is a factorization, but 9T, 91 are not coupled factors. (Cf. (1), p. 138, Definition 3.1.3.) (iv) The class of the factor 91 is (III) or (II,,) according to whether ,u(S) is finite or not.

PROOF: Ad (i), (ii): Literally the same as the proof of (1), pp. 208-209, Lemma 13.4.1.

Ad (iii) : Literally the same as the proof of (1), p. 209, Lemma 13.4.2. Ad (iv): For A e 91., A definite, we have also A e 9R,, A {X7(x); Y E 9)I (but

xY(x) 0 for ly 4 go!), and we define again To(A) = f xl(x) dcx. Then the

considerations of Lemmas 4.2.1., 4.2.2, apply literally in 91 (because for a e $3o ju (M) = u(Ma), and for all other a e 9 we have xa(x) 0). Hence D91(E) =

To(E) for all projections E e 91. Hence we see, as in the Corollary after Lemma 4.2.2, that A91 belongs to the classes (I) or (II).

If 91 were of class (I), then A91' would be a normal factor by (1), p. 184, Lemma 11.2.2, and hence 9OT, 91. would be a coupled factorization by (1), p. 183, Lemma 11.1.2. This contradicts (iii) above, hence 91 is of class (II).

Now D6,j(I) = To(I) = f 1 djx = AL(S), hence 91. is of class (III) or (II.), according to whether IA(S) is finite or not.-


Thus Lemma 4.4.2 shows, if we avail ourselves of the above Examples (a), (p), that non-coupled factorizations of the classes (II), (III.) and (II1), (III) exist. This shows that no analogue of (1), p. 182, Theorem X (which holds for coupled factorizations) holds for non-coupled factorizations. Indeed, the only restriction on non-coupled factorizations is that no factor can be of class (I) (by (1), pp. 183-184, Lemmas 11.1.2, 11.2.2, cf. the above proof of (iv) in Lemma 4.4.2). We could also show the existence of non-coupled factorization of class (III), (III), but we shall not discuss this now.

INSTITUTE FOR ADVANCED STUDY.

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Von Neumann-On Rings of Operators III