Inequivalence and equivalence of c e r t a i n kinds of non-normal operators By Ping Kwan Tam B.Sc, Chinese Un i v e r s i t y of Hong Kong, 1964 B.Sc. (S p e c i a l ) , U n i v e r s i t y of Hong Kong, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS ' We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1970-In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced deg ree a t t he U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada ( i i ) Supervisor : Dr. Donald J.C. Bures Abstract * This thesis i s concerned with the problem of unitary equivalence' of c e r t a i n kinds of non-normal operators. Suppose [m, K, G, g l—> U ] i s an ergodic and free C-system, with G abelian. Let m = m_ 8 1, n = R(U g 0 V g : g e G), and l e t a. = R(m, n) = R[m, K, G, g >-> U ] be the von Neumann algebra constructed from [m, K, G, g i—> U e] according to von Neumann. We compute : (1) the group A(a; m, n) of a l l automorphisms,: of a which keep m pointwise f i x e d and keep n i n v a r i a n t , and (2) the group•' .A(m, a; n) (resp. G(m, Ci; n)) of a l l automorphisms of•'• .'- which extend to automorphisms (resp. inner automorphisms) of Ci keeping n pointwise f i x e d . These ca l c u l a t i o n s lead us to compute G' H [G] and G' (where !("] i s the f u l l group generated by G) . We show that f o r an abelian and ergodic G on an abelian m. G' H [G] = G . In the course of th i s i n v e s t i g a t i o n we obtain several i n t e r e s t i n g r e s u l t s . For example we see that such £ G i s automatically free on m_ . For a large class of tensor algebras (and i n p a r t i c u l a r f o r a large class of m u l t i p l i c a t i o n algebras) we succeed i n determining G' . (For the p a r t i c u l a r cases of m u l t i p l i c a t i o n algebras we only use measure-theoretical arguments.) These r e s u l t s are applied to solve the problem of unitary equivalence of c e r t a i n kinds of non-normal operators. F i n a l l y f o r most of the i n t e r e s t i n g thick subalgebras fc i n the l i t e r a t u r e , we construct numerous u n i t a r i l y non-equivalent operators A , 7 i t h R(Re A) = E . ( i i i ) Acknowledgement The author i s indebted to h i s supervisor, Dr. Donald J.C. Bures, for h i s suggestion of the topic of t h i s t h e s i s , and f o r h i s valuable assistance and encouragement throughout i t s preparation. He i s also g r a t e f u l for the f i n a n c i a l assistance of both the National Research Council of Canada and the Mathematics Department of the U n i v e r s i t y of B r i t i s h Columbia. F i n a l l y i t i s a pleasure to acknowledge the patience, the care and the p r o f i c i e n c y of Mrs. Y.S. Chia Choo i n typing t h i s t h e s i s . (iv) Table of contents page _ 1. Introduction 1 2. The basic set up 6 3. The c a l c u l a t i o n of A(<2; m,n) . 9 4. Operators distinguishable by means of A(a; m,n) 13 5. The c a l c u l a t i o n of A(ifl,a; n) and G(m,a; ft) 15 6. The c a l c u l a t i o n of G* H [G] 20 7. The c a l c u l a t i o n of G' 24 8. Operators distinguishable by means of G' H [G] and G1 44 •9. Examples of non-equivalent operators of various type with R(Re A) thick and of various type i n R(A) 49 Appendix 79 References 1 86 / .1 1. Introduction The following (so-called unitary equivalence) problem i s of paramount importance i n the theory of operators : given two (bounded l i n e a r ) operators A 1, A 2 on a (complex) H i l b e r t space H, to determine whether or not they are u n i t a r i l y equivalent, i . e . whether or not there i s a unitary operator U on H such that U*A1U = k^ . For normal operators t h i s question i s completely answered by the c l a s s i c a l m u l t i p l i c i t y theory [20 or 10]. Many authors, i n p a r t i c u l a r Brown [3], Pearcy [15], Deckard [8], Radjavi [19], and Arveson [1, 2], considered the problem f o r non-normal operators and have obtained various s i g n i f i c a n t r e s u l t s . However most of their, r e s u l t s (cf. [22]) deal only with operators which are of type I i n the following sense [21] : an operator i s of type I (II , H^, III) i f the von Neumann algebra generated by A i s of type.I.(resp. 11^, 11^, I I I ) . For non-normal operators of type I the problem i s already known to be d i f f i c u l t , and the known r e s u l t s are f a r from exhaustive; the author i s therefore very pleased i f he has obtained (as he so believes) some i n t e r e s t i n g r e s u l t s f o r some kinds of operators of more general type. The problem of unitary equivalence i s c l o s e l y connected with the following p rob lam of algebraic equivalence ; giv-:a two operators A^ > A 2 on the H i l b e r t space H, and denoting by Cl^ , a.^ the von Neumann algebras generated by A,, A^ r e s p e c t i v e l y , to determine whether or not they are a l g e b r a i c a l l y equivalent, i . e . whether or not there i s an (algebraic *-) isomorphism (j> of CL onto a such that <}> (A ) = A . In f a c t , i f a.^ ='-'a. i s a fac t o r , i t i s well-known that the two concepts coincide. So we concentrate on the algebraic equivalence. Let us o u t l i n e at t h i s point a 'sieving'.program f o r the above problem of algebraic equivalence, and put the present work into perspective. For s i m p l i c i t y l e t us c a l l two operators equivalent when they are a l g e b r a i c a l l y equivalent i n the above sense. F i r s t we examine whether d and Ci^ are isomorphic. I f they are not, then and are non-equivalent; i f they are, we can assume that = , and proceed to the second stage. Secondly, assuming that A 2 generates the same von Neumann algebra a as A^ does, • • • i and denoting by (m^) the von Neumann algebra generated by the r e a l part Re (resp. Re A 2) of A^ ^ (resp. A 2 ) , we examine whether [«.', m' ] and [a, m2] are equivalent, i . e . whether there i s an automorphism <J> of a such that <)>(m ) = . When a i s a type I f a c t o r , the c l a s s i c a l m u l t i p l i c i t y theory [see 20] provides a complete s o l u t i o n to t h i s question. For type II fact o r <X, t h i s question has been examined and some r e s u l t s have been obtained by Bures [ 5 ] . If the answer to t h i s question i s negative, then 'A and A 2 are non-equivalent; otherwise we can assume = and proceed to the t h i r d stage. T h i r d l y , assuming that A 2 generates the same von Neumann ,; algebra a as does, and that Re A^ generates the same von Neumann algebra <r; as Re A 2 does, we examine whether . Im A^ = a(Im A.^ ) . f o r some • a £ A(a; wi) , where . Im A i s the imaginary part of A and A(o.; m) i s the group of a l l automorphisms of Ci which leave m i n v a r i a n t . The computation of A(a; rrt) i s i n general very d i f f i c u l t and only fragmentary information i s a v a i l a b l e . [see §3 of t h i s t h e s i s . ] I f the answer to the above question 3. is negative, then A^ and A2 are non-equivalent; otherwise, we can assume that Im = Im A^ and proceed to the final stage. Finally, assuming the ;-same conditions as in the third stage, and in addition that Im A = Im A^ = T, we examine whether there is an automorphism <f> of a "such that 4>(Re A^) = Re A^ and <f)(T) = T. Obviously A^^ and A^ are equivalent i f and only i f the answer to the above question is affirmative. To settle this question one needs to compute the group A(m, a; ft) of a l l automorphisms of ' m which extend to automorphisms of a keeping ft pointwise fixed, where ft denotes the von Neumann algebra generated by T. For a large class of [a, m, ft] this A(m, a; ft) is determined in §§5,7 of this thesis. As we have seen in the last paragraph, we can assume without much loss of generality that a 1 = a 2 . In that case, the algebraic equivalence is one inside one and the same algebra Ci . By analogy to the classical situation, we are also interested in the problem of inner equivalence, i.e. whether or not there is a unitary U in a such that U*A1U = A2 . In this thesis we consider the special case when R(Re A^ = R(Re A2) = m , Im A '•= Im A2 = T and a - R(m, T) . For the same class of [a, m, ft] as in the last paragraph, we have succeeded in calculating the group G(m,a;ft) of a l l automorphisms of m which extend to inner automorphisms of a keeping, ft pointwise fixed, and settled the afore-mentioned question completely in §§5,6 . We have not pretended.that this program is an easy one; indeed a l l questions listed above are very difficult. After a l l we cannot and do hot expect a simple solution to the general problem. As an application of our results, we construct in the last section numerous examples of non-equivalent operators (of type II ^, and III). In fact, for a large class of [E, a], where E is thick in a(a can be a factor of type U j , Hoo> HI etc.) [see 5], we construct a family (A^) of pairwise inequivalent operators such that R(A^) = a, R(Re A^) = E , the Im A^'s are identical, and the Re A^'s can be chosen so that they are unitarily equivalent to each other. In the course of the investigation of A(m, a; n) we have deter-mined the commutant G' of a group G of automorphisms of m (see §7 and the appendix) for a large class of G . ,[G' is the group consisting of automorphisms of m which commute with each element of G .] Also we have succeeded in showing that an abelian and ergodic group of automorphisms of an abelian von Neumann algebra is necessarily free [cf. §6]. These results are of independent interest. ^ Finally a few words about the contents. In §2;we have the basic set up : a is the von Neumann algebra constructed from,a free, ergodic C-system .[m; K, G, g »—> Ug] according to von Neumann and Dixmier [13, 8], m = m 8 1, and n = R(U 8 V : g e G) [For details of notations, cf. §2 o g below.]. In section 3 we compute the group A(<X; ni, n) of a l l automorphisms of a. which keep m pointwise fixed and keep n invariant. This result indicates that A(a; m) can be rather complicated [Note A(a; m, n) cz A(o.;m)]. In §4 we present simple examples of operators which are distinguishable (up to unitary equivalence) by means of the calculation of A (a.; m, n) in 5. §3. In §5 we compute A ( m , a; n) and G(m, a; n), the importance of which became clear in the preceding paragraphs. This calculation leads us to compute G ' C\ [G] ( [ G ] being the f u l l group generated by G ; cf. §5 below for detailed explanation) in §6, and G ' in §7. The results contained in §6 are very general; those in §7 is rather specific. Thus we present in the appendix a result which generalizes those of §7 to a large family of tensor product algebras; this arrangement is due mainly to the tone and the machinery of this thesis. Then in §8 we apply the results of §§6,7 to operators. Finally in §9, for almost a l l interesting thick subalgebras E conceived in the literature [5], we construct numerous (unitarily) non-equivalent operators A with R(Re A) = E . \ 2. The basic set up In t h i s thesis we s h a l l work mostly i n the framework out l i n e d i n D e f i n i t i o n 2.1 ( i i ) below. Though i n general our notation and terminology i s that of Dixmier [9], we f i n d i t h e l p f u l to use the following standard (yet d i f f e r e n t from Dixmier's) notations and d e f i n i t i o n s . D e f i n i t i o n 2.1 [cf. .4] . ( i ) The system [m, K, G, g >-> U„] i s c a l l e d a C-system i f m i s — o — a maximal abelian von Neumann algebra on the H i l b e r t space K, i f G i s a group of automorphisms of m_, and i f g j—> U i s a unitary representation of G on K such that U 2(M)U* = g(M) f o r each M e m . ( i i ) Suppose [m, K, G, g »—> Ug] i s a C-system. Define KQ to be the H i l b e r t space with an orthonormal basis ( < J > » ) n- indexed on G . Define H = K 0 Kg , m = m 8 1 K , n = R(U g ® V g : g e G), the von Neumann algebra generated by { Ug 8 Vg : g e G } i n L(H), where Vg i s the unitary operator on Kg which maps <f>b to ^ g l / • Define a.[m, K, G, g >-> U g] = R(m, n) . This construction of a i s o r i g i n a l l y due to von Neumann [15], and further developed by Dixmier [9, pp. 129-137]. ( i i i ) The C-system [ni,-K, G, g •—> Ug] i s free i f for any g £ G , not the i d e n t i t y , there i s a family (E^). of projections of m. s u c h that l ± £ l E i = 1 a n d §( Ei) Ei = 0 f o r a 1 1 i G . I - T h i s i s not the o r i g i n a l d e f i n i t i o n of freeness due to von Neumann [15] or Dixmier [9]; b u t - i t i s proved i n [4, §4; 5, §7] that these two d e f i n i t i o n s of freeness are equivalent. We f i n d i t more convenient to employ the above d e f i n i t i o n i n th i s thesis. 0 (iv) The C-system [m, K, G, g >-> U ] i s ergodic i f © m H' { U g : g e G }' = C . (v) We s h a l l say that, the C-system [m_, K, G, g t—> u ] i s abelian i f G i s abelian. Throughout t h i s thesis we s h a l l consider only free and ergodic C-systems (or rather the algebras Cl, m, n constructed from such. C-systems) , and we s h a l l s t i c k to the notations set above without further explanation. We remark i n passing that i f the C-system i s free and ergodic, then m i s maximal abelian i n a [this i s a well-known f a c t ; c f . 9, p.133, Lemma 2], that (U & 8 V„)m(lL 8 V C T)* = m always, and that m C\ n = C always. In 6 g & & order to see the l a s t e quality, l e t e be the i d e n t i t y of G ,. <}> the natural isomorphism from K onto K 8 [<$>e] given by : <j>(x) = x 8 <pe ([<|>e] being the closed l i n e a r subspace generated by (j>£ i n Kg) , E the p r o j e c t i o n from K 8 Kg onto K 8 [ $ e ] , and f i n a l l y f o r each A e a , define A e to be the operator i n L (K) such that A £x = (<J> ^ EA<j>) (x) f o r each x e K . Then i t i s s t r a i g h t forward to compute that (Ug 8 V g ) e = 1 i f g = e, = 0 ,i f g i- e, and that (M 8 1) = M for each M e m . Thus f o r any A e n , A1' ' e C and therefore f o r any M.8 l " e m C\ n : ' ' M = (M0 l ) e e C We conclude that m C\ n = € . . Before we close this section let us introduce one more definition which will be very useful in the sequel. Definition 2.2 Let Ci => m be von Neumann algebras. Then (E, F) fits in [a, m] i f E, F are von Neumann subalgebras of m such that for any automorphism <j> of a with cj>(E) = F : <j>(m) = m. \ An obvious example of such (E, F) is : E' H a = F' C\ a = m . In this thesis we shall consider operators A-^, A2 on H with R(A^) = R(A2) = d, and such that (R(Re A^, R(Re A2>) fits in [a, m], or that (R(Im A ), R(ImA2)) fits in\ [a, n] . 3. The c a l c u l a t i o n of A(a; m,n) We begin with the d e f i n i t i o n of A(a; mtn) . D e f i n i t i o n 3.1 We denote by A (a; m,ft) the group of automorphisms of d which keep m pointwise f i x e d and leave n i n v a r i a n t . C l e a r l y A(a; m,n) i s a subgroup of the group A(a; m) of a l l automorphisms of a which leave m i n v a r i a n t , mentioned i n §1. Lemma 3.1 For each a e A(&; m,ft) there i s a character c : G — - > C such that f o r a l l g e G : a(U g 0 V g) = c ( g ) ( U g 0 V g) . Proof. As a e A.(a; m,n) ' i t keeps m pointwise f i x e d and leaves ft i n v a r i a n t . Now f o r every M e m : a(U g 0 V g)Ma(U g 0 V g ) * = a ( ( U g 0 V g)M(U g 0 V g)*) = (U g 0 Vg)M(U g 0 V g ) * . Hence (U„ 0 V„)*a(U g 8 Vg) e mr O n = m C\ n 10. by the remark preceding D e f i n i t i o n 2.2. Therefore there i s a complex number c(g) such that o ( U G 0 V G ) = .c(g)(Ug 0 V G ) . It follows r e a d i l y that the mapping g »-> c(g) i s a character on G . That completes the proof. D e f i n i t i o n 3.2 Suppose a i s a von Neumann algebra on the H i l b e r t space H, and that U i s a unitary operator on H. Suppose UaU* = Ci . Then U i s s a i d to induce the automorphism of a given by : A c a i—-> U A U * e a . 5 Lemma 3.2 For each character c : G — > C there i s a unitary operator W£ on Kg such that the unitary 1 0 Wc induces an automorphism a c e A (a; m,n) with x-a c ( U g 0 V = c ( § ) ( u g 8 V g) , g e G . Proof. Let c be a character on G. Define a unitary operator Wc on Kg by :• Wc(<j>g) = c ( g H g , g £ G . Then obviously 1 0 Wc ' commutes with m . Furthermore f o r any g, h £ G , v/e have : 11. W c V g W ^ h = W cV gclh)<? h = c(g)Vg<f>h . Thus WcVgW* = c(g)V g , and (10 W c)(U g Q V g ) ( l 8 W c)* = c ( g ) ( U g 0 V g) . Hence the unitary 1 0 Wc induces an automorphism a c e A (a.; m,n) with a c ( U g 0 V g) = c ( g ) ( U g 0 V g) , g E G . That completes the proof. V Remark. Lemma 3.2 holds true with no r e s t r i c t i o n on the C-system as we do not need any property, l i k e freeness, of i t i n the above proof. D e f i n i t i o n 3.3 For a character c on G l e t ac denote the automorphism of a induced by the unitary 1 0 Wc i n Lemma 3.2 . Theorem 3.3 A(a; m,tt) = { a c :. c a character on G } . Proof. By Lemma 3.2,. each c c e A(&; m,n). By Lemma 3.1 for each cr e A (a; m,n) there is a character c on G such that a\n = c?c|n' . But a\m• = ac|m , both being the identity map on m . Since a = R0n, n), so a = cr .. That completes the proof. 13. 4. Operators d i s t i n g u i s h a b l e by means of A(<Z; m,n) The following d i r e c t consequence of Theorem 3.3 i s • u s e f u l i n operator theory. ' Theorem 4.1 ' Let A and A^ be two operators on H such that ^(A^) = R(A 2) = a , Re A } = Re A 2 R(Re A ) = m , and (R(Im A^ , R(Im A 2 ) ) f i t s i n [a, ft] ( c f . D e f i n i t i o n s 2.1, 2.2). Then the following statements are equivalent : (i) A^ and A^ are u n i t a r i l y equivalent, ( i i ) A^ and A^ are ( a l g e b r a i c a l l y ) equivalent, ( i i i ) Im A = a„ (Im A ) f o r some character c 'on G . z. c \ [For d e f i n i t i o n of a £ , c f . D e f i n i t i o n 3.3] Proof. The theorem follows d i r e c t l y from Theorem 3.3 and the observations that Ai and A^ are equivalent i f and only i f there i s a a e A(<£; m,n) such that Im A^ = a(Im ^ ) » a n ^ that each a c i s implemented by a unitary operator on H. • / We now i l l u s t r a t e Theorem 4.1 by the following simple, example. : / Let m be L^O, 1].. acting by m u l t i p l i c a t i o n on K = L„[0, 1] (cf. 14). Let D be the group of dyadic r a t i o n a l s i n [0, .1] under addition mod 1. For each d e D, let- be the automorphism of m given by : 14. T d M f = M d [ f ] > where Mf e m_ is the multiplication by f e L^tO, .1], and d[f] is the function in L^tO, 1] given by : d[f ] (y) = f (y - d), ye [0, 1]. Let G = { xd : d e D } . Let U d be the unitary operator on K given by : (Udh)(x) = h(x - d), h e L 2 [ 0 , I], x e [0, 1]. Then [14] the system Qn_y K, G , T d >-> Ud) is an abelian, free and ergodic C-system. Let g be a strictly monotone, continuous and real-valued function ' . defined on [0, 1]. Then Mg 0 1 is self-adjoint, and i t generates m (i.e. R(Mg 0 1) = m). Suppose j> ad(Vd & Vd) and £ Pd(ud'® Vd) are d£D deD self-adjoint, and suppose each of them generates n . Then by Theorem 4.1 we have : The operators Mg 0 1 + i £ a d(U d 0 Vd) and Mg 0 1 + i £ ^ ^ d ® vd^ 1 deD deD • >• axe unitarily equivalent i f and only i f for some character X on D, '"•-•9* ••"•"":'->' 1 a d(U d 0 Vd) = a ( £ 0 d ( u d 0 V d » " ' '' deD A deD i.e. i f and only i f , a d = X(d)8d for a l l d e D . 15. 5. The c a l c u l a t i o n of A(m,0.; ft) and G(m,a; ft) For convenience as w e l l as for l a t e r references l e t us introduce-the following d e f i n i t i o n s . * D e f i n i t i o n 5.1 [cf. 5] Let m be an abelian von Neumann subalgebra of the von Neumann algebra a . ( i ) A(m) = { a : a i s an automorphism of m } ; and an m-group i s a subgroup of A(m) . ( i i ) G(m, a.) = { a z A(m) : there i s a unitary U z <X such that a(M) = UMU* f o r a l l M e m } ' - ' ( i i i ) A(m, a) = { a e A(m) : there i s an automorphism Y of a such that y|m = a } . (iv) , For a subset G of A(m) , . G' = { a e A(m) : ag • go f o r a l l 6 e G } . ' (y) . Let a, -g e A(m).. We say. that a and g agree on a p r o j e c t i o n ' ' '"' F of m i f • ' a (M) = g (M) f o r a l l M e m with FM = M . We denote by E(a, g) the largest p r o j e c t i o n of m on which a and g agree. (vi) Let S 'be a subset of A(m). For each a e A(m) define. E(a, S) = sup{ E(a, g) : g e S } , 1 6 . and let . [S] = { a e A(m) : E ( a , S) = 1 }. . Call S fu l l i f [S] = S. We shall need the following result. Lemma 5.1 [cf. 5, 21] Under the assumptions of Definition 2.1 (i)-(iv), we have (i) G(m, a ) = [ G ] , where G~ = { a e A(m) : for some g e G, a(M 0 1) = g(M) 0 1 for a l l M e m } . (ii) For each a e; G' there is a unitary operator Wa on H such , that Wa (M 8 1)'W* = a (M) 0 1 , for a l l Mem, and Wa(Ug 0 Vg)W* = U g 0 V g , for a l l g e G . Proof. Part (i) is a well-known result, and i t is proved in [5, 20]. Suppose now a e G ' . Since m is maximal abelian in L(K), by [9, p.241] there is a unitary operator Y on K such that YMY* = a(M) for a l l Mem. Define a unitary operator Wa on H by : Wa(x 0 <bg) = (UgYU|x) ® <f-g , x e K, g e G . Then ' W*(x 0 <f ) = (U Y*U*x) Q <j, and for a l l Mem 17-[Wa(M'8 -l)W*](x 0 <j>g) = [Wa(M 0 l)](U gY*U|x 0 ^ g) -• = (UgYU|MUgY*U*x) 0 'c|)gv'-;--[(gag-' )M 0 l ] ( x 0 <J>g) = [a(M) 8 l ] ( x 0 <f)g) , also C W a ( U g 0 V W S ] ( x 8 *h> = [ wa< ug 0 V g) ] ( U h Y * U g x 8 cj>h) = W a [ ( U g h Y * U * x ) 0 <f>gh] = U g h Y U * h [ U g h Y * U g x ] 0 * g h - < Ug x ) 0 +gh = (U g 0 V g ) ( x 0 <J)h) .. . Thus 1 Wa(M 0 1)W* = a(M) 0 1 , for a l l M e m , and w a(U g 0 Vg)W* = U g 0 V g, for a l l g e G . Before we introduce the next theorem, r e c a l l that A(m,a; ft) i s the group of a l l automorphisms of m, which extend to automorphisms of a keeping ft pointwise f i x e d , and that G ( m , a ; ft) i s the group of a l l automorphisms of m,; which extend to inner automorphisms of CI keeping ft pointwise f i x e d . 18. Theorem 5.2 With the assumption of D e f i n i t i o n 2.1 ( i ) - ( i v ) , we have : . A(m,a; ft) = G7 , and • G(m,a; ft) = G7' (1 [G] , where f o r S <=A(m), S = { a e A(m) : for some s e S, a (M 8 1) = s (M) 8 1 for a l l M e m } . Proof. Suppose a e A(m,a; ft). Then a extends to an automorphism oT of a- with a|m = a and H"(Ug 8 V g) = U g 8 V g f o r a l l g e G . Let a be the automorphism of m_ such that f o r a l l . M e m_ : o_(M) 8 1 = a(M 8 1) . Then f or a l l g e G , and for a l l M £ m : \ ag(M) 8 1 = a(g(M) 8 1) a(U gMU| 8 1) a [ ( U g 8 V g)(M 8 l ) ( U g 8 Vg).*] (U g 8 V g)a(M 0 l)'(U g 8 V g ) * = [ga(M)] 8 1 Thus a e G' and a E G Suppose, on the other hand, that a, e G' . Let" cx be as above. Then a. e G' . So by Lemma 5.1, there is a unitary operator Wa on H such that Thus we see that a extends to an automorphism of Ci keeping n pointwise fixed. So the first equality of the present theorem is proved. The second equality follows directly from the first and Lemma 5.1 (i). That completes the proof. v Wa(M 8 1)W* = a(M) ® 1 = a(M 8 1) and Wa(Ug 8 Vg)W^ = U g 8 V g 20. 6. The c a l c u l a t i o n of G' A [G] In view.of Theorem 5.2 we s h a l l compute G' C\ [G] i n t h i s s e c t i o n . Before we do that we need a few d e f i n i t i o n s and an a u x i l i a r y r e s u l t . D e f i n i t i o n 6.1 Let m be an abelian von Neumann algebra, and F an m-group. Then F i s ergodic i f M e m , f(M) = M f o r a l l f e F M e C. F i s free i f for any . f e F,..;not. the i d e n t i t y , there i s a family (E^) o f . projections of m such that ^ E i = 1 and f ( E i ) E i = 0 f o r each i . 0 Remark. I t i s r e a d i l y seen that the C-system [m, K, G, g >-> U j i s ergodic (or free) i f and only i f G i s . Also i t i s well-known and could be e a s i l y proved that i f F i s ergodic, then f o r any two non-zero projections P 1, P 2 of m , there i s an f e F such that f(P )P f 0 . Proposition 6.1 Suppose that m i s an abelian von Neumann algebra, and f i s an ergodic and abelian m-group. Suppose that 3 i s i n F T . Then i f ctj and a 2 are i n F with E(g, a ) ^ 0 and E(g, a 2 ) f 0, we have : E ( 3 , ax) = E ( 3 , a 2) . • Proof. Let 3 agree with on a non-zero p r o j e c t i o n P i of m(i=l,2). Since' F i s ergodic there e x i s t s . a e F such"that Q = a ( P 1 ) P 2 f 0 . Now i f M e m with o(M)Q = a(M) then 3(M) = (M). So f o r M e m with MQ = M we have f i r s t . 21. B(M) = a 2(M) , and secondly, . g(M) = (a(3) ( a " 1 (M)) • •••/ = a^M) , where we have used both that 3 e F' and that F i s abelian. Thus we see that and agree on a ( P l ) P 2 . That i s , any non-zero p r o j e c t i o n (of m) on which 3 agrees with a 2 majorizes a non-zero p r o j e c t i o n (of . , m) • on which agrees with a 2 • •Therefore E(3, a 2 ) [ l -' E(a1',-,a2) ] = 0, . . or • ' E(f3, a 2) <_ E ( a l f a 2) . By the d e f i n i t i o n of Efa^, a 2) we obtain E(3, a 2) <_ E(0, a i ) . The reverse i n e q u a l i t y i s obtained by reversing the roles of and a2 ', and we conclude that . E(3, ax) = E(3, o 2) . We s h a l l also need the following r e s u l t of Bure's [5]. Lemma 6.2 [5, Proposition 4.3] Suppose that a and g are automorphisms of an abelian von Neumann algebra m . Then there e x i s t s a family (E.;) 22. of projections of m such that •I E i = 1 - E(a, 3) and • (a(E i)) (gCEj.)) = 0 for each i . Now we can prove our next theorem. I t should be pointed out that part ( i i ) below i s proved i n [7] under the superfluous assumption that F i s free, and by completely d i f f e r e n t techniques. Theorem 6 . 3 Suppose that m i s an abelian von Neumann algebra, and F i s an ergodic and abelian m-group. Then : (i) F i s fr e e . ( i i ) F i s maximal abelian i n [F]. ; ( i i i ) F' n [F] = F. (iv) 3 e F ' E ( 3 , a) f 0 f o r at most one a £ F. Proof. Ad ( i ) . Let e be the. i d e n t i t y of F, and l e t 3 e F\{e} . Since "E(3, 3) = 1. and 3 $ e, we have E(3, e) f E(3, 3). Now.as F i s abelian, 3 e F' and so, by Proposition 6.1, E(3, e) = 0 . By Lemma 6.2, there e x i s t s a family (E^) of projections of m such that • I E i = 1 and g(E i)E ; j. = 0 for each i . 2 3 . So F- i s free . Ad ( i i ) . . Let 1-^ be an abelian subset of [F] containing F. Let 3 E F x . Then as F x i s abelian and F, g e F' . Now g £ [F] also„ so sup{ E ( 3 , a) :. a £ F } = 1 , or sup{ E ( 3 , a) : a £ F and E ( 3 , a) ^ 0 } = 1 . By Proposition 6.1 th i s means that f o r some a 0 £ F, E ( 3 , a Q ) = 1 i . e . 3 = a Q . _ So 3 e F. We conclude that F-^ = F. Thus F i s maximal abelian in^ [F]. Ad ( i i i ) . As F i s abelian we obviously have F' Pi [F] F. The above proof of ( i i ) shows i n f a c t that F' (~\ [F] cr F. Thus we have F' ft [F] = F. A d (iv) . Suppose, that E(g, c^) ^ 0 and E(g, a 2) f 0 for , a 2 £ F. Then by Proposition 6.1, and agree on the non-zero p r o j e c t i o n Q = E ( 3 , c ^ ) = E ( 3 , c * 2 ) . Now l e t (E.^) be any family of orthogonal projections i n m such that a^' ct2(E-j_)E^ = 0 for each i . Let = QE^ . Then we have Q ± <_ Q and Q ± <_ E ± so that Q ± = a* a 2 ( Q i ) Q ± <_ a] 1 a 2 ( E i ) E i = 0 for each i . As Q ± = 0E± and Q f 0 , so .\ E i f 1. Now by (i) , F i s free . Thus a',1 a 9 = e, i . e . a, = a0 . This completes the proof. 2 4 . 7 . The c a l c u l a t i o n of G ' In t h i s s ection we s h a l l treat a number of s p e c i a l cases of the .. s i t u a t i o n considered i n the § 2 , and compute the commutant G ' of c e r t a i n group G of automorphisms. [cf. Propositions 7 . 1 , 7 . 2 , 7 . 3 and 7 . 4 below.] These r e s u l t s w i l l be used i n the next two sections. These cases are concerned with the m u l t i p l i c a t i o n algebras on some spaces. The r e s u l t s contained i n t h i s s e c t i o n can be obtained by r e a l i z i n g the concrete algebras considered here as tensor products of s u i t a b l e abelian W*-algebras with respect to s u i t a b l e normal states, by r e a l i z i n g the group of automorphisms as product groups, and then by appealing to general r e s u l t s f or tensor product' algebras established i n the appendix, but we prefer to prove them here by purely measure t h e o r e t i c a l arguments. 7 . 1 Let T = [ 0 , 1 ) with addition mod 1 ( i . e . the c i r c l e group), or H with usual ad d i t i o n (and i n both cases, with usual topology). . Let m be L o c X T ) acting by m u l t i p l i c a t i o n on L 2 (T) [cf. 1 5 , Chap. I l l ] , and l e t D be a dense subgroup of T. For each x e T, denote by T x the automorphism of m given by : ( t x f ) ( y ) = f ( y - x) , for any f e 1^(1) , and any y e T . N Proposition 7 . 1 { x,.: d e D } ' = { T x : x e T } . 25. Proof. Suppose a e { x d : d e D } 1 - . Since D i s dense i n T, . {' x d : d e D } i s ergodic on m [cf. 14, Lemma 13.2.1]. Thus a preserves the trace induced by the Lebesque measure X on m . Now l e t A = [0, e] where 0 < e < 1, and let, F be a measurable set such that " X-p = <*(XA) • - » , • , This F i s determined to within a set' of measure zero. As X(A) = e, so X (F) .•= e . We s h a l l see that f or some x e T, F = [x, x + e] modulo sets of measure zero. 'In the case T = [0, 1], F i s obviously bounded,. In the case . T=IR, F i s e s s e n t i a l l y bounded. For i f otherwise there w i l l be integers m, n with |m - n|^> 2 such that B = [m, m + 1 ] f l F ^ 0 and C = [n, n + 1] f l F $ 0 . But then for some d z D with |d| > 1 , T d(C) H B f 0' . [Here, and i n what follows, we i d e n t i f y x d with the t r a n s l a t i o n by d. Also we s h a l l use a to denote the automorphism a induces on the r i n g of measurable subsets of T, and i d e n t i f y sets which d i f f e r by a set of measure zero.] Now we have a"' (B), a*1 (C) <=. [0, 1] so that f or any d e D with M > i , : Td(as (C>)' A a"' (B) = 0 , 26. and so T d(C) A B = 0 , a c o n t r a d i c t i o n . Thus i n any case F i s e s s e n t i a l l y bounded, and therefore we can choose and f i x a bounded F. We now associate a point x k e T f o r .any bounded, non-zero measurable subset K of T as follows. Consider the set I of points, y of TR such that (-co , y] f\ K has measure zero. I i s c l e a r l y non-empty and bounded above, so we may denote by x k i t s sup'remum. Note that (-co, x^ .] H K has. zero measure as Xj i s the l i m i t of a sequence of points i n ( -oo , x k) A I. Observe also that we always have x k e T, and that f o r any 6 > 0, [x k, x k + 6] H K ^ 0 . To prove the l a s t statement of, the f i r s t paragraph, l e t us consider -: the, case ' T,= E. f i r s t . Let x denote x for the F mentioned i n the a f i r s t and second paragraph. Let F Q = F A (x + e + 2, co ), and l e t • F n = F- A [x + e + x + e + 2] for each p o s i t i v e integer n. I f for some n >_ 0, F n i s of p o s i t i v e measure, then there are <5 with 0 < <S < min ( £ > ^ ^ [ ) 5 and d e l ) with d >_ z such that x d ( F ,'n [x, x + 6]) a- F n f 0 . But a'1 (F f| [x, x + 6]) and a"' (F ) are both subsets of A so that f o r any • d e D with d > s we have T d[cT' (F A Ix, x +:$])} A a"1 (F n) = 0 , and so, x d ( F A [X, x +.63) fl F n = 0 , 27. a con t r a d i c t i o n . Thus each F n i s of measure.zero and F = [x, x + .£] modulo sets of zero measure; i . e . a ( X r n -i) = X r / • L U , £ J [X,X+£ J Now we turn to the case T = [0, 1). For the sake of convenience, l e t us take e < 1/3 . Let F be as i n the f i r s t paragraph, and y be X p . I f B = F \ [ 0 , y + e] i s of zero measure, then we have a ( X r Q £ j ) = Xr i • So we may assume B i s of p o s i t i v e measure. We s h a l l see Ly,y+eJ y = 0 i n th i s case. I f x = y + E , then by an argument s i m i l a r to that employed i n the preceding paragraph we see that there are measurable subsets A1 v, A 2 of A , and d £ (e, 1 - e) f l D such that x d ( A 1 ) H A £ 0 , which i s absurd. So x > y + e . Now suppose y > 0 and we s h a l l draw some con t r a d i c t i o n from t h i s assumption. . Let C, = a - ' ([y + e, x^]) , -and -.. ; G2-= a" ([0, y]) . ' ' '!'' : - - ' - r ' ^ : '^•'••'v-.':' Then f o r any p o s i t i v e 8 we have / [1 - <S, 1] C\ C x j 0 and [1 - 6, 1] f l C 2 f 0 . For i f otherwise, we would have, say, [1 - <50, 1] fl Cj = 0 for some p o s i t i v e 6 Q , i . e . C c: [e, 1 - <50], and therefore T ^ C C ^ ) f l [0, E] = 0, i . e . r d ( [ y + E, x ]) H F = 0 f o r a l l d e D' H (0, <SQ), which i s absurd by the d e f i n i t i o n of x^ . A s i m i l a r argument also applies to the assumption [1 - <50, 1] fl C 2 = 0 . Thus we have 28. [1 - 5, .1] A C1 J 0 and [1 - <5, 1] H C2 f 0 , for any positive 6 . Now we can choose a 6Q > 0 such that xd([i - <s0, j]) rt u - « 0, i] = o , for a l l d e [e, x,,] H B. However there must be a d e [e, x,,] A D such that • T d(a([l - <50, 1] H C2)) H a([l - 60, 1] fl Cx) ^ 0 , i.e. Td([i - 6 0 , i] n c 2 ) n ([i - 6 G , i] n c ^ . / o , a contradiction. Hence y = 0. Now we shall show that x > 1 - e . We know already that x,, a — B > y + s = e . If x B < 1 - e , then by an argument similar to that employed i n the case T = R we see that there would be a d € (s, 1 - E ) C\ D, and measurable subsets A' , A" of [0, e) such that Td(A') n A" i 0 , which is false.. So > 1 - e . , Finally we shall show that F = [0, e + x^ - 1] y [x B, 1] (mod. measure zero set). If F\([0, e + - .1] \j [x^, 1]) is of measure zero, then we are done. Otherwise by the definition of x^, one of the sets B^ = F A [e + x_ - 1 +—, ,ej, n> 1/(1 - x R) , must be of positive measure. 29. As e < x . - x R . < 1 - e ( r e c a l l e < 1/3), an argument s i m i l a r to that B n of above would lead to a cont r a d i c t i o n . Thus F.= [0, e + x^ -.1] \j [xg, 1] to within a set of zero measure. Let x - x^ then we have n a ( X [ 0 , £ ] ) = T X ^ [ 0 j £ ] ) • So f a r we have established that i n both cases of T, for a given £ « (0, - j ) , there i s an x e T such that a ( X [ 0 , £ ] ) = T X ^ [ 0 , £ ] ) • As a x d = x^a for any d £ D, we have a ( x [ d , d + e ] ) = T x ( x [ d , d + e ] ) for any d e D. As a and x x are automorphisms i t follows that a ( X [ 0 , d ] ) * T x ( x [ 0 , d ] ) for any d e D O (0, E ) . As the ordering i n JR i s archmedian, D i s dense i n T and a and x x preserve countable sums of c h a r a c t e r i s t i c functions, i t follows that a ( X [ 0 , t ] > = Tx<X [ 0 f t ]> , f o r any t e T. Thus a and x x agree on every c h a r a c t e r i s t i c function i n Loo (T). I t follows now from .th- normality and l i n e a r i t y of a and x x that 30. a(M) = T X(M) for any M £ m , i . e . a = T X . So we have { x d : d £ D }' ^ { x x : x e T } . The reverse i n c l u s i o n i s cl e a r , so the proof i s complete. 7.2 Let m be I^QR) acting by m u l t i p l i c a t i o n on L 2QR) . For each non-zero r e a l number r, define s r be the automorphism of m given by : ( s r f ) (x) = f (r"' x) , f e ItoCR) , ' x e H . • Proposition 7.2 { s r : r e (Q • , r ^ 0 } ' = { s r : r e 3R , r ^ 0 } . Proof. Suppose' that a £ { s r : r e Q , r ^ 0 }' '. We f i r s t prove that e i t h e r a(B+) cz 8 + or a ( B + ) c r B_ , where B + denotes the set of a l l measurable subsets of the p o s i t i v e reals H + , and B_ denotes the set of a l l measurable subsets of the negative reals ]R_ . Suppose a (B +) j£ 8 + . Then there i s a non-zero A^ ^ £ 8 + such that a ( A ) H 3R_ ^ 0 . By replacing A^ by a subset i f necessary we can assume that ct(A^)d.3R_ . Let A^ be an a r b i t r a r y non-zero member of B + . Since { s r : r e <Q+ , r f 0 } i s ergodic on ]R+ (with the usual measure and measurable subsets) by an obvious a p p l i c a t i o n of Lemma 13.2.1 of [14], there must be a r a t i o n a l r > 0 such that s R ( A 1 ) n A 2 f 0 . 31. Thus s r[a(A')] A a(A £) f 0 . Therefore a(A 2) (~\ JR_ f 0 for any non-zero . A 2 E B+ . I t follows then a(B+)cr B_ . Thus we have e i t h e r a ( 8 + ) c : B+ or a (B+) c= BJ . We consider the case a (B+) c= B+ f i r s t . In t h i s case we have : of'(8+) <= B+ , a (l^CRjf.)) = , and C ' I J ^ Q R ) i s a n automorphism of ItoCE^.) (Here an f e L^CRf.) i s i d e n t i f i e d with the g e L^dR) where g = f " ori'']R+ and •'.-. g = 0 elsewhere.)' . Now define $ : L^GR.^)—> L^QR) by : . (<i>g)(x) = g(e x) , x e 1R , g e L^O^) . This i s well-defined because i f A i s a zero set i n 1R+. then log (A) i s a zero set i n It . In fa c t <f> i s an isomorphism and (j)"' : L M (JR) — > QR+) i s given by : (cj)-' f ) (x) = f ( l o g x), x £ ]R+ , f £ L^OR) . So the map a = <|>a<J>*' : LooQR) — > L^flR) ^ s a n automorphism.. I t i s e a s i l y checked that s r = <j>sr<j>~' = ^^.og r ' a n < ^ ^ r = ^ o r a n y r e ' As { log r : r e Q+ ,. r ^ 0 } i s dense i n It, i t follows from Proposition 7.1 that St = T , log t f o r some t e 1 + . This means a ^ ^ ^ = cfT1 T l o g t <J> = s t|L 0 0(|B +) . Therefore '•' a = s t . Next we consider the remaining case a(B+) cr. 8_ . In t h i s case 32. we have a.(I t oCR +)) = I ^ Q O , and a 11^ QR+) : L^ QR+) —> I^QRJ i s an isomorphism. Now define : L03OR_) — > L^OR) by : tyg)(x) = g ( - e x ) , x e JR , g e I ^ O R j . Again ij> i s well-defined, and i t i s an isomorphism. Put a = ipa<jf' , where <jfl i s given i n the preceding paragraph. Then a i s an automorphism of La,(R) . Also i t i s e a s i l y checked that s = \bs dT1 = T, for any - r - r T log r J r e $+\{0} . Thus for any r e Q+NjO} CtT, = CCS _ log r - r = i/>acj>~' ijjs_r<jr' = ^asrq>~' = s^ra<J>"' = T, a log r So as i n the preceding paragraph we conclude that 33. for some t e . Let h be an arbitrary element of L^QR^). Then ,'. : a(h)(x) = 0|T» T l Q g t*)(h)<x) . / = ( T l o g t*)(h)(log(-x)) = <j) (h) (log1!) = h(-4) = s_t(h)(x) . * Thus ' a | I* QR+) = s_t|Loo0R+) , and a = s_ t ' We see therefore (\ s r : r e Q , r ^ 0 } ' c { s r : r e TR, r f 0 } . The reverse inclusion is clear, and our proof is complete. 7.3 Let (Xj[_, S^ , u^) be a a-finite measure space, be L^ CX^ , Sj_, u^) acting by multiplication on L2(X^, S ^ , u^), i = 1,2 . Let G - ^ be a group of automorphisms of m^ , each element g^ of which is induced by a point transformation, denoted again by g•, ofL X. . Supppse that each G - is countable and ergodic on . Let m -be L o o O ^ x x 2 ' x ^2' y l X y2^ acting by multiplication on L2(X^ x X^ , S 1 x S^, y 1 x y , ) . For each pair g e G . g e G , denote by a,, s the automorphism of m given by : 1 1 2 2 (e ,e J 34. ( a ( g j f X x . y ) = f C g ^ x ) , g ^ y ) ) , (x,y) e J ^ x X j , , . f e l ^ C ^ x x,). Let denote the group of automorphisms of which commute with each element of G ^ , and suppose that each g| e G ! i s induced by a point transformation, again denoted by g! , of X i . Use the notation a , , fo r (g',g') e G ' x G ' , i n exactly the same way as a , N . 1 2 1 2 ,(g 1»g 2^ Proposition 7.3 . For each -:.'V Proof. Suppose, a e. { a . •• (g ,e ) e G x G . } ' ••, • " \ g^ > 82' 1 / i . z f E L^Cxp, l e t <f>fe I t o(X 1 x X ^ be defined by : (<j>f)(x,y) = f(x) , (x,y) e Xj x 5^ . • \ • • Then <)> : L^ ( X j ) —> L^ , (X^ x X^) mapping f to <f>f i s a monomorphism. Let S = ^ ( I t o C X , ) ) • Then i t i s not hard to see that S p= { g e ItoCX-^ x X ^ ) : for almost a l l x £ X^ , g(x,y) i s equivalent to a constant function of y E X 2 } . Now we show a maps 5 onto S . I t s u f f i c e s to show that a(S)(=- S . (For then we have also a"' ( S ) CZ S , i . e . S C a ( S ) . ) Let g £ S and consider a (g). As g E S , a m \g = g f o r each g £ G ? (where 0 35. denotes the i d e n t i t y element of ), and so a <q', % ) a ( g ) = a ( g ) f o r e'ach g 2 z G 2 • We s h a l l see that t h i s l a s t property implies a (g) e S . Now f o r each g^ e G 2 , there i s a measure zero set Eg of such that f o r each x e X \ E , the set 1 S2 h • F x - , { y £ \ '• a ( o g^ )<».(8)(x,y) a(g)(x,y)} i s of zero measure. Let E = \j E . Then, since G i s countable, g 2 e G 2 S2 E i s of zero measure. Furthermore f o r each x z ^ \ E, the set g 2 e G 2 i s of zero measure, and f o r each y e F x (x e E) we have : (*) a ( 0 jot (g) (x,y) = a(g) (x,y) , for a l l g^ z G 2 . This l a s t property (*) implies that f o r each f i x e d x e \ E.. a (g) (x,y) i s equivalent to a constant function of y e X 2 . For i f otherwise we claim there would be subsets A x , of ^ \ F x , each of p o s i t i v e measure, such that (**) a ( g ) ( x j 7 l ) f a.(g)(x,y 2) for y e A x and y^ z A x . To prove t h i s l a s t a s s e r t i o n (**), l e t 36. x e X^\ E be such that the r e a l (or imaginary) part a ( g ) 1 ( x , •) of a(g)(x, •) i s not equivalent to a constant function i n X , and l e t h 2 x be the function on defined by : h x(y) = \ o(g)1(x, y) , y e X2 F x 0 . , y £ F x For each n £ Z, l e t B n = [n, n + 1]. I f for a p a i r of d i s t i n c t m, ri e Z, • both h H (B ) and h"' (B_,) are of p o s i t i v e measure, then we are done. • x m x n k » ..Otherwise, there' i s a ' n Q e Z such that h ' (Bn- •) . i s of positive" measurey,-,-• o and that h"' (B n) i s of zero measure f or a l l n £ Z\{n Q} . Now we can apply a s i m i l a r argument to h„ and B„ . Such a procedure must come to o an' end a f t e r f i n i t e l y many steps, f o r otherwise, because of the compactness of B , h would be equivalent to a constant function. So (**) i s o x established. Now by the er g o d i c i t y of G^ , there i s a h^ e G^ such that A x = h 2 (AX)A* * 0 . Thus f o r y £ A x we have y £ A* , h2'(y) £ A x and a(0,h ) a ^ ) ( x , y ) = a(g)(x, h 2 ( y ) ) f ct(g)(x,y) , which contradicts (*). Therefore we see that f o r 'each f i x e d c:(g)(x,y) Is equivalent to a constant function of y £ X 2 . the proof that a(S) = S . x £ X x\ E, This completes 37. Denote now a = <J>-1 a<j> : L^CX.,).—> (X^) . Then a i s an automorphism of ( X 1 ) . Now f o r each E G^ . and each g e 5 we have <()g1 = ° (g 0)^ ' (here 0 denotes the i d e n t i t y of 6^), and ... •'; <fl a ( > 0 ) g = g^"' g . So we have ag 1 = g a for each g^ e G^ ., Thus a = g' f o r some g^ e G^ , i . e . .° | S."°(8 1',0)I S • Dually by s e t t i n g T = { g e L 0 0(X 1 x.X 2) : for almost a l l y e X£, g(x,y) i s equivalent to a constant function of x e X'1 } , we have a|T = ct,_ ,N|T 1 (o,g2')' f o r some g^ E G^ . Thus f o r any measurable rectangle V of . X x X^ we have Since a, a, i t\ are automorphisms of m = L (X x X ), by s i m i l a r A g^ .»=>2 0 0 1 2 38. arguments as those employed i n §7.1, we conclude a = a.(gl,8 2) f o r some (g^ ,g^) e G1 x 6^ . Therefore we see that { a , •> : (g. , g ^ e G]L x G 2 } ' ^ a ( g ' g') : ^ i , § 2 ^ e x } . The reverse i n c l u s i o n i s c l e a r , so the proof i s complete. 7.4 Let be the add i t i v e group of two elements 0 and 1, SQ the ring of a l l subsets of 2^ , u Q the measure on (Z 2 , S Q ) assigning q to 1 and 1 - q to 0 where q e [y, 1] . For each n £ Z, l e t ^ = Z 2 , S n = S Q , and P n = y 0 . Let X = V ^ , 5' = ) { 5 n , and l e t (X,S,u ) nzZ neZ 4 be the completion of ^ y n on (X, S'). Let A = ]_]_' X n . Let m be neZ nzZ L O T(X, S; u ) acting by m u l t i p l i c a t i o n on L 2(X, S, U q ) . For each <5 z A the t r a n s l a t i o n i n X by 6 induces an automorphism of m (cf. [18]). For s i m p l i c i t y we write a n instead of ct~ , where 6 n e A (n e Z) i s n such that <5n(m) = 0 i f m ^ n , = 1 i f m = n . Proposition 7.4 • (i ) When <1 > |r > { ct n : n e Z } ' = { : S e A } . ( i i ) • When q = — , the t r a n s l a t i o n i n X by any x z X induces an automorphism ct x of m , and { a n : n e X } ' = { a x : x e X 39. Proof. Suppose a £ { a n : n e 2-}' . Let P n denote the p r o j e c t i o n of X onto X n , and l e t ^ = P"1 (1), B n = P^' (0) , C'n = a C ^ ) and D n = a ( B n ) . ,[Here C n, D n are chosen, and f i x e d , among a class of sets, ., the dif f e r e n c e of any two members of which i s of zero measure. Here-after e q u a l i t i e s i n v o l v i n g sets of X means e q u a l i t i e s modulo sets of zero measure.] Then as A n A.B n = 0, U \ = X, « n ( A n ) = B n , % ( B n ) = B n , ctjjjCAj^ ) = f or each integer m =r= n ,. and a^a = aa^ , we have C n A D n =0, C n u D n = X , a n ( C n ) = D n , and f o r each integer m f n , a m ( C n ) = C n and a m ( D n ) = D n . We s h a l l see these properties imply that e i t h e r C n = A^ or C n = B n (and therefore D n = B n or D n = A^ ^ resp.). For th i s purpose assume that there i s a point x e C n with P n(x) = 1 , and we s h a l l show that under this a d d i t i o n a l condition, C n = A^ . I t s u f f i c e s to show Aj^C C n (modulo sets of zero measure), as t h i s implies that B nC D n and f i n a l l y , together with other properties of Ap , B n , C n , D n , that A^ = C n . Suppose. A,^ C n (modulo sets of zero measure) . Then there i s a set E nc: An\ C n of p o s i t i v e measure. I f C nC A^ then Dnc: B n so that E nC Aj^X D n , which i s imposible as C n U D n = X . So there i s a point ' y e C n with- ? n ( y ) = 0 . • Now as i n addition we have x z C n with P n(x) = 1 , and as ct m(C n) = C n for a l l integer m ^ n , i t follows that .'.{'. z £• X : f o r some '-.z'-.-.e C n , P^('z) = P^ Cz') • for. a l l 'integer"' k"> n'-} ^ /C^ Since D n = a n ( C n ) i t follows that D nc: C n . Thus again we would have E nc: A n \ ( C n \j D n) , which i s absurd. So we have proved that i f there i s a point x e C n with P n(x) = 1 , then C C n and so 40. • C n ~ A i • Now s i m i l a r l y we can prove that i f there i s a point x £ C n with P n(x) =.0 then C n " B n • Thus we have : . a(A n) = A n e i t h e r (1) " { a(B n) = B n or (2) a ( V = B n a(B n) = A,, Now define an x e X by x(n) = • Then we have 0 f o r such n that (1) i s the case, 1 f o r such n that (2) is. the case, a(An) = A n + x for each n £ Thus we have a(S) = S + x for any S e 5 . [In p a r t i c u l a r the t r a n s l a t i o n by x maps a measure zero set to a set of zero measure.] ' 1 Consider now the case where q = — . In t h i s case, the Kakutani theorem [11] implies that the measure by v u ( z £ X) defined by : 41. yz(S) = y q(s + z), s e S i s equivalent to ]iq • Thus the t r a n s l a t i o n i n X by an a r b i t r a r y z e X. induces an automorphism az of m . Combining the r e s u l t of the preceding paragraph we have a = a x Now we turn to the case where q > y . In t h i s case, the Kakutani theorem [11] implies, that the measure u z , defined above, i s equivalent to u'q only i f z e A ; i n other words, the t r a n s l a t i o n i n X by z e X maps a measure zero set to a measure zero set only i f z e A. Thus, i n view of the r e s u l t i n a previous paragraph, we have x e A and a = a x . We see therefore { a n : n e S } ' c { ag : S e A } i n the f i r s t case ( i . e . when q > —) , and { ct n : n e 2 } ' c { a x : x e A } i n the second case ( i . e . when q = -j ) . As we have seen that i n the second case, the t r a n s l a t i o n i n X by an a r b i t r a r y x e X induces an automorphism a x of m , the reverse i n c l u s i o n s are c l e a r . Our proof i s thus complete. We now define a measure on [0, 1], and show that there i s an isometric isomorphism from L 2[0, 1] onto I<2 (X), which maps L ^ Q O j l ] , X q) onto L^X, u_). To t h i s end define .ty : X — > [0, 1] by : x . i / \ V 'x(n) It i s easy to check that i|» i s measurable. Define Xq on [0, 1] by 42. X^(B) = yqOr,"< (B))> B Lebesque measurable subset of [0, .1] Notice that X, = A, the Lebesque measure on [0, .1] because they agree on vk 1 ' " sets of the form [0, 2,. , —r~ ) • In general,' an isometric isomorphism .x-1 2 n i • <f> i s defined on L 2 ( [ 0 , 1 ] , X^) to L 2(X, y ) by (<J)f)(x) = f ( * ( x ) ) , f e L 2[0, 1 ] , x e X Since the set of a l l dyadic r a t i o n a l s i n [0, 1] i s a Xg-zero set, <f> i s indeed s u r j e c t i v e . C l e a r l y <j> maps L^QO, 1 ] , X^) onto I^CX, y q ) • X q) by Now we define, f or each n E Z , an automorphism 3 n of LooCtO.l]', en = "n* > where. a n i s defined previously. These 3n's a r e induced by the point ^transformation i|/a n^"' on [0, 1 ] , The ambiquity of ^a n^"' at dyadic <rationals , w i l l be s e t t l e d by s e t t i n g : f tya^-' )<-^) = ^ f o r k = 0, 1, 2 n With t h i s agreement we have O K r * " ' ) (y) = \ 1 . _ „ ,2i 2 i + l x y + — , ,xf y k (— , ——) 2 n 2 n 2 n 1 .. /2i+l 2i+2 N . . . y - — , i f y k (—— , — — ) , 1=0,1, ,n ,n ,2 n-] l y otherwise 43. We shall also refer to these point transformations by g n . The automorphisms . <J>"' c^<|> (x e X) are also induced by point transformations, even though their expressions are not as neat as the $n's However for convenience we shall refer to them by the following method. Fo each dyadic irrational number s let ct = ii/ct , .ii"- 1 where x(s) is the s x(s) T unique element in X such that ip(x(s)) = s, and where the ambiquity of a g at dyadic rationals is settled as before. For a dyadic rational t, let x(t) be the unique element in X\A such that iKx(t)) = t, and let a t = ^ a x ( t ) ^ ~ ' » with the ambiquity of afc settled as before. Then the automorphism (j)"1 ax<j> is either a finite product of the 3n's , or is induced by the point transformation a r for some r e [0, 1]. With these notations we have Corollary 7.5 Let m be L o o([0, 1], Aq) acting by multiplication on L2([0> 1]>. • Then an automorphism a of m commutes with a l l 3n's (n e Z) i f and only i f : / (i) in the case q = y , i.e. X^=A, a is either a finite product of the gn's , or is induced by. an for some r £ [0, 1] ; (ii) in the case q > ^ , a is a finite product of the Bn's 8. Operators dis t i n g u i s h a b l e by G' A [G] and G' i We summarize here the r e s u l t s obtained i n §§5, 6, 7 i n a form most 'suitable f o r applications to operator theory. Theorem 8.1 Suppose ' A 1 , A> are two operators on a H i l b e r t space H such that R(Aj) = R ^ ) = &[m, K, G, g »-> Ug] for some ergodic and abelian C-system [m, K, G, g >-> U g] , that' (R(Re A : ) , R(Re Ag)) f i t s i n [a, m 0 1], that Im A x = Im = .T, and that R(T) = R(U g 0 V g .: g e G) [cf. D e f i n i t i o n 2.1, 2.2 for not a t i o n s ] . Then k^. and A£ are u n i t a r i l y equivalent i f and only i f they are ( a l g e b r a i c a l l y ) equivalent, and that i s the case i f and only i f there i s an a e G' such that a(Re A x) = Re A 2 , where G' = { a e A ( m 0 1) : X f o r some s e G' , a(M 0 1) = s(M) 0 1 for a l l M e m } . Moreover A^ and A^ . are inner equivalent i f and only i f for some g e G : g(Re A :) = Re A 2 . Proof. The present theorem follows d i r e c t l y from Lemma 5.1 ( i i ) , Theorem 5.2 and D e f i n i t i o n 2.2. 45. Theorem 8.2 . Suppose that (i ) ni i s L^[0, 1] acting on K = L 2[0, 1] , D a dense subgroup of [0, 1] under the addition mod 1, Q the group of a l l automorphisms <on m induced by the t r a n s l a t i o n i n [0, .1] by d e D , and g e G 1 - > U_. the usual (cf. [15]) unitary representation of G on K ; or ( i i ) m i s L^OR) acting on K = L 2QR), D a dense subgroup of K, G the group of a l l automorphisms on m induced by the t r a n s l a t i o n i n TR by d e D , and g e G > U the usual unitary representation of Q on K ; © or ( i i i ) m i s L^QR) acting on K = L 2 0 R ) , G the group of a l l automorphisms s r on m given by : ( s r f ) ( x ) = f ( r - ' x) , f e L r o 0 R ) , x e JR , fo r r e Q with r ^ 0, and g e G *~> U the usual unitary representation of G on K ; or (iv) m i s L ^ X j x X 2, Sj^ x S 2, u 1 x U 2) acting on K = L £ (X1 x X 2, x S 2 > y 1 x u 2 ) , G the group of all'automorphisms a, . , (g , g ) e .Gj x G2 , of _m given by : ( a ( g ,g ) f ) ( x > y ) : = f ^ O O , . 8 2 ( y ) ) , ' (x,y) e x x x X 2, f e x X,,), where each G^(i=l,'2) i s a countable, abelian and ergodic group of automorphisms on m± [ = L^CX^ Sj., y ± ) acting on L, (X ±, S±, u-)] such 46. that each element e and each e G^ are induced by point transformations, denoted again by g^ ; g^ r e s p e c t i v e l y , of . Suppose that each G-; has a unitary representation g. »-> U e on L 9(X-, S., y . ) - . Let U, v , (g , g ) e G x G„ , be the unitary operator on L 2 (X-, x Y , » vg^ .* §2 ^ ' ' '. . . . ^ 5 i x 52 ' y i * y2^ § i v e n b y : • U ( g 1 ) g 2 ) f l , 2 " V l ) ( V 2 > ' where f]_ e L 2 ^ , y ^ , f 2 e L 2 (X 2, S 2, y 2 ) and ^ 2 ( x , y ) = f 1 ( x ) f 2 ( y ) , f o r a l l (x,y) e X^ x X 2 . Suppose G i s with the unitary representation °'(g1 ,g2) >~> U ( g x ,g2) * or (v) rn_ i s ^ [ O , 1] acting by m u l t i p l i c a t i o n on K = L 2 [ 0 , 1] , G = .{ 3g : S e A } with the usual unitary representation 3g •—> U^ . on K ( c f . the paragraph preceding C o r o l l a r y 7.5 f o r the d e f i n i t i o n of 3<$ ) ; or (vi) m i s L o o([0, l ] , v A q ) (q > ^) acting by m u l t i p l i c a t i o n on K = L 2 ( [ 0 , 1], Aq), G = { ; <5 e A } with the usual unitary representation and that the H i l b e r t space H = K 8 Kg and the u n i t a r i e s Vg (g e G) are 1 as defined i n § 2 [ D e f i n i t i o n 2.1], Suppose A^ , A 2 are operators on H such that R(A ) = R(A ) = a[m, K, G, g U ], . Im A1 = Im A 2 , R(Im Aj = R(U g ® V g : g e G) , and that (R(Re A ^ , R(Re A )) f i t s i n [a, m 8 1] . ' Denote R e = 0 1, being the multiplication by the essentially i i bounded measurable function . Then A.^ and are unitarily equivalent i f and only i f they are (algebraically) equivalent, and they are unitarily equivalent i f and only i f , in the case : (i) , f 2 = T^Cfj) for some r e [0, 1], where T r is the translation mod 1 by r ; (ii) ,' f^ = T r ^ i ^ ^ o r s o m e r e TR , where r r is the translation by (i i i ) , f 2 = s r ( f 1 ) for some non-zero real number r, where s r is defined by : (s rf) (x) = f(r"* x) , x e TR ; (iv) , f 2 = ^ ( f ^ for some (g[,g2) e G^ X G^ , where (a ( g, ) gpf)(x,y) = f( g;(x),g^(y)) ; (v) , f £ = a r(f 1) for some r e IR, or f 2 = 0 n 6 n ••• S n ^ i ) f o r some finite sequence of integers n^, n 2, •••, n m , where ar's are defined in the paragraph preceding Corollary 7.5 ; (vi) , f - 3 n g_ ••• 3n (f, ) for some finite sequence of integers 1 2 m n 1 ? n 2, n m . Proof. The present theorem follows readily from Theorem 8.1, Propositions 7.1, 7.2, 7.3 and Corollary 7.5 . 48. We now illustrate Theorem 8.2 by the following simple example. Let be L M OR) acting by multiplication on L 2 OR) , D the dyadic rationals in B. , and G the group of a l l automorphisms of m induced by translations (in IR ) by d e D . Suppose that J «d(ud ® V d)', deD a d e C , is self-adjoint, and i t generates R(Ud 0 Vd : deD). Let f ^ f 2 be two strictly monotone, continuous, bounded and real-valued functions defined on PR . Then by Theorem 8.2 we have : Mf 0 1 + i I a d(U d 0 Vd) and Mf 0 1 + i J a d(U d 0 Vd)' 1 deD 2 deD are unitarily equivalent i f and only i f for some r e ]R : f 2 (x) = f j, (x - r) a.e. x e H 4 9 . 9 . Examples of non-equivalent operators of various type with R(Re A) thick and of various type i n R(A) , In t h i s s ection we s h a l l construct numerous non-equivalent operators on the separable H i l b e r t space, by applying the r e s u l t s stated i n §8 and the theory of abelian subalgebras developed by Bures [ 5 ] . More p r e c i s e l y , we s h a l l prove the following theorem. [For the terminology see § 9 . 1 below.] Theorem 9 . 1 [cf. 5 , § 1 5 . ] -I. Given a p o s i t i v e integer n >_ 2, there e x i s t s a family ±£x (Card I the continuum) of pairwise u n i t a r i l y non-equivalent operators on the separable H i l b e r t space such that : 1 . R(A^) i s the h y p e r f i n i t e f a c t o r a on the separable H i l b e r t space, 2 . R(Re Aj.) = E f o r a l l i e I , 3 . E i s thick i n <X , of d e f i c i e n c y type I and uniform m u l t i p l i c i t y n, and E c i s regular maximal abelian i n a . I I . Given a p o s i t i v e integer n >_ 2 and a r e a l number x > n, there e x i s t s a family (A^) ^ ^ (Card I the continuum) of pairwise u n i t a r i l y non-equivalent operators on the separable H i l b e r t space such that : . 1 . R(A^) i s the h y p e r f i n i t e f a c t o r a on the separable H i l b e r t space, 50. - . ( ' • • • 2 . R(Re A i) = E f o r a l l i e I , 3. E i s thick i n a , of deficiency type I and uniform m u l t i p l i c i t y x, and E c i s regular maximal abelian i n a.'. ( • I I I . There ex i s t s a family (A.) (Card I the continuum) of pairwise 1 i e l u n i t a r i l y non-equivalent operators on the separable H i l b e r t space such that : 1. R(A^) i s the h y p e r f i n i t e f a c t o r a on the separable H i l b e r t space, 2. R(Re A ±) = E f o r a l l i e l , 3. E i s t h i c k i n a , of def i c i e n c y type and uniform m u l t i p l i c i t y oo , and E c i s regular maximal abelian i n a . IV. Given A : [1, oo ) — > [0, 1] non-decreasing, r i g h t continuous, and with A(2) = 0 and l i m A(x) = 1, there e x i s t s a family (A^)^ (Card I the continuum) of pairwise u n i t a r i l y non-equivalent operators on the separable H i l b e r t space such that : 1. R(A^) i s the h y p e r f i n i t e f a c t o r a on the separable H i l b e r t space, 2: R(Re A±) = E. f o r a l l i e l , 3. E i s thick i n a , of def i c i e n c y type I 2 and with m u l t i p l i c i t y function A , and E c i s regular maximal abelian i n a. . 51. V. Given a p o s i t i v e integer n >_ 2, there e x i s t s a family (^i) ±el (Card I the continuum) of pairwise u n i t a r i l y non-equivalent operators on the separable H i l b e r t space such that : il. - ' R(A ±) = a f o r a l l i e I , 2. a i s a f a c t o r of 1 type 11^ , 3. R(Re A.) = E fori a l l i e I , 4. E i s thick i n a. and i s of deficiency type I n , and E c i s regular maximal abelian i n a, VI. There ex i s t s a family . (Aj[)^ ej (Card I the continuum) of pairwise u n i t a r i l y non-equivalent operators on the separable H i l b e r t space such that 1. R(A ±) = d f o r a l l i e I , 2. a. i s a f a c t o r of; type 11^ , 3. R(Re A/) = E f o r a l l i e I , • \ 4. E i s thick i n & and i s of deficiency type 1^ , and E c i s regular maximal abelian i n 0. . VII. There ex i s t s a family ( ^ l ) i e i (Card I the continuum) of pairwise u n i t a r i l y non-equivalent operators on the separable H i l b e r t space such that 1. R(Ai) = a f o r a l l i e I , 2. a i s a f a c t o r of type -II , 3. R(ReA ±) = E for a l l i e I , . 52. 4.'- E i s thick i n a and i s of deficiency type I I I , and E c i s . , ' • " •'! • regular maximal abelian iri Ci . r. VIII: Given a p o s i t i v e integer n >_. 2,. there e x i s t s a family ( A i ^ - j _ e i (Card I the continuum) of pairwise . u n i t a r i l y non-equivalent operators on-the separable H i l b e r t space, such that : 1. RCA.^ ) = a for a l l i E I , 2. a i s a f a c t o r of type III , • 3. R(Re A i) = E for a l l i s I , 4. E i s thick i n a and i s of deficiency type I n , and E c i s regular maximal abelian i n a . IX. There e x i s t s a family ^i)±zi (Card I the continuum) of pairwise u n i t a r i l y non-equivalent operators on the separable H i l b e r t space such that 1. R(A ±) = 0. f o r a l l i e l , 2. a i s a f a c t o r of type I I I , 3. R(Re A ±) = E f o r a l l i e l ' , 4. E i s thick i n d and i s of deficiency, type 1^ , , and. E c : . i s . regular maximal abelian i n Ci . X. There e x i s t s a family ^i)±zj_ (Card I the continuum) of pairwise u n i t a r i l y non-equivalent operators on the separable H i l b e r t space such that 1. R(A i) = a f o r a l l i s I , 2. a i s a f a c t o r of type I I I , 53. "3. R(Re A±) = E for a l l i e l , ,4..."\E. is thick in Cl and is of deficiency type III, and .,Ec.:,'is regular maximal abelian in Cl . The detailed proof of the above theorem is given in §9.2 below. However the construction can be outlined as follows. We begin with a suitable C-system [m, K, G, g v-> Ug] and construct von Neumann algebras Cl , m and n as outlined in §2. Then we construct a suitable thick subalgebra E_ in m with R(E_ ® 1, n) = a . We then find for each of £ and n a self-adjoint single generator (i.e. a self-adjoint element S of E_ with R(S) = E_.) ; they will respectively be the real part and the imaginary part of our operator. By varying the real part of this operator according to the results of § 8 and by varying the E_ , m_ and G , we obtain numerous non-equivalent operators of type 11^ , 11^ and III on the separable Hilbert space. This section is divided into two subsections. In the first subsection we give a summary of what we need from the theory of thick subalgebras developed in [5]. In the second subsection we construct those promised operators. 9.1." We introduce here a few (algebraic) invariants and results from the the theory of abelian subalgebras developed by Bures in [5], which will be used in the next subsection. 5 4 . Definition 9 . 1 . 1 Suppose that a is a factor of type 1 1 ^ with the normalized, dimension function d, and E a fixed abelian von Neumann sub-algebra of a. . (i) If P is a projection of E c = E' H a , define : C(P) = inf{ E : E a projection of E with E >_ P } . (ii) If P xs a non-zero projection of E c , define : s(P) = d(C(P))/d(P) . ( i i i ) If E is a non-zero projection of E , define : r(E) = sup{ s(P) : P a projection of Ec with 0 < P <_ E } (iv) If E is a non-zero projection of E , define : m(E) = inf{ r(F) : F a projection of E with 0 < F <_ E } . (v) A projection E . of E is said to have uniform multiplicity (with respect to [ E , <x]) x i f r(E) = m(E) = x . E is said to have uniform multiplicity x in a i f r(l) = m(l) = x . (vi) It has been proved in [ 5 , Proposition 1 . 1 4 ] that for each real number x >_ 1 there, exists a unique projection E x of E such that : ..; '• ."'v. (a) r(E x) <_ x , ^ / and (B) i f F is a projection of E with 0 < F <_ 1 - E x , then r(F) > x. We define the multiplicity function of E as the function A from [ 1 , co ) to [ 0 , 1 ] given by : 55. A(x) = d(E x) f o r a l l x > 1 . D e f i n i t i o n 9.1.2.. Let m be a von Neumann algebra and Q a f u l l m-group (cf. D e f i n i t i o n 5.1(v)). (i ) A . G-trace i s a trace OJ on m+ which s a t i s f i e s ; .:, • . • " w(a(M)) = w(M) for a l l a e G and a l l M e m+ . ( i i ) G w i l l be c a l l e d f i n i t e (resp. s e m i - f i n i t e ) i f , f o r every T e n ? with T > 0, there e x i s t s a f i n i t e (resp. s e m i - f i n i t e ) normal G-trace m s a t i s f y i n g w(T) > 0 . ( i i i ) G w i l l be c a l l e d p r o p e r l y - i n f i n i t e (resp. of type III) i f there e x i s t s no non-zero f i n i t e (resp. s e m i - f i n i t e ) normal G-trace. .(iv) A p r o j e c t i o n E of m w i l l be c a l l e d G-abelian i f a(F) <_ F + (1 - E) for a l l a e G and for a l l projections F of m with F <_ E. (v) G w i l l be c a l l e d continuous i f every non-zero p r o j e c t i o n of. m f a i l s to be G-abelian. (vi) I f E and F are projections of m , write E ~ F (G) to mean that there e x i s t s a family (a,-, E.,-). T with each J 1 1 l e i ct^ e G^ and each E^ a p r o j e c t i o n of m , such that ; : y. T E.. = E and T a,(E,-)' = F. 56. ( v i i ) G w i l l be said to be of type I i f , f o r every non-zero p r o j e c t i o n E of in , there e x i s t s a G -abelian p r o j e c t i o n E^' of fn s a t i s f y i n g 0 < < E . »(viii) G w i l l be said to be of type II (resp. ILjo) i f G i s continuous and f i n i t e (resp. continuous, s e m i f i n i t e and properly i n f i n i t e ) . (ix) Let n be a c a r d i n a l number. G w i l l be sa i d to be of type I n i f there e x i s t s a family ( E i ) i e x °^ G-abelian projections , :v\ of • m such that £ = 1, the. cardinality.' p , f I - ' Is -n',.''.. and E ± ~ Ej (G) f o r a l l i , j e I . D e f i n i t i o n 9.1.3 Let E be an abelian subalgebra of a von Neumann algebra a (not n e c e s s a r i l y a f a c t o r ) . E i s sa i d to be of deficiency type I (resp. type I n , type 11^ , type 11^ and type III) i f so i s the E c c-group D, which consists of a l l automorphisms of E c c keeping E pointwise f i x e d . (Note that E c c E ( E ' H a ) ' n a , that E c c i s abelian and D i s f u l l . ) D e f i n i t i o n 9.1.4 An abelian von Neumann subalgebra E of a i s c a l l e d thick i f E c c , i s maximal abelian i n a . Note that D e f i n i t i o n s 9.1.1, 9.1.3 and 9.1.4 are (algebraic) i n v a r i a n t s of E with respect to a i n the sense that i f E 1 and are abelian subalgebras of a and a 2 r e s p e c t i v e l y , and i f there i s an isomorphism <j> of , a 1 onto a 2 . with <j) (E ) = ,E 2 " , then •£ has one of those properties i f and only i f so has E 2 . We now turn to methods of producing thick subalgebras having various i n v a r i a n t s . Lemmas 9.1.2 to 9.1.8 57. are taken from [5], and on these lemmas our construction of thick subalgebras i n §9.2 w i l l be based. We need two more notions before we can introduce these lemmas. D e f i n i t i o n 9.1.5 Suppose that m. i s an abelian von Neumann algebra, H i s an m-group, a e A On) (cf. D e f i n i t i o n 5.1 ( i ) ) , and P i s a p r o j e c t i o n of m . We w i l l say that a i s strongly orthogonal to H on P ( i n symbols, a i H on P) i f there e x i s t s a family (P..-). T °f projections of m such j- i e J_ that sup > P and : i e i • [(a5)(P i)][5'(P i)]P = 0 for a l l V. e H and a l l i e I. We define E ' ( a , H) to be 1 - P ( a , H) where P(a, H) i s sup{ P r o j e c t i o n P of m : a i H on P } . D e f i n i t i o n 9.1.6 Suppose a i s a von Neumann algebra, ScrA(a) and S e a . We define : S° = { A e a : a(A) = A f o r a l l a e S } , and S° = { a E A(a) : a(A) = A f o r a l l A e S } . Lemma 9.1.1 Suppose that m is" an abelian von Neumann algebra, H i s an m-group, a e A(m) and P i s a p r o j e c t i o n of m . Then a I H on P i f and only i f . there e x i s t s a family-." ( E ^ . -j of projections "of H° such 58. that sup Pi>_ P and [ a ( E i ) ] E i P = O . f o r each i e l . i e l Lemma 9.1.2 Suppose that [m, K, G, g '»-> Ug] i s a C-system. Let a = a[m, K, G, g i—> Ug] . Suppose now that H i s . an w-group s a t i s f y i n g : E'(g, H) = E(g, 1) for a l l g e G , Then E = H° 0 1 . i s t h i c k i n a with E c = m 0 1 . . Corollary 9.1.3 Suppose £ e A (m_) be such that e i t h e r one of the following holds : (i) £ n = 1 f o r some n e 2 ; ( i i ) there e x i s t s a family ( Q k ^ e z °f p r o j e c t i o n s 1 of m_ such that I k £ Z Q k = 1 and ?(Q k ) = Q k + ]_ f o r a l l k e % . Suppose also that f o r any integer . m and any g e G : E ( ? m , g) <_ E ( l , g) , where 1 i s the i d e n t i t y of G . Then E = £° 8 1 i s thick i n a with E c = m 0 1 . . Lemma 9.1.4 Suppose that m i s an abelian von Neumann' algebra, that n i s a p o s i t i v e integer or oo , and that £ e A(m). If n i s f i n i t e assume that g n = 1 . Assume that there, e x i s t s a family (Qj_) • of projections n 59. of m such that - l ± e Z = 1 a n d ^^i) ~ ^ i + i f o r a 1 1 1 E z n • ^ W e n take 2ZOT = 2> .) Then (5°)° i s of type I n . , : , \. .-/[;;• " Lemma 9.1.5 Suppose that the conditions of Lemma 9.1.4 hold . Suppose 'also that E = £° i s thick i n a f a c t o r a of type I I 1 with E c = m , and denote the normalized dimesnion function by d. Suppose the following condition holds : d[5(PQ ±)] = [d(Q. + 1)/d(Q.)][d(PQ i)] for a l l projections P .of m , and a l l i e 2£n . Then E has uniform m u l t i p l i c i t y x, where x = sup{ l/d(Q^) : i e 2 } . In p a r t i c u l a r , i f n = OD , then E has uniform m u l t i p l i c i t y co . Lemma 9.1.6 Suppose that m i s an abelian von Neumann algebra, that' 2 £ £ A(m) s a t i s f i e s . •£ =1, and that E = £° i s thick i n a fac t o r of type I I j with E ° = m . Denote by d the normalized dimension function on a .'• Suppose also that there e x i s t s a p r o j e c t i o n Q of m such that : £(Q) = 1 - Q „ and d[£(PQ)] >_ d(PQ) for a l l projections P of m . F i n a l l y , suppose that f o r each x >_ 1 there e x i s t s a p r o j e c t i o n E x of E such that, f o r a l l projections P of mi. d[£(PQE x)] <_ (x - DtdCPQE^J and d[5(PQ(l - E x ) ) ] > (x - l ) [ d ( P Q ( l - E x ) ) ] whenever PQ(1 -•E x) + 0 . Then A , defined by': . A(x) = d(E x) for a l l x > 1 , i s the m u l t i p l i c i t y function of E . 60. Lemma 9.1.7 Suppose that m2 i s a l g e b r a i c a l l y isomorphic to L^fO, 1] a c t i n g by m u l t i p l i c a t i o n on 1^ [0, .1], that i s of countable decomposi -b i l i t y type [ i . e . there i s a countably i n f i n i t e f a m i l y ± z % °^ p r o j e c t i o n s of m. w i t h - J . E. = 1 , and any f a m i l y ( F - ) . of orthogonal p r o j e c t i o n s of m • i s countable], that m = m1 0 m2 and E = m1 8 £ . Then E° i s of type I I I . Let m, ft be a b e l i a n W*-algebfas, and H an ergodic ft-group. (1 0 H)° = m 0 <C , where 1 0 H = { c t e A ( m 0 t t ) : . a = 1 8 h f o r some h e H } ( c f . [ 9 ] , p. 56). We s h a l l a l s o use the f o l l o w i n g n o t i o n and lemma from [ 5 ] . D e f i n i t i o n 9.1.7 Suppose that m i s a von Neumann alge b r a , that a e A(m), and that a) i s a tr a c e on m + . We w i l l say that a i s l o c a l l y non-to-p r e s e r v i n g i f there i s no non-zero p r o j e c t i o n E of m such that . to (a (EM)) = u (EM). f o r a l l M £ rr& . We say that a preserves to i f to (a (M)) = to (M) f o r a l l M £ m+ . Lemma 9.1.8 Then , Lemma 9.1.9 Suppose that to is a trace on the von Neumann algebra m and that a > 3 e AOn) . If a is locally non-to-preserving and 3 preserves a) , then".; ' V - ; U ' , : •"' V E(a, g) .- 0 Definition 9.1.8 Let m be an abelian von Neumann subalgebra of the von .'• Neumann algebra ci . Then m is regular in a i f . R(U : U unitary in a with U%UCm) = a . 9.2 In this subsection we shall follow the outline in the beginning of §9 to construct the promised operators, and thus prove Theorem 9.1. We shall adhere to the use of notations m_, G, a, etc. therein. Note that (cf. [5]) with the conditions we already have on G , a. will be a factor of type II 1 , 11^ or III i f so is the type of G • Furthermore (cf. [5]) a. will be the hyperfinite factor of type 1^ on the separable Hilbert space i f G is type IIj and abelian. By a theorem of von Neumann [13], the abelian von Neumann algebra R(Ug 8 Vg : g e G)' on the separable Hilbert space H has a self-adjoint single generator. So our first task is to construct a thick abelian subalgebra E of. a so that R(E, Ug 8 V g : g e G) = a . We now establish a lemma which will be useful forjthis purpose. •'• • '• • ! • • .'• '. ' ' • Lemma 9.2.1 Let £ e A(m) . Suppose that there is a nen-zero projection Q of; m such that for every non-zero projection P of m with P < Q , there exists a non-zero projection R of R(£°, Ue. : g e G) (X m with R <_ P •„" Then a l l projections of m are In R(£°, U : g e G) and, O R(?° 0 1, U g @ V g : . g e G) = a . Proof. Since a = R(m 0 1, U 0 V : g e G) i t s u f f i c e s to prove that a l l projections of m_ are i n R (§0, Ug : g e G). Let P be a non-zero, pr o j e c t i o n of m . Let R be the la r g e s t p r o j e c t i o n i n R(£°, Ug : g e G)' f l H m_ with respect to the property that R < P . Assume that P - R f 0 . As G i s ergodic on m_ , there i s a g e G such that Q[U g(P - R)U|] ^ 0. By assumption there i s a non-zero p r o j e c t i o n Q Q of R(^°, Ug : g e G) fl H m such that Q 0 <_ Q[U g(P - R)U*]. But then 0 f U|(Q 0)U g <_ P - R , and U*(Q 0)U g ! i s a p r o j e c t i o n of R(£°, U g : g £ G) A m .. This contradicts the choice of R. Therefore P - R = 0 , and P £ R(£°, U g : g e G) . That completes the proof. We can now prove Theorem 9.1. Proof of Theorem 9.1 We s h a l l prove the theorem by constructing the required operators according to the o u t l i n e given i n the beginning of § 9. The construction i s an inmitation of that of the corresponding thick sub-algebras i n [5], with some modification to s u i t the present purpose. Paragraph indexed by I etc. w i l l serve to e s t a b l i s h statement I etc. of Theorem 9.1 . Since i n each case the m w i l l be L^CX) acting by mul-t i p l i c a t i o n on L 2 (X) f o r some o*-finite measure space X, we specify only the group of " i n v a r i a n t " transformations, with the unitary representation 63. g > Ug of G on L2(X) understood as the usual one in our case (cf. [15]). These G's will be abelian, ergodic and free on' m_ . We shall write R(E,G) instead of R (E, U g 0 Vg : g E G) , and write m = m_ & l j ^ . For f E ^"(X), wj.ll be the operator in m corresponding to multiplication by f. I. Take m to be I^fO, 1] acting on [0, j ] , G = { T x : x p-adic rational in [0, 1] } , where p is a fixed prime >_ 2 such that p does not divide , n, and T x is the automorphism of m_ given by (x xf)(y) = f(y - x) for any f e ^[0, 1] and y e [0, 1] (addition in [0, 1] is mod 1) . It is well-known that G is of type IIj . Let £ be the automorphism of m_ given by : - y) > y e •£ (£f)(y) = • f(y . ye. , i] f ( i - y) , y e [0 where f E LJ0', .1] . Then £ n = 1 , and 5(Q±) = Q i + 1 for i E Z n , where Q- is the projection of rn corresponding to the characteristic function of . [— , ^ —] . Observing that — is never a p-adic rational,, i n ' n 6 n r ' is easy to verify that. E ( ? r a , t x ) = 0 , for a l l integers m and a l l p-adic rationals x'e" (0', 1) '•- . By. Corollary 9.1.3, Lemmas 9.1.4, 9.1.5, the subalgebra E = £° 0 1 of m is thick in a , of deficiency type n , with uniform multiplicity n and E c = m. To show that R(E, G) = a , let x be a p-adic rational in (0, "2^ -) ,, and let Pj be the projection of m_ corresponding to x> Let ' Px = ? i +. g (Px) + .-.+ 5 n" 1(P 1) • Then Pj e ?° and 0 < (Pj) < Q0 » a n d V i - x ( V i s i n R (^°> U j g e G) H m . So by Lemma 9.2.1 R(E, G) = a. In order to have a self-adjoint single generator 0 1 of E (Mj £ m : the multiplication by f £ L^fO, 1]) we may take any strictly monotone, continuous and real-valued function g defined on [0, —] , and i n let f = g, + 5(g) + v + 5 n _ 1 ( g ) , where g is extended to [0, 1] with 0 as its value on [0, 1] \ [ 0 , ^ ] Thus in particular .if for each strictly positive real number r we denote by g„ the function x t-> x r on [0,.—] and let 65. then each Mf 0 1 is, a self-adjoint single generator of ' E . Let T be ' r ! ' " a self-adjoint single generator of the abelian R(U_ 0 V : "g e 6) , and let A r be the operator on H such that Re A r = M^ 0 1 , and Im A r — T. r Then by Theorem 8.2 (A ) is a family with the desired properties. . II. Take m to be L^tO, 1] acting on L 2[0, 1] , G = { r g : g p-adic rational in [0, 1] ,} , where p is a fixed'integer >_ 2" .'4.G'*:-'is of type • II 1 . Select , ( 5 k ) k e Z s u c h t h a t 1 = 6Q < fil < .... < S n_ 2 < 6n_± , and . /. _ S, = x Lke2L k nThis is possible as x > n . \Let ^xk^keS ^ e a P a r t i - t : ^ o n °f 1) : 0 = x Q < x1 < < xn_± <. 1 = Xn , such that x^+j ~ x k = ^k^x ^ o r ^ e ^n * Then we define an automorphism £ of m by . : 6 k-1 (?f) ( x k + y) = f ( x ^ + - — ^ y) for a l l y £ [0, S k/x] , where f e L^tO, 1] . (We take x., = X r ^ and <$_, = 5 ^ .) It is easy to see that for m =1, 2, • • •, n - l,.£m is locally non-A-preserving ( A Lebesque measure). By Lemmas 9.1.3, 9.1.4, 9.1.5, the abelian algebra 66. E = £° 0 1 i s t h i c k - i n a w i t h E c = m , a n d . i s o f d e f i c i e n c y t y p e I n and u n i f o r m m u l t i p l i c i t y x . .. . To. p r o v e t h a t R ( E , G) = CL , l e t Q k , ? l , P 2 be r e s p e c t i v e l y " t h e p r o j e c t i o n s o f m c o r r e s p o n d i n g t o t h e c h a r a c t e r i s t i c f u n c t i o n s o f [x^ +^1"! ' • V^ .*] and [3/4x , 1/x] , and l e t and P2 = P2 + g(P2) + c 2(P 2) + ••• + ? n 1(P 2) Select a p-adic rational g so that g e (l/2x , $^/2x) . Then-By Lemma 9.2.1 we see that R ( E , G) = a . \ In order to have a self-adjoint single generator Mf 0 1 of E. we may take any strictly monotone, continuous and real-valued function g defined on [0, x^] , and let f - g + 5(g) + ••• + ? n _ 1(g) , where g is extended to [0, 1] with 0 as its values on [0, 1]\[0, x1 ], Thus in particular i f for each strictly positive real number r we denote by g^ the function ,x>-> x r on [0, x^ ] and let f r = g r + 5 ( g r ) + C 2 . ( g r ) + + 5 n " 1 ( g r ) , then Mf 0 1 is a self-adjoint single generator ,of E . Let T be a r self-adjoint single generator of R(Ug 0 Vg : g e.G) and let . A r be the * -operator on H with Re(Ar) = Mf 0 1 and Im(Ar) = T . Then by Theorem 8.2 r the family ,(Ar) has the desired properties. III. Take m to be ^[0, 1] acting on 1^ [0, 1] , G = ' { T g : g dyadic rational in [0, 1] } . Let ( x j be defined by : ' n • 1 o ' T 1 n-1 2n for n e Z Define an automorphism £ of m by : (?f)(x) = • f(x) . , X E [ x 0 ) Xj) u I D \ Then clearly £ m is locally non-A-preserving (A : Lebesque measure) . Thus by Lemmas 9.1.9, 9.1.3, 9.1.4, 9.1.5, the abelian subalgebra E = 5° 0 1 is thick in a. with E c = m , and is of deficiency type 1^ and uniform multiplicity co . To prove that R ( E , G) => a , let ?l , ?^ be respectively the projections of m corresponding to the characteristic functions of [0, 1/8] .68. and [3/8, 1/2]. . Let P i " P i + •• I » i B 1> 2< keZ where for each k e Z , n, is the automorphism of m given by f ( x k - l + 2 ( x - x k)) » x e [x k , x ^ ) (n kf)(x) = { x e [0, 1] \ [x k , Then Px and ? 2 are in 5' Now T 3 / 1 6 < V W = ° > .• T3Vi6< Pl> P2 * ° whe re is the projection of . m_, corresponding to the characteristic function of [xQ , x^] . Th us 0 f V % 6 < P i . > P 2 ] - < \ ' So by Lemma 9.2.1, R(E, G) = a For each strictly positive real number r, denote by g r the function x x r on [0, 1/2] and x w> 0 on [1/2, 1] , and let f r •-. § r + I n k(g r) .. keZ 69. Let T be a s e l f - a d j o i n t s i n g l e generator of R( u g ® V g : g e G) and l e t A be the operator on H with Re A r = Mf © 1 and Im A = T . Then by Theorem 8.2, (A r) i s a family with the desired properties. IV. Take m = L°°([0, l ] 2 ) on L 2 ( [ 0 , l ] 2 ) , and take G = { T , : v.x,yj x, y dyadic r a t i o n a l s i n [0, 1] } , where T ^ ^ i s the t r a n s l a t i o n i n [0, l ] 2 by (x, y) (mod 1) . C l e a r l y G i s of type II . Let (J) : [0, 1] —> [2, O D) be defined by : <j> (y) = inf{ t >_ 2 : A(t) >_ y } . Define the subset S. of [0, l ] 2 by : • S = { (x, y) e [0, l ] 2 : x < [<j> (y) ]"' } . Define the automorphism £ of m_ by : f f([<|)(y)r' +.[<J>(y) - l ] x , y) for a l l (x, y) e S , (Cf)(x, y) = • - I f([<f.(y) - 1] ' [x - (^(y))"' ], y) f o r a l l (x, y) 4 S , where f e ^ ( [ 0 , l ] 2 ) . Let Q be the p r o j e c t i o n of m corresponding to S . Then c l e a r l y 5 i s l o c a l l y non-A-preserving (A : Lebesque measure), ? 2 =1 and £(Q) = 1 - Q . Thus by Lemmas 9.1.9, 9.1.3, 9.1.4, the abelian von Neumann algebra E = £° & 1 i s t h i c k i n a with E c = m , and E i s of deficiency type 1^ . For. each x >_ 1 let E x be.the projection of m_ corresponding to ' [0, ..1]. x [C-, A (x) ]. Then:it;lsVeasy to see, that the conditions of Lemma 9.1.6 are satisfied, and we conclude that A is the multiplicity function of E . In order to prove that R (E, G) = a, let x Q = l/<j>0) • Since A (2) = 0 and since A is right continuous, x Q < 1/2 - 6 for some <5 e (0, 1) . Let be the projection of m_ corresponding to { (x, y) : 0 </x <_ fi/* (y), y e [3/4, 1] } . L e t Pj, = ?l + 5 (Pi) , Then Pj e 5° and 0 < T ^ ^ (P^) <_ 1 - Q . Thus by Lemma 9.2.1 R(E, G) = a Let Mf• (resp. T) be a self-adjoint single generator of £° (resp. R(Ug 8 Vg : g e G)), where f e I^QO, l ] 2 ) and f >_ 0 . For each strictly positive real number r, let g r = ( f ) r and let A r be the operator on H with Re A r = Mg 8 1 and Im A r =.T . Then by Theorem 8.2 the family (Ar) has the desired properties. V. Take m = ^([O, 1] x 3R) acting by multiplication on L 2([0, 1] x IR) and take G = { T ^ x j : x p-adic rational in [0, 1], y rational in IR } ,. where p is a fixed prime >_ 2 such that p does not divide n . Clearly G is of type 11^ 71. In order to define our thick.subalgebra l e t us r e c a l l the automorphism £ of I . Let E, 1 be the unique automorphism of the present m such that f o r any e L^fO, 1] and f o r any f 2 e L^ (R): : ' . g ' (f )(x, y) » [ ^ f X ^ l f (y) , where ^ 2 (x, y) = f 1 ( x ) f 2 ( y ) , f o r a l l (x, y) e [0, 1] x ]R . Now i t i s easy to see that for exactly the same reasons as i n I, the abelian algebra E, - ® 1 i s thick i n a with E c = m , and i s of d e f i c i e n c y type I n . In order to prove that R(E, G) = a, we f i r s t n o tice that a l l projections of m_ corresponding to [0, 1] x S (S C H measurable) are i n . (g')° . I t then follows that a l l projections of m are i n R((£')°, Ug g• e G) because a s i m i l a r property holds for £ . Thus R(m 8 1, U„ 8 V : o g . g e G) '= R((5')° 8 1, U g 8 V g : g e G) , i . e . a = R(E, G) . Let (resp. T) be a s e l f - a d j o i n t s i n g l e generator of (£')° (resp. R(U g 8 V g : g e G)), where f e I^QO, 1] x IR) and f >_ 0 . For each s t r i c t l y p o s i t i v e r e a l number r, l e t g r = ( f ) r and l e t A r be the operator on H with Re A r = M 8 1 and Im A r = T . Then the family (A r) B r has the required properties. acting on L 2 ( [ 0 , 1] x ]R) , and [0, .1], y r a t i o n a l i n H } . VI. . Take m to be L^([0, ,1] x ]R) G = { T , \ • x dyadic r a t i o n a l - i n (x,y) x C l e a r l y G. i s of type. 11^ . 72. To define our thick subalgebra l e t us r e c a l l the automorphism E, i n I I I . Let £' be the unique automorphism of the present m_ such that f o r any ^ e [ 0 , 1] and any f £ e 1 ^ O R ) : ? ' ( f )(x, y) = [ ( C f , ) ( x ) ] f 9 (y) , where f (x, y) = f (x)f (y) , f o r a l l (x, y) e [ 0 , 1 ] x K. . Now i t i s 1 >2 1 2 easy to see that £ ' i s l o c a l l y non-A-preserving "as so i s £ (A : Lebesque measure). Thus f o r exactly the same reasons as i n I I I , the abelian algebra. E = ( £ ' ) ° % 1 i s thick i n a with E c = m, and i s of def i c i e n c y type 1 ^ . In exactly the same way as In V above, we can prove that R (E , G) Let (resp. T) be a s e l f - a d j o i n t s i n g l e generator of (£')° (resp. R(U g ® V g : g e G ) ) , where f e L M ( [ 0 , 1 ] x ]R) and f >_ 0 . For each s t r i c t l y p o s i t i v e integer r l e t g r = ( f ) r , and l e t A r be the operator on H with Re A r = M g 8 1 , and Im A = T . Then the family (A r) has the required properties. VII. Take m to be L^OR2) acting on L 2 (E 2) , and G = { a, ( r i s r 2 ) r non-zero r a t i o n a l , r„ r a t i o n a l "V ' where a, , i s the J- ^ - ' ( r r ) automorphism of m_ defined by : ( a ( r j r ) f ) ( x > y) = f ( r j x> y - r 2 ) > f°r a l l x, y e TR , 73. where f e OR2 ) . Now the functional, u defined on m_+ (to [0, GO ] ) u(f) = | r ^ r - . f ( x ) y)dxdy -m 2 , . • . : . .is a semi-:finite, normal G-trace on' m_ • So G i s of type 11^ ,, . Also G ' is"ergodic"(and abelian). Thus a i s a f a c t o r of type 11^ ", acting on the separable H i l b e r t space. Let H, = { T , s : r r a t i o n a l } , where m i s a f i x e d i (r,mr) ' i r r a t i o n a l p o s i t i v e number, and. T , \ i s the automorphism of m defined (r,mr) r — by : ( x ( r , m r ) f ) ( x ' y ) = f ( x " r» y " m r ) for a l l x, y e ]R and f e L^ QR2) . Let E = H? 0 1 . v 1 To see that E i s thick i n Ci , we l e t , f o r r e a l numbers a , b with a < b : E , = { (x, y) e TR : mx .+ a < . y < mx + b } , 3. y D ~™" ~~~~ and l e t P & b be the p r o j e c t i o n of m corresponding to E a b • T n e n f o r any a , , e G , not the i d e n t i t y : •E , a / v (E ) — > 0 strongly as. a —> b from the l e f t . As a,b ( r ^ ^ ) a » D / . ' . E'a b z H° we conclude by Lemma 9.1.1 that E ' ( a ( r r j > ^ ) =0 . So by Lemma 9.1.2, \-E is thick in CL with E c = m . We use Bure's argument to determine the deficiency type of E . Let m = • be L^ OR)' acting on. QR). Then by rotation-it is seen that there is an isomorphism of m_ with .8 which takes onto 1 8 ^ for some ergodic -group ... It then follows from Lemmas 9.1.8, 9.1.7 that H° , is of deficiency type III in m_ So E is of deficiency type III in m . . • ( In order to see that R ( E , G) - Cl , let Qa,b = Ea,b A(2- ,l) ( Ea,b ) ' Then i t is easy to see that each Q , (a < b) is of positive Lebesque measure and that within each open circle C inside Q there is a Q , 1 .2 a,b .-• within C for some suitable a, b satisfying a < b . Thus we see that the projection in m corresponding to any open measurable subset of Q ~ 1 ' 2 is in R(H°, U : ge G) , and therefore (Lemma 9.2.1), R ( E , G) = a . O / Let S (resp. T) be a self-adjoint single generator of H£ (resp. R(U„ 8 V„ : g e G ) ) . For each r e H let a = T . . on m and let A_ 8 g r (r,r) — r be the operator on H such that Re A r = «r(S) 8 1 and Im A r = T . Then : R(Re Ar) = H° 8 1 as a r commutes with each element of H i . By Theorem 8.2 the family (Ar) has the desired properties. 75. VIII. Take m to be I^CtO, l ] 2 , \ x y) acting on 1^ ([0, l ] 2 , A x y), G = { a / ~\ : x p-adic rational in [0, .1], and 6 a finite sequence of ( X , Q ) positive integers } , where y can be any one of the measure y q(q e (y,1)) on [0, 1] defined in the paragraph preceding Corollary 7.5 ( \ is the Lebesque measure), and a , ^ is the automorphism of m_ given by : ( X ) 5 ) f ) (y» z) = f (y - x» efi (?)) » y> z e [o, i] , where is as defined in the paragraph preceding Corollary 7.5. Then G is abelian and ergodic (and free) on m . Also G is of type III [14 or 7] so.', a is ..a factor of type III on the separable Hilbert space. Totdefine our thick subalgebra let us recall the automorphism £ in I . Let g' be the unique automorphism of the present m_ such that for any f x e ^([0, 1], A) and for any ±2 e.Loo([0, 1], y) : \ k'if )](y, z) = [CsfcXy)]!. (z) , 1 >2 1 2 • i . where f (y, z) = f (y)f (z) , for a l l y, z e [0, 1] . Now i t is easy 1 »2 • 1 2 to see that for exactly the same reasons as in I, the abelian algebra E = (£')•" 0 1 is thick in a. with E c = m , and is of deficiency type I n . i • For almost verbally the same reason as in V, we have R ( E , G) = ft . \ •) • • Let (resp. T) be a self^adjoint single generator of (£')° (resp. R(U ® V : g e . G ) ) , where ..f e ^ ( [ 0 , l ] 2 , \ x y) and f > 0 . 76. For each s t r i c t l y p o s i t i v e r e a l number r, l e t g r = . ( f ) r , and l e t A r' be the operator on H with Re A = M„ 0 1 and Im A_ = 1 . Then the family r :' B r (A r) has the required properties. IX. Take m to be I^CfO, l ] 2 , A x y) acting on ^ ( [ O , l ] 2 , A x y ) , and G = { a, r \ : g dyadic r a t i o n a l i n [0, .1], and <$ a f i n i t e sequence of p o s i t i v e integers } , where A > y are the same measures used i n VIII above, and a • . i s the automorphism of m_ given by : ( a ( g > 6 ) f ) ( x , y) = f ( x - g, 3 6 ( y ) ) , for a l l x, y e [0, . 1] and f e ^ ( [ 0 , l ] 2 , A x y) (g g being as i n VIII above). For v e r b a l l y the same reasons as i n VIII, a i s a fa c t o r of type II I on the separable H i l b e r t space. R e c a l l the automorphism £ i n I I I , and define the automorphism V on the present m by (for f j e L r a([0, 1], A ) , f 2 e L w C[0, 1], u)) : [ C'(f ) ] ( x, y) = [ ( 5 f i ) ( x ) ] f 2 ( y ) , x, y e [0, 1] , 1 »2 where f (x, y) = f 1 ( x ) f 2 ( y ) , x, y e [0, 1] . Then by exactly the same 1 >2 reasons as i n I I I , the abelian algebra E s ( ? ' ) ' 8 1 i s th i c k i n a with E c = m , and i s of deficiency type 1^ . Furthermore the same argument used i n V applies here and we conclude that R(E, G).= a . 77. Exactly as i n VIII above we can e a s i l y obtain a family of operators on H with the desired properties. X. Take m\ to be' 1^ ,QR2 x [0,1] , A x y ) acting on L (JR2 x [0, 1] , X x y ) and. G - {. a > . \ : r. s t r i c t l y p o s i t i v e r a t i o n a l , r 1 '^ 2 r a t i o n a l , 6' a f i n i t e sequence of p o s i t i v e integers } , where a , x . i s the automorphism of m_ defined by : ; (a ( r > r j g ) f ) (x, y, z) = f (r x"Vx, y - ^ , B g (z )) , f o r a l l f e LjJR2 x [0, 1], X x y ) t and a l l (x, y, z) £ m2 x [0,' 1] (The y, and 6^ are as those i n VIII above, and X i s the Lebesque measure on IR 2 .) Then as i n VIII we see that CL i s a f a c t o r of type I I I on the separable H i l b e r t space H . Fi x a p o s i t i v e i r r a t i o n a l number m and l e t K\ = { a r : r r a t i o n a l }, where a • i s the automorphism of m_ defined by : ( a r f ) ( x , y, z) = f ( x - r, y - mr, z) fo r a l l x, y e 1 , z e [0, 1] and f E 1^ 0R2 x [0, 1], X x y) . Then i t follows f o r reasons s i m i l a r to those i n VII that E ^ c t , N , K,) = 0 ( r } , r : ,<$ ) 1 ' f o r every (± , r 2 , <5 ) d i s t i n c t from the i d e n t i t y . Therefore E = Kj 8 1 i s t h ick i n a with E c = m . As i n VIII we see that R ( E , G) = a . Now we use Bure's argument to determine the deficiency type of E . By r o t a t i o n 78. of axes there is an isomorphism of m_ with, 8 m2 taking onto 10 1^ ; ..where m1 . is L^ QR x • [0, .1],X x y ) - acting on OR x [0,1] , X x y), m2 , is • Ife, OR , X) acting on L2QR ,X) (X being the Lebesque measure on E, ), and- is an ergodic -group. . It then follows from Lemmas 9.1.8, 9.1.7 that E is of deficiency type III . For each real number s let a s be the automorphism of m defined by : [a sf ] (x, y, z) = f(x - s, y - s, z) , x, y e IR, z e [0, 1], , where f e L^ QR2- x [0, 1], X x y) . Let S (resp. T) be a self-adjoint single generator of the abelian K° (resp. R(Ug 8 Vg : g e G)) . Then each I a g(S) is a self-adjoint single generator of K£ since a g commutes with each ctr e . Let Ag be the operator on H such that Re As = as(S) 8 1 and Im A s - T . Then the family (A s) g g ] R has the desired properties. v So we.complete the proof of Theorem 9.1 . / 79. Appendix We have seen in § 5 that G' plays an important role in the study -of A'dm,ai. ft) • In this appendix we shall compute G' when m = ^ ^ j ^ ^ * ^ ) and G ^ ± » u^) (where each m.j_ is an abelian WA-algebra, co ^ is a normal state on n\j_ with to^(l) = 1 , and each G^ is an m^-group.). These notations will be explained later (cf. Definition A.l below). The main result below (Theorem A.4) generalizes those results (Propositions 7.1, 7 2, 7 . 3 , 7.4) we obtained in § 7 . In order to have a set of standard notations let us briefly recall von Neumann's (infinite) tensor product of a family of von Neumann algebras. Let I be an arbitrary index set, and for each i e I, let a.^ be a von Neumann algebra acting on the Hilbert space H^ , and let x^ e H^ with |] X j J | = 1 . Then by von Neumann's construction (see- [12]) we have the . tensor product H = Jd^ (H^ , x^) (in von Neumann's terminology, incomplete direct product) of (H•) . T with respect to (x,-). _ . There exist canonical isomorphisms A^ »—> A^ from L(H^) into 1(H) The tensor product 8. T ( a . , x.) of («••). with respect to (xn-). T is defined to be the ieI i i • i u l 1 le I von Neumann algebra on H generated by the set { A^ : i e I , Aj_ E } . We now introduce the following definitions. Definition A.l [6] Let a be a W*-algebra . (i) A .representation d) of a on a Hilbert space K is an 8 0 . isomorphism of CL onto a.von Neumann algebra acting .on K . (ii) For a representation ^ of a (on K) , and a normal state y on CL : S(<j>, y) 5 { x e K : (<KA)x|x) = y (A) for a l l A £ a } . ( i i i ) For normal states y, v on a , define : [y(l) + v ( l ) ] 5 i f either S(<j>,y) or S(cj),v) is empty, d^Cy.v ). = • [ ' inf{ || x - y|| : x e S(<f>,y), y e S((j>,v) } otherwise, for any representation tj> of CC , and define : d(y , v ) = inf { d^ (y , v ) : $> a representation of CL } . (iv) Suppose that I is an arbitrary index family, and suppose that for each i e I, Cl^ is a W*-algebra, u ^ is a normal state on Cl^_ with co^(l) = 1 . We say that a W*-algebra CL , together with (a^) ^ , -where is an isomorphism from Cl^ into CL , is a tensor product of with respect to (^±)±£x ^ t n e following condition holds : For every family (CJK , x^)^^ where <i> ^ is a representation of and x^ £ S (<J> ^ , > there is an isomorphism $ of a onto i f l ^ i ^ i ) ' x i ? ' w i t h ^(ai(A.)) = ^ ( A ^ for a l l A^ £ and for a l l i e - I . It has beenproved [6] that the tensor product CL of ^ i ^ e i with respect to C w i ^ i ' e i exists and is unique up to isomorphism preserving 81. the injections (an-)- T • We write 'a, = .8 (a., co-.) . (v) Let a, with canonical injections (a±)±ej_ » ^e t' i e t e n s o r product of ^ i ^ i e x w:"-tn respect to ^ u i ^ i e i • ^et • §i ^e a n automorphism of • If there is an automorphism g of a such that ga± = a . ^ for a l l i e l , we write g = 8. _g. , and say that .®Tg,- exists on a . l£l i ' ^ I E I i .Note that..8 Tg. , i f i t exists, is.uniquely determined by the (g-t) . T . • (vi) For each i e I let be a W*-algebra, co ^ a normal state of a.-^ with (0^(1) = 1 , and G^ a group of automorphisms of . Define i | I ( G i , -u.±) = { a e A( i| I(a i, OK)) : a = ^ j g i for some g ± e 1 7 6 ^ } , ieI i and Jl(G i > co±) = { a e A( i| I(a i, co±)) : a = . . f ^ for some (g±)±£l e JJ_ G± i e l ' i e l where JJ" G^ and J_[_ are, respectively, the direct product and the ie l isI \ weak direct product of the family ^±^j_ej °f g r o uP s' It follows readily from the definition that i f x ^ i ' w i ^ i - s a subgroup of A(J3-r(aj_, c O j ^ ) ) > and that JJ_ (G^, to^) is a subgroup of ie l . " I E I ( 6 i ' ,Ui> ' We shall need a special case of the following result from [6 and 16], e.g. when e a c h - i s abelian .-• 82. Lemma A . l [6, 16] For each, i e l l e t . .be. a W*-algebra, a normal state of ' with u)^(l) = 1 , and an automorphism of Then .8 Tg. exi s t s on .® T(a., UJ.) i f and only i f xel x xel x' x' } • » . 7. _[d(u)., u.«g.)] 2 <; co . ^x e l x 6 i Corol l a r y A.2 With the notation of Lemma A . l we have i t l < 6 i ' w l > ='{ i f l S i ' ( 8 ± ) l e I e IT Gi with Z l e I [ d ( U l , V g i > ] 2 < 0 0 "> ' i xel •and/ 11 (C^, o,±) -{ . t l g . : < g ± ) e \j_G. } . . xel i e l • ; ' -,V ' ' Furthermore i f I = {1,. 2 } then .8_(a., co.) can be i d e n t i f i e d with x e l i i a-, 8 02 » a n d i f l ^ i ' = 11 (G-> to .) = G: 8 G2 under the i d e n t i f i c a t i o n . i e l We s h a l l also need the following r e s u l t from [5]. Lemma A. 2 [5] . Let m, ft. be abelian W*-algebras, and H an ergodic n-group . Then ' { A e m 8 n : ( 1 8 h)(A) = A f o r a l l h £ H } = m 8 C . Proposition A.3 Suppose that I i s an a r b i t r a r y index set, and that f o r each I e I , \ i s an abelian W*-algebra, u i - a normal state of m± with to. (1).= 1 , and G. an ergodic m,--group.. Then Ji (G., co.) i s • ^ 1 .~^r x x xel ergodic on ifjOr^, w ±) . •'' ' 83. Proof. . Let $ . (j e I) be a representation, o f jn . on H. with , • J - •. s 3 3 x. e S(f> ., co •) •• Let a . be the canonical i n j e c t i o n of L (H.) into J J. 3 3 *J .® T(/£H-), X ± ) . Let G = _[]_ (S.,<o.) act on .0_(<j>. (pi .) ,< x.) . For each xe i. . T- x x xe J- x x x xel f i x e d j e I , l e t 3 xel\{j } Y x v x' x' Let G ° = { A e .JIJG » x ± ) : g ( A ) = A f o r a l l g e G } . Then by Lemma A.2, G °c C ® n. so that (G°)'=> L (H.) • I t follows that 3 ( G ° ) ' 3 KIL), x ±) . By a well-known theorem (see [4] or [12]), . 8_(L(H.), x.) = L ( . 0 T ( H . , x.)) , so we have ( G ° ) ' = L ( . 0 T ( H . , x.)) , i . e . xe l x x x e l x' x x e l x' x G ° = C ', and G i s ergodic. Theorem A. 4 Suppose that f o r each i e l , i s an abelian W*-algebra, co i a normal state on with w^(l) = 1 , and G I an ergodic m-group. Then xel Proof. Let m be tensor product of Cm.). ,. with respect to (co.). ^ , x xel . x x e l and f o r each i e l , l e t be the canonical i n j e c t i o n from m i n t o m . Suppose h e ( J_j_ (G., to.))' , i.e.-h. i s . an automorphism of m with i e l h ° ^ i e l 8 ^ = ^ i t l g i ' > ° h f o r e a c h ^ i e l e . - U - G i ' L e t M j e m j b e a r b i t r a r y . Then, f o r any (g"l). T e J_J_ G. with g"! .= e the i d e n t i t y - 1 i e l 1 J 3 of G . : 84. ( i l l g i ) [ ( h ° ' a j ) ( M j ) ] = h [ ( ( i l l g i ) ° a j K M j ) ] = h[(a o g^)(M ')] = (h o a )(M ) , i.e. (h o a.)(M.) is a fixed point of .8Tg*l e II GJ with gi = e. 3 3 i e l 6x ± ^ 1 - J ' 3 Now i t is known (cf. [12], Theorem VI) that there is an isomorphism A from .0T(m., co.) onto m. 8 (. T8/.-,(m., <*>.)) such that ie l i ' i j vxel\{j} v x' x (A o cc.)(M.) = M. for any M. em. , 3 3 3 3 3 where M. »-> M. is the canonical injection from into m. 8 '(. T8 ,. (m. ,co.)) . 3 3 - ^ 3 xelN{j) i i But then we have for any (g"3). , e I I G. with g"? = e. : J &x xel ."4r 1 1 1 xel J J [1 8 (. T8,.,g^)] o A = A o ftgj , xel\{j} 6x / j x e l 6 i ' so that A[(h ° a.)(M.)] is fixed by any such 1 8 (. 8r.-,g~}) . Now by J 3 xelMjJ- x Lemma A. 3 J l (G., co.) is ergodic on . T8/. -,(m., co.) , so by Lemma A.23 iel\{j} : 1 . 1 xel\lj) x x A[(h o a.) (M.') ] e m. 8 C . 3 3 3 Thus (h ° a )(M.) e A ' (m ) = a.(m.) . This shows that .85. since h" e ( 11(G., c o . ) ) ' also we have . i e l x 1 h[a Am )•] r a. Cm.) J J J j Thus we may'put h = a" » h 0 a . Then h. is an automorphism of m . 3 3 3 3 3 such that h o a . = a . o h . , J J J and that for any g. e G. , .. J J h. . g.=g.o h. by a direct simple calculation. Therefore we see that • (II (G., c o . ) ) ' cr . f l(G:, co. ) But obviously we have ie l and , , u.) CZ (.^(G. , u.)) ' Thus we have (11 (V yr-; (t^v -))• = .1^, co.) 86. References W.B. Arveson, Subalgebras of C*-algebras, Acta Math., 123(1969), 142-224. ;W.B. Arveson, Unitary inva r i a n t s for compact operators, B u l l . Amer. Math. S o c , 76(1970), 88-91. A. Brown, The unitary equivalence of binormal operators, Amer. J . Math., 76(1954), 414-434. D. Bures, Certain factors constructed as i n f i n i t e tensor products, Compos. ' Math., 15(1963), 169-191. D. Bures, Abelian subalgebras of von Neumann algebras, p r e - p r i n t , to be published i n the Memoirs of Amer. Math. Soc. D. Bures, An extension of Kakutani's theorem on i n f i n i t e product measures to the tensor product of s e m i - f i n i t e W*-algebras, Trans. Amer. Math. S o c , 135(1969), 199-212. M. Choda and H. Choda, On extensions of automorphisms of abelian von Neumann algebras, Proc. Jap. Acad., 43(1967), 295-299. D. Deckard, Complete sets of unitary inva r i a n t s f o r compact and trace-class operators, Acta S c i . Math., 28(1967), 9-20. J. Dixmier, Les algebres d'operateurs dans l'espace h i l b e r t i e n , 2° ed., G a u t h i e r - V i l l a r s , P a r i s , 1969. P.R. Halmos, Introduction to H i l b e r t space and the theory of s p e c t r a l m u l t i p l i c i t y , Chelsea, New York, 1951. S. Kakutani, On equivalence of i n f i n i t e product measures, Ann. of Math., 49(1948), 214-226. J . von Neumann, On i n f i n i t e d i r e c t products, Compos. Math., 6(1938), 1-77. J . von Neumann, Zur algebra der funktional operatoren und theorie der normalen operatoren, Math. Ann., 102(1929), 49-131. J . von Neumann and F.J. Murray, On rings of operators, Ann. of Math., 37(1936), 116-229. ' J . von Neumann and F.J. Murray, On rings of operators I I I , Ann. of Math., 41(1940), 94-161. 87. [16] C. Pearcy, A complete set of unitary invariants for operators generat-ing finite W*-algebras of type I, Pacific J. Math., 12(1962), 1405-1416. [17] S.D. Promislow, Semi-metrics on the normal states of a W*-algebra, Ph.D. Thesis, Univ. of British Columbia, 1970." [18] L. Pukanszky, Some examples of factors, Publ. Math. Debrecen, 4(1956), 135-156. [19] H. Radjavi, Simultaneous unitary invariants for sets of bounded operators on a Hilbert space, Ph.D. Thesis, Univ. of Minnesota, 1962. [20] I.E. Segal, Decompositions of operator algebras II, Memoirs of Amer. Math. Soc, 9(1951), 1-66. [21] I.M. Singer, Automorphisms of finite factors, Amer. J. Math., 77(1955), 117-133. [22] N. Suzuki, Isometries on Hilbert spaces, Proc. Jap. Acad., 39(1963),' 435-438. [23] N. Suzuki, Algebraic aspects of non self-adjoint operators, Proc Jap. Acad., 41(1965), 706-710.
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Inequivalence and equivalence of certain kinds of non-normal...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Inequivalence and equivalence of certain kinds of non-normal operators Tam, Ping Kwan 1970
pdf
Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.
Page Metadata
Item Metadata
Title | Inequivalence and equivalence of certain kinds of non-normal operators |
Creator |
Tam, Ping Kwan |
Publisher | University of British Columbia |
Date Issued | 1970 |
Description | This thesis is concerned with the problem of unitary equivalence of certain kinds of non-normal operators. Suppose [m, K, G, g ↦ U [subscript g]] is an ergodic and free C-system, with G abelian. Let m = m [symbol omitted] 1, n = R(U[subscript g] [symbol omitted] V[subscript g] : g є G), and let a = R(m, n) = R[m, K, G, g ↦ U[subscript g]] be the von Neumann algebra constructed from [m, K, G, g ↦U[subscript g]] according to von Neumann. We compute: (1) the group A(α; m, n) of all automorphisms of α which keep m pointwise fixed and keep n invariant, and (2) the group A(m, α; n) (resp. G(m, α; n)) of all automorphisms of m which extend to automorphisms (resp. inner automorphisms) of α keeping n pointwise fixed. These calculations lead us to compute G' [symbol omitted] [G] and G' (where [symbol omitted] is the full group generated by G). We show that for an abelian and ergodic G on an abelian m G' [symbol omitted] [G] = G . In the course of this investigation we obtain several interesting results. For example we see that such [symbol omitted] G is automatically free on m. For a large class of tensor algebras (and in particular for a large class of multiplication algebras) we succeed in determining G'. (For the particular cases of multiplication algebras we only use measure-theoretical arguments.) These results are applied to solve the problem of unitary equivalence of certain kinds of non-normal operators. Finally for most of the interesting thick subalgebras E in the literature, we construct numerous unitarily non-equivalent operators A ,with R(Re A) = E. |
Subject |
Functions |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-31 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080511 |
URI | http://hdl.handle.net/2429/34965 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
Download
- Media
- 831-UBC_1970_A1 T34.pdf [ 4.12MB ]
- Metadata
- JSON: 831-1.0080511.json
- JSON-LD: 831-1.0080511-ld.json
- RDF/XML (Pretty): 831-1.0080511-rdf.xml
- RDF/JSON: 831-1.0080511-rdf.json
- Turtle: 831-1.0080511-turtle.txt
- N-Triples: 831-1.0080511-rdf-ntriples.txt
- Original Record: 831-1.0080511-source.json
- Full Text
- 831-1.0080511-fulltext.txt
- Citation
- 831-1.0080511.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
data-media="{[{embed.selectedMedia}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0080511/manifest