UBC Theses and Dissertations

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UBC Theses and Dissertations

Abelian von Neumann algebras Kerr, Charles R. 1966-12-31

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ABELIAN VON NEUMANN ALGEBRAS  hy  C h a r l e s R. K e r r B.A., Washington S t a t e U n i v e r s i t y , 1962  A THESIS SUBMITTED I N PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS  i n t h e Department of Mathematics  We a c c e p t t h i s t h e s i s as conforming required  t o the  standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1965  In the  presenting  r e q u i r e m e n t s f o r an  British  for reference  and  the  be  granted  by  Library  study.  for extensive copying of  p u r p o s e s may  in partial  advanced d e g r e e at  C o l u m b i a , I agree t h a t  available mission  this thesis  the  Head o f my  cation  for f i n a n c i a l gain  w i t h o u t my  written  Department  o  permission,.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada * Date  the U n i v e r s i t y  of  shall."make i t . f r e e l y , agree  this thesis for  It is.understood  this thesis  of  I further  his representativeso of  fulfilment  Columbia,.  that  scholarly  Department  that  or  c o p y i n g , or  shall  per-  not  be  by publi-  allowed  T h i s t h e s i s c a r r i e s out some o f c l a s s i c a l i n t e g r a t i o n theory  i n the context  o f an o p e r a t o r  algebra.  The s t a r t i n g  p o i n t i s measure on t h e p r o j e c t i o n s o f an a b e l i a n von Neumann a l g e b r a .  T h i s y i e l d s an i n t e g r a l on t h e s e l f -  a d j o i n t operators  whose s p e c t r a l p r o j e c t i o n s l i e i n t h e  algebra.  F o r t h i s i n t e g r a l a Radon-Nikodym theorem, a s w e l l  as t h e u s u a l convergence the or ems ,C'I;S proved . The  methods and r e s u l t s o f t h i s t h e s i s g e n e r a l i z e , t o  non-commutative von Neumann A l g e b r a s [2, 3j> 53-  (1)  J . Dixmier  L e s A l g e b r e s d'Operateurs dans l ' E s p a c e Hilbertien.  (2)  H.A. Dye  P a r i s , 1957.  The Radon-Nikodym theorem f o r f i n i t e of o p e r a t o r s ,  rings  T r a n s . Amer. Math. S o c ,  72, 1952, 243-230. (3) ' F . J . Murray and J . von Neumann, On Rings o f O p e r a t o r s ,  Ann. Math. 57, '  1936, 116-229. (4)  F . RIesz and B . v. Sz.-Nagy, Functional Analysis,  (5)  I . E . Segal  New Y o r k , 1955.  A non-commutative e x t e n s i o n integration, 1953, '401-457.  of abstract  Ann. o f Math. (2)  57,  iii  TABLE OP CONTENTS page  Introduction Examples  1  a von Neumann A l g e b r a  .  6  . . . .  Weak Compactness o f t h e UniformlyClosed B a l l  15  S e l f - A d j o i n t O p e r a t o r s i n a von Neumann A l g e b r a  .  ...  .  . . .  Unbounded O p e r a t o r s  .  . . .  21 49 69  Measure Theorems Measure on Simple F u n c t i o n s  ..  72  Measure on Bounded S e l f - A d j o i n t Operators  . . . .  86  .  Measure on Unbounded O p e r a t o r s Absolute Continuity  .  .  .  . .  .  . .  . . . .  .  .  96 102  1 INTRODUCTION  Let set  H  "be a complex H i l b e r t space.  o f bounded l i n e a r t r a n s f o r m a t i o n s on  o v e r t h e complex f i e l d . i s the unique  Let  H.  L(H)  L(H)  I t has an i n v o l u t i o n  be t h e  i s an a l g e b r a  T —»T*, where  T*  L(H)-member which s a t i s f i e s (Tx, y ) = (x> T*y)  for  a l l x, y  in  H.  I n what f o l l o w s (a)  L(H)  w i l l be t o p o l o g i z e d t h r e e ways..  The weak t o p o l o g y on  neighborhood o f  A e L(H)  L(H): I n t h i s topology  each  must c o n t a i n t h e i n t e r s e c t i o n o f a  f i n i t e f a m i l y o f s e t s o f form N(A,x,y,e) = where  x,y e -H (b)  A s L(H)  and  [n  : T e L ( H ) , | ( A x , y ) - ( T x , y ) | < ej ,  € > 0.  The s t r o n g t o p o l o g y on  L ( H ) : Each neighborhood o f  contains the i n t e r s e c t i o n of a f i n i t e f a m i l y of sets  o f form N(A,x,€) = where  x € H (c)  £T : T e L ( H ) , ||Tx-Ax|| < e j ,  and  e > 0.  The u n i f o r m t o p o l o g y on  L(H): This i s the m e t r i c  t o p o l o g y i n d u c e d by t h e o p e r a t o r norm. These t h r e e t o p o l o g i e s a r e comparable: Weak <=  Strong c  Uniform,  so  that i f "5  S c  L(H) "5"  (weak)  (strong)  Under each o f these algebra,  are  t h a t i s , the  (X,Y)  -»  (a,X)  —•  aX  X  —^  AX  X  —>  XA  continuous !•  of  R,  all  S e  A,  If  A  (AB)S  that  and  =  R c L(H),  B  are  + BS  = A(SB)  (A+B)  T  L(H)  and  R c L(H)  S*T  subset  i s a topological  scalar  then  T  In  R', L(H)  a. the  commutant  such  that  TS*  =  and  R*  i s a l w a y s an  in  R>,  then f o r  = SA  + SB  = S(A+B)  algebra:  S e R  S(AB). AB  commute w i t h  S*.  Thus  (A+B)  and  R'.  Note t h a t and  L(H)  R. any  also i n  in  If  = TS  (A+B)S = AS  are  X, Y  set of a l l operators  For  Similarly,  topologies  (X+Y)  for  ST for  (uniform).  operations  Definition.  i s the  =>  is in  R  1  R'  F u r t h e r m o r e , R*  always c o n t a i n s i f and  only  i f  i s closed i n  the  identity  T*  is in  L(H)  operator, R'.  u n d e r any  of  the  above t o p o l o g i e s .  T o p r o v e t h i s n o t e t h a t f o r S e R,  i s m e r e l y t h e s e t on which t h e continuous f(T)  ss'  g(T)  = S*T - T S *  are  equal  two  closed  so t h a t  sets.  =D  R'  £s  and h e n c e t h a t  Is the i n t e r s e c t i o n o f  Finally  Is]' ,  (SeR)  i s a closed s e t .  2. R c L(H)  functions  ST - TS  t o zero,  R»  Definition. such t h a t  A v o n Neumann a l g e b r a i s a s e t  R = (R')' -  R".  Such a s e t i s , i n view o f p r e v i o u s closed  symmetric  subalgebra  of  L(H)  remarks, a s t r o n g l y  which contains  operator.  On t h e o t h e r  algebras.  T h i s f o l l o w s f r o m t h e " v o n Neumann d e n s i t y  which says  that  hand, a l l s u c h a l g e b r a s  R" s= rT (weak) = H f o r every  symmetric  J>. is  Theorem.  any bounded  then  £S  T € R  R  self-adjoint  i f and o n l y  a r e v o n Neumann theorem"  (strong)  subalgebra If  the i d e n t i t y  R  with  identity.  i s any v o n Neumann a l g e b r a operator  i f E_ € R  with r e s o l u t i o n  fora l l  a.  and  T  Proof.  If  £i £ T  R  T € R  then  >  {T}« 2  R',  so t h a t fTJ"  Since  E  e  &  c  R" = R. E  € R  f o r a l l a.  On t h e o t h e r hand, a dE , a  — 00  where t h e r i g h t - h a n d s i d e i s a - u n i f o r m l i m i t o f a g e n e r a l i z e d sequence o f l i n e a r c o m b i n a t i o n s  of  E .  Now  R c TT(uniform) <= E(weak) = R" = R, so t h a t  R  i s uniformly closed.  Thus  jjs ^ C R  implies  TeR.  T h i s p r o v e s theorem J>. 4.  Definition.  If  R  i s any von Neumann a l g e b r a  i s the set of u n i t a r y operators i n 5.  Theorem•  bounded o p e r a t o r  T  If  R  i s a von Neumann a l g e b r a , t h e n t h e  belongs t o  R  i f and o n l y i f T  commutes  1  R=((R«) )'  (a)  U  Since  ( R ' )  U  R.  w i t h every u n i t a r y o p e r a t o r i n R , i n o t h e r words  Proof.  R  U  c R '  5  ( fR'O ) ' 3 R" = R  (b)  U  On t h e o t h e r h a n d , e v e r y o p e r a t o r I n t h e u n i f o r m l y c l o s e d symmetric unitary  algebra  G  may be w r i t t e n a s a l i n e a r  operators i n  = 1/2  1  H are  2  self-adjoint  operators i n  1  + i H  (since  and  2  2  G  In  G and  ||H||<1,  then  + iH i s uniformly  Thus i f an o p e r a t o r operator i n  G  i s self-adjoint  U « Vl-H G  then  (A+A*)  Next i f H  in  t h a t i f A € G,  = i / 2 (A*-A)  A = H  is  T  c l o s e d ) and i s u n i t a r y .  commutes w i t h e v e r y u n i t a r y  G , i t commutes w i t h a l l o t h e r members o f  G, t h a t i s  G' and  of  G.  To s e e t h i s n o t e f i r s t E  combination  i n particular  C(R'-)V c ( R ' V = Combining  (b) a n d ( c ) y i e l d s ( a ) .  (c)  6  An example o f a von Neumann a l g e b r a . (X, n )  Let  be a a - f l n l t e measure space.  L ( X , u) i s ro  the s e t o f measurable f u n c t i o n s w h i c h a r e bounded a l m o s t where i n X.  L ( X , |a) i s a H i l b e r t 2  Given formation  t e L  T  on  every-  space.  there i s a corresponding  linear trans-  L : 2  (Tg)'(x) = t ( x ) g ( x ) for  g e Lg. Let  M  be t h e f a m i l y o f l i n e a r t r a n s f o r m a t i o n s so i n d u c e d  (1)  M  i s a f a m i l y o f bounded l i n e a r t r a n s f o r m a t i o n s o f  CO  Lg  onto i t s e l f .  To prove this, let g e L . 2  J  |Tgj  2  d|i  X  =  f  .  X  Itg| dU 2  < iitiif /  i g i dM 2  X (where  ||t|| = i n f  k : \x x : | t ( x ) | > k  =  Thus  Ilgjl  2  Tg e L g and l|Tg|| <  so t h a t  .JitII?  T € L(L ). 2  | t„  g  ,  <  -  = 0  )  7  This l a s t i n e q u a l i t y implies llTll <  lltl^  To show t h a t e q u a l i t y o b t a i n s , l e t € > 0 fx  :  |t(x) | >  lltH,  has p o s i t i v e measure, and s i n c e subset.  S  of f i n i t e  2  € j  i s cr-finite,  p o s i t i v e measure.  i s t i c f u n c t i o n of the s e t l|Tg|| = = ff  u  -  l || tt gg |  = C  S. d  2  Then  Let  g  t h i s s e t has a be t h e c h a r a c t e r -  g € L g . Moreover  u  | t g | du + 2  |t(x)|  be g i v e n , t h e n  f  |tg|  2  du  du > ( l l t l l ^ - e )  2  2  MS  s  =  (lltll^ - e )  2  llgll . 2  Thus f o r a l l e > 0, l|T|| >  Hence  ||T|| _> Htj|  .  ||t||  -  w  €  .  Combining t h i s w i t h t h e p r e v i o u s i n e q u a l i t y  yields T T h i s means t h a t  = M  t and  00  L  a r e i s o m e t r i c c o p i e s o f one another.  8  (ii)  M  i s an a b e l i a n a l g e b r a .  To p r o v e t h i s l e t respectively to  s  and  ([S+T]g)(x)  Thus  S+T  t  c  T  be M - o p e r a t o r s c o r r e s p o n d i n g  i n L CO.  (Sg)(x) + (Tg)(x)  o  s(x)g(x) + t(x)g(x)  -  (s(x) + t ( x ) ) g(x) s+t  i n L^, so  S+T e M.  i s any complex number, t h e n  ([cS]g)(x)  whence. cS  and  =  corresponds t o  If  S  =  c(Sg)(x)  -  c's(x') g ( x ) ,  corresponds t o  c s ( x ) € L^, so  cS € M.  Next,  Since  (STg) (x)  =  (S[Tg]) (x)  -  =  s ( x ) [Tg] (x)  =  s(x) t ( x ) g ( x ) .  s t e L , ST e M.  From t h i s i t i s o b v i o u s t h a t  ST = TS. Thus  M  i s an a b e l i a n a l g e b r a :  M c M' .  (iii)  M  i s a symmetric a l g e b r a .  To c a l c u l a t e unique  T*  n o t e f i r s t t h a t i t i s by d e f i n i t i o n t h e  L ( L ) o p e r a t o r such t h a t 2  (Tf,g) = ( f , T*g) for a l l f,g e Lg, that i s / t f g dy  =  / f(T*g)du  or 7  for a l l  f [ t g - ( T * g ) ] du = 0  f ge L . 3  T h i s means t h a t  2  f(x) [t(x) " i W almost everywhere  t(x)  forail  tr^T -  a l m o s t everywhere  (a)  (7*g)  (x)]  f,g. e L , hence t h a t 2  ( W ) (x) =  forall  =0  0  g e Lg.  Thus  (x) = t(x) -gTxT and T*g in  L . 0  Thus  T*  T h i s shows  =  Tg  corresponds t o T e L . T*  symmetric s u b a l g e b r a o f  e x p l i c i t l y and p r o v e s t h a t L(L ). 2  M  i sa  10  M = M»-  (iv) Since sion l e t  M  A € M*.  A  These p r o j e c t i o n s measurable  f e  L  2  X.  =  (x)  get  the  the  reverse  inclu-  projections i n  characteristic  M.  functions  of  Thus  E e M  =  implies  (EAf)  and  To  AE  a l l projections  where  i n d u c e d by  subsets of  (AEf)  M».  commutes w i t h a l l t h e  are  AE for  M c  i s abelian,  E  (x)  that  = e(x)(Af)  corresponds to the  (x),,  characteristic  function  e € L . If  now  both  e  and  f  are  characteristic functions  s e t s o f f i n i t e measure t h e n e,f  e  L  n  2  L  M  and (Aef)(x) Since  where t h e  E^  = e(x)(Af)(x) is  X = (J  (ietu)  are |U E  Thus, f o r  ( A f ) ( x ) =Y  disjoint <  i  f  (  E  ±  and  co  as  i e u )  f(x)(Ae)(x)  cr-finite,  (X'^i .) 1  =  above,  ) i ( e  x  )  of  11 (where  e^  i s the  characteristic  = £  (Ae.f)  = ^f(x)  E^)  (x)  (Ae )  (x)  ±  I (Ae )  = f(x)  function of  ±  (x)  Let  a(x) = J(ieu)) ( A e ^ ^ x ) Then (Af) whenever  f  i s the  (x) = a ( x )  f(x),  characteristic  (*)  function of a set of  finite  measure. This  a ( x ) , "being t h e p o i n t - w i s e l i m i t  measurable f u n c t i o n s ,  is itself  Indeed, the f u n c t i o n S = Then and  S cp  has  a subset  i s the  S»  >  >  ||A||  L  (X,uJ,  llcpll.  for let  ||A||j.  function of  That i s / I M  is in  o f f i n i t e measure.  characteristic  of  measurable.  a(x)  [x • | a ( x ) |  o f a sequence  S*.  If then  nS'  i s positive,  cp e L  p  and  12  T h i s i s a c o n t r a d i c t i o n , hence a(x)  s L  o  and  Hali^  US' = 0  |aS = 0,  Thus  < \\A\\.  The l i n e a r i t y o f the o p e r a t o r (*) t o the e a s e where Since  and  (X,p)  f u n c t i o n s i s dense i n  f  A  Immediately extends  i s a summable simple f u n c t i o n .  i s a - f i n i t e , the s e t o f summable Lg.  Hence g i v e n  there e x i s t s a summable simple f u n c t i o n  f e L s(x)  2  and  simple e > 0,  such t h a t  2||A||  Then llAf - a f I) < ||Af - as || + ||as - af|| •p ||Af - As || + ||as - af|| (since  (As)(x) = a ( x ) s ( x ) ) < llA|| l|f-s|| + l l a l l ^  ||f-s||  < 2||A|| ||f-a|| . Thus  ||Af. - a f || < €.  But  €  almost everyhwere f o r every  I s a r b i t r a r y , hence (*) must h o l d f e L . 0  Since  a e L , t h i s means  that A € M. Thus' . M = M . 1  :  Now,  because  M = M',  M  I s maximal a b e l i a n , i . e . M  is  p r o p e r l y contained i n no symmetric subalgebra which i s a b e l i a n .  15  If M C N, where  N  i s a b e l i a n , then N c N»  and M'  or  = N , 1  combining M  = M'  D N ' S N S M ,  s& t h a t M = N. More t o t h e p o i n t i s t h e o b s e r v a t i o n  that  M' = M implies that M" = H» so t h a t M = that i s , M 6. R  M»  = M",  i s a von Neumann a l g e b r a . Definition.  Let R  be a von Neumann a l g e b r a .  Then  I s t h e s e t o f p r o j e c t i o n s i n R. If  d e f i n e d on  R  i s any von Neumann; a l g e b r a , t h e n a measure can be  R . P  14  7.  Definition,  real function  extended  (i)  m E _> 0  (ii)  m on  i s a measure on P R such t h a t  for a l l  I f ^E^J  E e R  R  p  i f m  i s an  P  i s any f a m i l y  of mutually orthogonal  P R -members,  l\  m  E  i  ^  %  m  '  (Theorem 27 w i l l show t h a t ^  8. and  E  ±  =  sup [E ]  e R ). P  ±  Definition.  A measure  m  i s finite i f m I < » P P E e R there i s P e R  s e m i - f i n i t e i f f o r every non-zero  such t h a t 0 < F < E  and  m F < <». P  9. then  n  for a l l  Definition.  i s absolutely E e R  P  If m  and  n  a r e two measures on  continuous w i t h respect t o  such t h a t  m  R ,  i f nE = 0  m E = 0. P  I n what f o l l o w s i s abelian  -  a s e m i - f i n i t e measure  m  on  R  - R  i s extended t o an i n t e g r a l on t h e s e t o f a i l  s e l f - a d j o i n t o p e r a t o r s ( n o t n e c e s s a r i l y bounded) whose s p e c t r a l P r e s o l u t i o n s l i e i n R . T h i s i n t e g r a t i o n t h e o r y i s developed f a r enough t o p r o v e a Radon-Nikodym theorem f o r s e m i - f i n i t e measures,  15  WEAK COMPACTNESS OP UNIFORMLY CLOSED BALL  10.  Theorem. B  The u n i f o r m l y c l o s e d b a l l  = [A : AeL(H)  and  ||A|| _< l j  i s compact i n the weak topology o f L ( H ) . Proof  b e g i n s w i t h a lemma.  Lemma.  I n i t s weak topology  a subset o f the product  where  C,  C  \  e p  t  L(H)  ( ,y)  —*  W  e  X  »  H  x  I >  H  The homeomorphism i s the e v a l u a t i o n map  '  P/  €  :  i s a copy o f the complex p l a n e .  Proof.  where  i s homeomorphic t o  space  = nT (x,y)  P  L(H)  =  f  e  5 ( >y)> Ax  i s the p r o j e c t i o n o f  onto  C/  v  and  A e L(H). T h i s map i s one-to-one: that  If  A  and  in  L(H)  are such  e(A) = e ( B ) , then (Ax,y) = P ( , ) e ( A ) = ( , ) ( B ) = x  for  B  all x  and  Both subset o f  e  y  p  y  in  and  e  x  y  (Bx,y)  H, t h a t i s A = B.  e"  1  a r e continuous:  Let  5  be a f i n i t e  H x H, l e t € > 0 , l e t A e L ( H ) , l e t S(c,c)  sphere o f r a d i u s  e  about  the complex number  c, and l e t  be the  16  Z  x  =  [T  : TeL(H)  and  |(Ax,y) - ( T x , y ) | < E  for  ?-2  The In  x  s e t s o f form  Z (A,JF,e)  form  (x,y)  [S(P  y )  form  1  t h e weak t o p o l o g y o f e  •ff^  =n» 'y)«) P ( i ,  a l l  ( X j y )  e>J  e(A),  «)]..  a neighbourhood  basis f o r A  L ( H ) . The s e t s o f t h e f o r m  [ L ( H ) ] n Z (A,5,€) 2  a neighbourhood  basis f o r  relativized  e[L(H) ].  Now,  to  since  e[Z } = [e(T) 1  p  (  y) ( ) e  X  j  : TeL(H)  A  e(A)  «  and for  ( X j y )  a l l  = e[L(H)]  n  - P  ( x  ^  y )  e(T)|  < €  '  and  p  for  e(A)  ( x , y ) e 3^  l (x,,)«W'- (.,y)<")'l < p  topology o f  (Ax,y), |P  m e [ L ( H ) ] n £w : w e fp  i n the product  a l l  e  (x,y) € ^ \  Z . 2  That i s e[Z ]  = e[L(H)] n Z ,  1  2  ' i •  so t h a t  e  and  e"  1  are continuous.  T h i s proves  t h e lemma.  17  This e[iB]  lemma i m p l i e s  that  x  arid  y  ( ,y)  =  x  in  H  £  '  2  (x,y)  i s a closed  let i s a complex number  z  and bounded  and i s t h u s compact f o r e v e r y theorem t h e p r o d u c t  s u b s e t o f t h e complex p l a n e ,  x  and  y.  Hence b y t h e T i h o h o v  space  compact. Now  I  e[is] c z j £ ,  P  e(A)  M  since f o r  be p r o v e n i f i t i s shown t h a t  s e t i n t h e compact  First  let  HarII  l|A|! !|x|| ||y|| < l|x|| ||y||  Thus t h e theorem w i l l a closed  A € ©  | = | ( A x , y ) | < ||Ax||  <  numbers.  hWJ .  = T T f ( f (x,y) : ( x , y ) e H x E]  $ also  ?  <.NI  • M  is  i s compact i f and o n l y i f  i s compact. Por  is  8  x, y , z  space $  be i n  H  e[fi]  . and l e t  a,b  be  complex  Let X (x,y,z) 1  = {w:we  ^ ,  P  ( x + y  ^  z )  (w) = P  ( X j 2 )  (w)  +  P  +P  (  ( y j Z )  (w)j  X (x,y,z) 2  = {w:wc#, P  ( x  ,  y z )  (w) - P  ( x  , )00 y  x  ,  z  )  (-)j  18  X (x,y,a,b) 5  =  The  projection  Hence  u  v  j  of ^  being  >  agree, i s closed closed  :  P^  X (x,y z), 1  [w we$ , P  ^  onto  = aTJ P  ( >Y)  ( x 5 y )  (w) / e  i s continuous.  X  fora l l  x, y , z .  Similarly  Xg  functions  and  X^ a r e  i n 3> .  P  (x+y,z) ( ) e  A  f o r any  so t h a t 2  P  i n  'ft,  (A(x+y;,z)  =  ( A x , z ) + (Ay,2)  (x,z) ( ) e  e(A) e X^(x,y,z)  X (x,y,z).  A  -  P  A  +  fora l l  P  (y,z) ( )> e  x,y,z.  A  Similarly  e(A) i s  Also  (ax,by) ( ) e  A  =  (  =  ab~ (Ax,y)  =  so t h a t  (w)  t h e s e t o n w h i c h two c o n t i n u o u s  Note next t h a t  in  ( a X j b x )  a  ^  A  a  P  x  (x,y) ( )' e  e ( A ) e X^ ( x , y , a , b ) e[ft] c  ^y)  A  f o r every  x,y  : x,y,zj  n flfx^  and  a,b.  Thus  J  where J=  C]{\i  x,y,zj  n/°]{x  2  : x,y,a,bj  1 9  which i s a c l o s e d s e t .  "Actually  e[©]  (x y,z)^)  P  +  P  = (x,z)(<P)  (x,y z)( P).= c  P  that  i s ,  H.  since  +  P  =  ( x , y ) ^ ) ^  functional  P , («p) (x  y)  Moreover  P,  e  „\(cp)  functional  on  H  (f ( x , y ) . 7  and  Hence  f  such  bilinear  i s a bounded  f u n c t i o n a l s there  representation  =  (Fx,y)  and  11*11 But t h i s i m p l i e s  P  (  X  ;  y  )  M  =  theorem f o r  e x i s t s an operator  that f(x,y)  bilinear  ||f|| < 1 .  T h i s means t h a t by t h e R i e s z bounded  then  (x,z)M  cp d e t e r m i n e s a b i l i n e a r  f(x,y) on  c  P  cp e  P  (x,y)( P)  (ax,hy)(^) = ^  , for i f  + (y,z)(^  P  +  P  exhaust s  ||ftl<l.  that  - * ( x , y ) = <Fx,y) = P  ( x > y )  e(P)  F.. e'L(H)  20  for a l l and  x,y,  cp = e(F)  so t h a t  and  cp € e['(j].  Thus  c e[u]  .  so e[8] As  =  stated before,  the f a c t t h a t  e[©]  *f. the  compactness of  B  now  f o l l o w s from  i s a c l o s e d subset o f the compact space ^  T h i s proves theorem 1 0 .  .  21  BOUNDED SELF-ADJOINT OPERATORS IN A VON NEUMANN ALGEBRA  11. R  Definition.  If R  i s a von Neumann a l g e b r a ,  i s the s e t o f s e l f - a d j o i n t o p e r a t o r s i n R. With t h i s d e f i n i t i o n theorem 3 rephrases  and o n l y i f the s p e c t r a l  resolution  Any f a m i l y o f bounded ordered by a r e l a t i o n S _< T  _<  of  T  as "T e R  S  i f  P // i si n R .  s e l f - a d j o i n t operators i s p a r t i a l l y  d e f i n e d as f o l l o w s ,  i f and o n l y i f  (Sx,x) _< (Tx,x) f o r all I n most cases t h i s example,  _<  x  i n H.  does not f u r n i s h a l i n e a r o r d e r i n g ; f o r  the o p e r a t o r s  E  and  I-E, f o r a p r o j e c t i o n  E, a r e not  comparable.  12.  Definition.  I f 3 c L(H)  o p e r a t o r s , then whenever i t e x i s t s sup  i s a family of self-adjoint  ,  ^  i s the s m a l l e s t s e l f - a d j o i n t o p e r a t o r which m a j o r i z e s U-member.  If R  1  every  i s a von Neumann a l g e b r a then sup 3 , R  whenever i t e x i s t S j i s the s m a l l e s t  S R -operator m a j o r i z i n g  every  member o f  3 c R  .  inf  The o p e r a t o r s  3. ,  i n f3 R  are analogously d e f i n e d . If If  F_  F  a  S  and- T  are i n  R  then  S  i s the spectral resolution of  S-T S-T,  is in  R  S  also.  then by theorem 3  P  e R  .  If  consequently, the f o l l o w i n g 13.  R  i s abelian, F  then  S,T  commute w i t h  commutes w i t h b o t h  S  and  T.  S-T, a n d ,  This permits  definition Definition.  be t h e r e s o l u t i o n o f  If  S-T.  (S UT)  S,T  e R  = T F  +  0  S(I-F ) 0  SF  Q  (S  (SUT)  R, l e t F  Then  (S D T) = T ( I - F ) +  Obviously  f o r abelian  S  and  0  (SnT)  U T ) * = F *T* + Q  are i n  R.  Furthermore  (I-F )*S* 0  = F T + (I-F )S Q  = TF  = (S  0  Q  +  S(I-F ) Q  U T)  S so t h a t  (S u T) e R 14.  Theorem.  .  I n t h e same way  (S D T ) € R  Under t h e assumptions  (S U T) = sup f s , T ? . R  o f d e f i n i t i o n 13,  Proof.  Since  R  i s abelian,  [F j<3R<£R. 0  The f o l l o w i n g p r o o f shows t h a t {FQI'  (Stff)  which m a j o r i z e s b o t h F o r any  S  i s the smallest operator i n  and T.  x e H,  (S U T x,x)  -  (T x,x)  = ( T F x , x ) + (S(I-P )x,x) Q  Q  i  - (TF x,x) - (T(I-F )x,x) G  Q  -'((S-T)(I-P )x,x) > 0 0  F . Thus  by d e f i n i t i o n o f (S  U  T) > T.  Similarly (S U T) > s  A e £ 1'  Now l e t  F  loe  9X1  0  > S  A  A  0  P  e r a  '  t o r  such t h a t  > T .  Then A  =  F  0  A F  •+  =  Q  +  (I-P )AP 0  FQACI-FQ)  FQ A F Q +  (I-FQ)  Q  + (I-F ) 0  A(I-FQ)  A(I-P ) 0  2k  (since  A e  S  (SyT)  = -  f o r any  i  n  c  e  TFQ + S(I-F ) 0  o  p  T P  o  ( - (J X  +  F  si^o)'  x € H (Ax,x) - (SuTx,x)  =  ( P Q A F ^ X )  + -  (I-F )A(I-P )x,x) 0  (P TP x,x) + Q  = ((A-T)F x,F x) 0  Q  0  ((I-F )S(I-F )x,x) 0  + ((A-S)(I-F )x,  0  0  T h i s l a s t l i n e i s non-negative,  since  A  0  (I-F )x) . Q  majorizes  S  and  T.  T h i s proves theorem 14, I n the same way, 15.  Corollary (SOT)  16.  Theorem.  i n f £S,T}  =  If R  n e c e s s a r i l y a b e l i a n ) and  i s any von Neumann a l g e b r a (not  3 c R  i s d i r e c t e d upward and bounded  above by the s e l f - a d j o i n t o p e r a t o r g belongs t o R .  S , then Q  Proof.  F o r a l l F e gr l e t W(F)  V(F) =  £T : T e  the s e t 3  and  T _>  FJ  sup 3  e x i s t s and  be the weak c l o s u r e o f  FQ e 3  Choose 3  and l e t  = V(F ) = £T  : T e 3  Q  Q  Observe  that i f  £F^  and  : i e nj  T _> F | Q  i s any f i n i t e  ^-subset,  then  (~) [V(F ) 2 i e n j ^ 0 . 1  To p r o v e t h i s n o t e t h a t T-^ € 5  such  l 1  k < n, t h e r e  T  Obviously  T  n  o  T  T  k + 1  - k+1 F  >_ F^  n  F  exists  k+1  T A  #  i s d i r e c t e d upward t h e r e  that  T  For  since  T  l-  F  e 2*  such  kfl -  f o r a l l 1 € n,  e f ]  l •  T  for a l l f i n i t e  {w(F ) ±  [ V(F ). : 1 e n ±  : i e nj  jL 0  families ^F  i  : i e n j  T h i s means t h a t ¥(F)  k  so  fortiori D  : F €' 3>Q  £ ^ Q  that  |  exists  26  has t h e f i n i t e i n t e r s e c t i o n p r o p e r t y i n B = £ r : T € L ( H ) , ||T|r<max [W^l, Since  8  \\S \ Q  i s weakly compact D If  {w(F) A  (ii)  0  i s i n t h i s i n t e r s e c t i o n , then since V(F) c 3 c R  weak c l o s u r e o f (i)  =0  : F € 3? ^  A  A  i s i n the  g  i s self-adjoint  A > F  for a l l F € 3  and hence  Q  A _> F  f o r a l l F € 3>Q (iii) Actually  A  A € R i s t h e o n l y o p e r a t o r i n t h e i n t e r s e c t i o n and i s  moreover t h e supremum o f  3.  For i f  T  i s any  self-adjoint  o p e r a t o r w h i c h m a j o r i z e s e v e r y 3-member, t h e n i n p a r t i c u l a r , . f o r a l l S e ¥(F)  T>S  f o r a l l F e 3.  T h i s p r o v e s theorem  16.  17.  R  Corollary.  If  Hence  T > A.  i s any von Neumann a l g e b r a , and  s  3 c R  i s d i r e c t e d downward and bounded below by t h e  adjoint operator  S , then  ( i n f 3)  self-  e x i s t s and b e l o n g t o  R .  27  Proof.  The s e t  -3?  is  =  £-F : F e 3 J  d i r e c t e d upward and bounded above by  -S . Q  Hence 1 7 f o l l o w s  from 1 6 .  18.  Definition.  F o r sequences o f s e l f - a d j o i n t  S  operators,  f  w i l l mean t h a t  S  n  S  n+1  and S |  S  (strong)  w i l l mean t h a t  s t n  and S = l i m (strong) The  S . n  expressions  S T are d e f i n e d  19. convergence  S(weak)  S  ^-'S ( u n i f o r m )  analogously.  Theorem.  F o r monotone  are equivalent: i f f T I  sequences,  s t r o n g a n d weak  i s a monotone  sequence,  28  then T = lim(weak) if  and  T  n  only i f T = lim(strong)  Proof.  Since the  T  R  strong topology  i n c l u d e s t h e weak  topology T = lim(strong)  T  n  implies T = lim(weak)  For the converse, T  t  n  T  T  n  suppose  (weak).  Since  T  G l  < n  <  T  T n  il  <  K  T  >  >  where K = max Now  f o r any  (T-T )x||* n  =  x  ^||T !|, 0  in  H  |((T-T )x, n  < ((T-T ) x, 2  n  (T-T )x)|  2  n  (T-T )x)((T-T )x,x) n  n  (since  .._> 0  T-T  )  . ((T-T ) x,x)f(T-T )x x) 3  n  n  3  < ||(T-T ) x|| ||x|| ((T-T )x,x) 5  n  n  < !!T-T|| ||xl| ((T-T )x,x) 5  2  N  < (since  n  ( 2 K ) ||x|| ((T-T )x,x) 3  2  n  ||T-Tl| < ||T|| + HTJ < n  2K)  Since T = l i r a (weak) T , there e x i s t s  N(e,x)  such t h a t  n J> N  implies  A and hence t h a t ||(T-T )x|| < e. n  Thus T  n  f T (strong)  and 19 i s p r o v e n . Remark.  I n view o f S„n t S  19 Tn I T  w i l l be w r i t t e n w i t h t h e u n d e r s t a n d i n g t h a t t h e convergence i s  30  b o t h weak a n d s t r o n g .  20.  Theorem.  I f  R  i s abelian with  S , S, T  i n  R , S  then i f  St  S,  n  then a l s o  \ (S  (s  Proof 21.  n  n  U T) f  (S U T)  n T)  (s n  t  T)  d e p e n d s o n two lemmas: Lemma.  I n any  R  fsj  .  S = sup  Proof.  f ^ s  i  with  S , S e R , S  i f S  n  f S  d i r e c t e d upward and bounded above.  s  n  Hence C = sup  Sj  I  S e x i s t s and b e l o n g s t o  R  b y 16.  Hence  S C . I f e q u a l i t y does n o t h o l d h e r e t h e n t h e r e must e x i s t such  that (Sx,x) >  (Cx,x),  x € H  then  and, s i n c e  s t n  there e x i s t s  N  s,  such t h a t  (Sx,x) > ( S x , x ) n  n J> H  implies  > (Cx,x),  c o n t r a d i c t i n g the f a c t t h a t  C _> S  for a l l  n  n.  Hence  S = C = sup £ s j n  T h i s proves 2 1 .  22. ST  and  Lemma.  I n any  R, i f £  s  n  ^  c R^  i s bounded  above  , then (by 1 6 )  N  S = sup fs J n  e x i s t s and belongs t o  R . Moreover, S  Proof. fS }c B = n  Since  {A  n  f S  Observe t h a t  HAH  : A € L(H),  < max  [& J  J J i s weakly compact,  n  n  a  s  f ||S!|, Q  a  ||s|ljj\  subsequence  such t h a t  f o r some  C e s .  Since £ $ ^ ^ n  k  c R, S  C € R  S  also.  ^"  s n  ^^  32  Now f o r any  x.e H,  ( n(k) ' ) ? ( S  x  x  C x  ' )' x  Hence a l s o (S x,x)t  (Cx,x)  n  Since a l l the operators  involved are s e l f - a d j o i n t  s T>c. n  By 21 t h i s i m p l i e s t h a t G = sup ' f s }  = S  n  Thus s and  n  f  S  22 i s p r o v e n . Now t o complete t h e p r o o f o f 20. Since (S U T) > S > S  Q  and (S u T) > T, obviously (S u T) > ( S  n  u T)  (a)  Furthermore, (S  n + 1  u T) > S  n  +  1  >  S  n  33  and (S  UT)>T,  B + 1  so t h a t  ( n+l s  U  T  (S  n  )  2. ( n s  u  T  ) •  That i s U T)  t  0>)  By (a) and (h) and 16, t h e r e e x i s t s C = sup  f (S  n  U T)J  (S U T) > ( S  n  UT)  C e R  S  such t h a t  Now  implies  that (S U T) > C By 21, t h e f a c t s  implies  (c)  that  n  t  s  that S - sup  {S J , n  hence t h a t C > S,  .  since c  f o r a l l n.  > (  U T) > S,n  s n  Moreover, C _> ( S  so t h a t hy 14, s i n c e  n  > T,  C e R  e > (s u  Combining  U T)  9  T)  ( c ) and (d) y i e l d s  C « (S U T), that i s ,  (S U T) = sup Thus by 22  (S U T) f ( S U T) n  To p r o v e t h e second a s s e r t i o n o f 20 n o t e t h a t  T)  (s  n  n  < s  (s  n  n T) < T,  n  <  S,n+1  and  so t h a t by 1 5 , s i n c e (s  n  (s  n  (S  f| T) € R , S  n  n T) < (S,n+1 0 T) .  Thus fl  T)  t .  That t h i s sequence tends weakly t o  ( S Cl T)  from the f o r m u l a (S+T) = S F  Q  + S(I-P ) + T F Q + T(I-P )  = SF  Q  + T(I-P >  =  n  T h i s proves  20.  (S  0  Q  T) +  (S  0  + T F Q + S(I-P ) 0  U T)  .  follows i  SIMPLE FUNCTIONS I N  R. S  SP 23.  Definition.  R  i s t h e s e t o f a l l hounded o p e r a t o r s  of the form P S = •£ a  i E i  ,  1=1 where  p  I s a positive integer,  a^  i s r e a l , and  ^E^J i s a  P  family of  R -members such t h a t P 1=1  ( t h i s i m p l i e s t h a t t h e E^  24.  { n] T  S  r S P  Theorem. s u c h  T  Proof.  n  are mutually  If T € R  orthogonal).  t h e n t h e r e i s a sequence  t h a t  f T  (uniform).  Choose  d > 0  and l e t P  be any f i n i t e  point  set p a r t i t i o n i n g the i n t e r v a l [-d-||T|1, ||T||]. P : -d-||T|| = a If for a l l real  E  a Q  Q  <  < ... < a ^  < a  i s the spectral r e s o l u t i o n of  p  = ||T|| T, t h e n  E„ e R a  a, and  a, , (E - E ) < T(E l - x a.,, a__ a i  ]  - E ±  a _ a  ±  ) < a, ( E - E ) 1 a . a _ / o  1  o  ±  ±  1  P  or, l e t t i n g  E. « E  - E a  1  a  i - l  E  i -  T  E  a _  ±  ±  i -  a  i  1  i  E  so t h a t L(P) «  <  P £ a ^ i=l P :£ T E  E  ±  ±  = T  1=1  P  < r i%^^' a  iT  L(P)  < T < U(P)  p)  that i s , .  How P  U(P)  - L(P)•-  Y  (a -a _ )E i  i  1  i  i=l <max  {a -a J  * (max  P) I ,  1  1-]  so t h a t 0  and  < T - L(P)  < (max  hence, T - L(P)H  < max  P  P)  38  Thus i f a sequence  of partitions of  £l> J n  [-d-||T||, ||T||]  i s chosen so t h a t l i m (max P ) = 0 , then 1|T  lim  -  L(P )||  =  n  0.  Hence T = l i m (uniform) L ( P ) . Now  c  ^L(P )J n  R , and i t s convergence t o T S P  be made monotone by r e q u i r i n g t h a t  P  n  £  P  n 1> +  for, l e t m  P'  where  a' / P, j _ i a  f a } u P, 1  < a' < a^, s a y . L e t  P L ( P - J )  -  £  (1*0  a _ 1  E .  1  ±  1=1 Then L(P)  =  L(P-j) +  -  L(P-J)  +  a  a j  J  _  _  L  1  (E  a j  "-E  a f  ).  may  <  L(P-j)  + a' ( E  +  =  that  a  j-l  ( a- "  L(P»),  is, L ( P ) < L ( P ' ) < T. T h i s p r o v e s 24.  - E  a  E  f t l  )  \  POSITIVE AND NEGATIVE PARTS OF AN R -OPERATOR S  Recall that I f T spectral resolution  i s any bounded s e l f - a d j o i n t o p e r a t o r w i t h  E . then cl  T = T  +  - T~,  where T  +  = T(I-E ) 0  T" = - T E  0  are p o s i t i v e o p e r a t o r s which commute w i t h  T  and w i t h each o t h e r j  and f o r w h i c h T T"  =  +  0.  The f o l l o w i n g theorem w i l l be i m p o r t a n t t o t h e e x t e n s i o n P S o f a measure from R t o R s<  25.  Theorem.  If T  •  •  .  i s any s e l f - a d j o i n t o p e r a t o r and  T = A-B, where  A  and B  a r e p o s i t i v e o p e r a t o r s t h a t commute w i t h  then A = T  +  + P-.  B = T" + P, where  P  i s a p o s i t i v e operator.  1  T,  Proof.  Since A - B  =  T  +  - T"  implies: A - T+  =  B - T",  i t remains o n l y t o show t h a t P  =  A - T+  is positive. If  E  i s the r e s o l u t i o n of  T, t h e n f o r any  x e H  ( P x , x ) ( A - T x , x ) = (A-T x,F x) + (A-T+XjI-Ex). +  +  Q  = ((A-T ) E x , E x ) + ( ( A - T ) ( I - E ) x , +  (since  E  commutes w i t h  Q  Q  0  +  (JME )x)  +  Q  T E +  Q  0  = T(I-E ^ 0  0  0  (since  T  +  0  =0)  = (AE x,E x) + ((A-T)(I-E )x, o  Q  A-T )  = ( A E x , E x ) •+ ( . ( A - T ) ( I - E ) x , (since  (I-E )x)  +  0  0  0  (I-E )x) Q  = T(I-E )) Q  = ( A E x , E x ) + ( ( A - [A-B| ) ( I - E ) x , ( I - E ) x ) 0  (since  G  0  Q  T = A-B) » (AE x,E x) + (B(I-E )x, ( I - E ) x ) . Q  0  0  T h i s l a s t l i n e i s non-negative since  0  A  and  B  are p o s i t i v e .  4:2 Thus  P _> 0  and 25 i s p r o v e n .  I m p o r t a n t Remark. If  A,B  are i n R , S  If  so i s  T e R , t h e n so a l s o a r e S  P.  T , +  T~.  43  SUPREMA AND INFIMA IN R Let  P  R  be any von Neumann a l g e b r a . I f attention i s restricted P P t o subsets o f R ••, then 16 can be improved: subsets o f R need not be d i r e c t e d t o have suprema and i n f i m a .  Moreover, the supre-  mum  ( o r infimum) o f d e f i n i t i o n 12 t u r n s out t o be the u s u a l  mum  (Infimum) o f a c o l l e c t i o n o f p r o j e c t i o n s .  26.  Theorem.  P then  P e R  P  If R  supre-  i s any von Neumann a l g e b r a and  = p r D £rng E : E e ffj,  and  P = infg  Proof.  To show t h a t P = inf  suppose t h a t  S  g ,  i s a s e l f - a d j o i n t operator  f o r which  S _< E  E  .  x e H  with  for a l l  Assume, moreover, t h a t there e x i s t s that (Sx,x) >  (Px,x).  ||x|| = 1  such  44  Now (Sx,x) < 1 and (Px,x) = 1  or  (Px,x)  =0.  Thus i t must he t h a t 1 _> (Sx,x) > (Px,x) = 0 But  (Px,x) = 0  implies that x £ rng E  f o r some  E  Q  e  , and hence t h a t  0 = ( E x , x ) >_ (Sx,x) > 0. Q  This i s a contradiction.  Hence  (Sx,x) < (Px,x) f o r a l l x e H,  that i s S _< P.  Since P _< E for a l l E e ^  , t h i s means t h a t P =  To show t h a t  infg  P e R, l e t U  he a u n i t a r y o p e r a t o r i n  R'  4 5  Then UE = EU for a l l  U*EU = E  UEU* = E  (a)  E e R .  If  P  F  i s any p r o j e c t i o n  such t h a t  F _< E  fora l l E ' ( ^ ,  then f o r a l l x e H, (U*FUx,x)  by  =  (FUx,Ux)  <  (EUx,Ux)  =  (U*EUx,x)  =  (Ex,x) ,  ( a ) , so that U*FU _< E  for a l l  E e^  .  Similarly,  UFU* < E for a l l  E e^  .  Thus, i n p a r t i c u l a r ,  UPU* < E for a l l E e Now  U*PU < E  .  P m i n f £ p , thus UPU* _< P  (b)  U*PU < P  (c)  Now by ( c ) (U*PUx,x) < (Px,x) for a l l  x £ H, hence f o r x = U*y, where  y e H :  (U*PU(U*y), (U*y)) < (PU*y,U*y), (UU*PUU*y,y) < (UPU*y,y) , whence (Py,y) < (UPU*y,y) for a l l  y € H.  Thus P _< UPU*  (d)  Combining (b) and (d) y i e l d s P = UPU* or PU = UP. Since  U  i s otherwise a r b i t r a r y ,  t h i s means t h a t  w i t h e v e r y u n i t a r y o p e r a t o r i n R'.  P  commutes  By theorem 5, P e R.  T h i s p r o v e s 26.  27.  Theorem.  If R  i s any von Neumann a l g e b r a and  cfs*Q  = p r [ ( J [ r n g E : E e£j  ]  47  (that i s ,  Q  i s the p r o j e c t i o n onto t h e c l o s e d subspace g e n e r a t e d  by t h i s s e t u n i o n ) , t h e n Q  and  sup^  Q = I - pr[(J  Proof.  by 26.  =  Q e R  £rng E : E egf  =  I - pr D  ^(rngE)- : E €  g>$  =  I - v^D  ^rng  e^ J-  =  I - i n f £l  1  (I-E) : E  - E : E e g  ]  X  3  j  Since P = i n f £l-E  : E  e^j  e  P Q € R  also.  I f now  S _> E  for a l l E € ^  , then  I - S < I-E for a l l E  hence by  26,  I - S _< P and S = I - ( I - S ) > I-P = so t h a t Q This proves  =  sup 27.  '  Q  .  E , P  48  Remark.  Now t h a t 27 i s p r o v e n , d e f i n i t i o n 7 o f a measure  P on  R  i s complete.  4  9  BT - OPERATORS  28.  Definition.  Let  R  be the s e t o f a l l s e l f - a d j o i n t  W  o p e r a t o r s ( i n c l u d i n g unbounded ones) whose s p e c t r a l r e s o l u t i o n lies in  R. P  I n l i g h t o f theorem J> R  L(H) n  =  S  R. N  P r o o f o f the f o l l o w i n g theorem may  be found i n F u n c t i o n a l  A n a l y s i s by,Rei s z and Sz.-Nagy (Ungar, New York, 1 9 5 5 ) , page  314.  Von Neumann's Theorem. Given (a) a sequence  : iewj  of projections  such t h a t  and (b) a sequence  £A^  : ieuu^  =  E^A^  o f bounded s e l f - a d j o i n t o p e r a t o r s  such t h a t A E i  for A  a l l i , then t h e r e e x i s t s a unique s e l f - a d j o i n t o p e r a t o r (which may AE  for  ±  ±  a l l 1.  D(A)  not be bounded) such t h a t  =  E AE i  =  i  E  i i i A  E  =  A  i  E  i  Moreover  =  £x  : x € H, r  llA^xll  2  <  »J  50  and f o r  x € D(A) .'ss ^  AX  29.  Theorem.  I f the hypotheses  i  A  Let R  E  i  x  *  he an a b e l i a n  von Neumann a l g e b r a .  o f von Neumann's theorem are strengthened:  [E j  cR ,  [ A j  P  ±  C R  S  ,  N  then  A e R . Hence i f A  i s any s e l f - a d j o i n t o p e r a t o r f o r which  E A c AE ±  f o r a l l i e uu,  ±  e R  S  where  N  then  A € R .  Proof.  The p r o o f i s taken up mainly w i t h d e r i v i n g the  spectral resolution of Let Since  R  P (i) a '  A  from those o f t h e A^.  be t h e s p e c t r a l r e s o l u t i o n o f  v  i s abelian, A  A  .  E  i A i  f o r a l l i , and F (i)E a  for a l l a  and i .  i  =  E i  F (i) a  A.  1  f o r each I .  Let F  = sup { P ( i ) %  a  : ieu^,  a  which i s a p r o j e c t i o n o f  hy 27.  R  Since the  E^  are mutually  orthogonal, P  a  whence o b v i o u s l y  P  "  'at ) 1  <:  a  To show t h a t  £  P  Q  E  i  i f a _< b . y i e l d s a r e s o l u t i o n o f the i d e n t i t y , i t  must be shown how the p r o p e r t i e s (i)  l i m (strong) F  (ii)  l i m (strong) F  (iii)  l i m (strong) P a -oa  b  = P  = I  0  Q  are i n h e r i t e d from the p r o p e r t i e s o f the  F (i)'.  To prove ( i ) , l e t  n for  x e H.  Then lim  Let  e > 0  =  =  be g i v e n .  x. Then f o r a < b,  Y (i€u>) | | ( F ( i ) - F j i ) ) E ^ l l a  2  = £(ien) ||(F (i) - F ( i ) ) E ^ J a  For  each  i c n t h e r e i s an a  j_  >  a >.  13  i  b  > a  such t h a t i f  a  then ||(P (i) a  (i)) E  F f c  l X n  l|  < ^  Hence i f a _< h < min  £a^ : i e n j ,  then ^(ien)|l(P (i) - P (i)) E a  b  i X n  ||  2  < e  2  that i s , ||F " a nx  - P.D x_ n"|| < €.  C o n s i d e r now t h e f o l l o w i n g :  | ||F x - F x|| a  <  IIP^  b  - F xJ| | b  ||F - F | | Ux-xJ b  b  Thus, s i n c e 0 < F this  b  - F  a  < I,  yields  | ||F x - F x|| a  b  -  ||F  aXn  - F x || | < ||x - x|| b  n  n  Now f o r n > N ,  IIx  -  x || < n  e/2,  so t h a t . 0  i H(V h) !l / lUVV ^ F  € <  x  2+  3  f o r ' n _> N . But t h e r e i s a l s o an  a(n)  such t h a t  a _< b < a ( n ) implies that  tKVa>J </ ' P  x  €  2  Thus i f a < b < a(n), ||(P -F )x|| < e / 2 + c/2 = e. b  Since  €  a  i s a r b i t r a r y , t h i s p r o v e s ( I ) . P r o p e r t i e s ( I i ) and ( I I I )  f o l l o w i n t h e same manner. Thus .  £ a P  :  -  00  <  a  < + °°J  i s a r e s o l u t i o n of the i d e n t i t y . I n o r d e r t o show t h a t the o p e r a t o r (iv)  A F  (v)  AF on  Q  ct  i s the s p e c t r a l r e s o l u t i o n of  i t remains t o show t h a t A c  ct  F  AF.  —  < a F D(A).  fora l l a  ct  and  A(I-Fj > a(I-Fj  54  To p r o v e ( i v ) note t h a t from t h e u n i q u e n e s s guarantee i n von Neumann's theorem, i f • x € D ( A ) , t h e n Ax  =  Y (ietu) A^E^x  Hence P Ax a  =  P  V a L*  =  Y  \  ±  '(sinee  M  E  i  VaW  i i  A  E  A  i  E  x  i  X  E F (i) = P (i) E ) ±  a  a  £ (since  E.A-E.x i i 1  ±  W a ^  V  A P (i) = P (i) A ) ±  a  &  =  ±  AP x. a  This proves ( i v ) . To p r o v e t h e f i r s t p a r t o f ( v ) , l e t x e D ( A ) . (AP x,x) a  =  Then  ^ (V (i) V i ) E  X  a  < ^ a(P (i) E.x, E x ) a  =  ±  a (F x,x) a  This proves the f i r s t a s s e r t i o n ;  p r o o f o f t h e second i s a n a l o g o u s .  Now t h e s e l f - a d j o i n t o p e r a t o r  A  has a r e s o l u t i o n  F '. cl  55  In virtue of ( i v ) F P. a b  1  for a l l real  a  and  =  F,'F b a  b.  Hence  are p r o j e c t i o n s . To show t h a t t h e s e a r e z e r o , suppose t h a t x e D(A) n r n g F Then s i n c e  (I-F »)  a  a  x e rng F , a ((A-aI)x,x)  < 0  by ( v i ) . That i s /  J  (b-a)d  1  b  -co  T h i s means t h a t F^  ( F ' x , x ) < 0,  (F-^'x,x)  i s constant  f o r b > a.  i s r i g h t continuous (F 'x,x) = (F 'x,x) b  for  b > a. But  x e r n g (I-F„') a (F »x,x) b  for  a  b '< a.  Hence  =0  a l s o , so t h a t  Thus s i n c e  56  (P 'x,x)  =0  b  for  a l l b.  Thus  x=0  and  Similarly  Hence  F  Q  < F  ct "™"  That i s , F  a  '  and  ct  F  St  ' < F . —~ cl  F  ci  i s the s p e c t r a l r e s o l u t i o n o f  T h i s proves t h a t The  so t h a t  A €  '= F '  f o r a l l a.  A.  R. N  29 f o l l o w s from the f i r s t :  second p a r t o f theorem  Since  E A c AE ±  e  ±  R  S  N the f i r s t p a r t o f 29 says t h a t there i s an o p e r a t o r  A'e  R  such  that EjA  c A'E  1  =  ±  AE  ±  By the uniqueness guarantee i n von Neumann's theorem, A=A'. A  Hence  R. W  €  T h i s completes the p r o o f  30.  Theorem.  domain and range i n  If H  T  —  R  x  (TE )* = TE i  i s a l i n e a r transformation  such that  E.T cz TE. s l  of 2 9 .  i  S  with":  f o r a l l ieuu , where  R  i s an a b e l i a n yon Neumann a l g e b r a and E  : ieuuj? c R , ^ P  ±  = I, N  then there e x i s t s a unique  T' e R  such t h a t  T' => T.  B y von Neumann's theorem t h e r e e x i s t s u n i q u e  Proof.  w h i c h i s s e l f - a d j o i n t and s a t i s f i e s E.T' c T'E. = TE. 1 — I i for a l l  i . D(T') =  £x  : x. e H, ^  HTE x|| i  2  <>J  and i f x e D(T») T'x By  =  T E  j[  29, T' e R . N  To show t h a t  T'  extends  Tx = C)^ ^) 2  (since  x  = I!  E.Tx  =  TE x  ^  T x  (a)  ±  E^T- c TE^)  Now s i n c e TE^  =  E TE jL  i  T, l e t x e D ( T ) . Then  ,  T'  the terms o f t h e sum (a) a r e m u t u a l l y  that i s ,  orthogonal.  Thus from (a)  x € D(T').  Now (a) a l s o i m p l i e s t h a t  Tx = T'x.  Therefore  T C T'  *  T T" e. R  i s the only  1  and  N  - operator  N  w h i c h extends  T, f o r i f  T c T», t h e n TE  f o r a l l 1.  R  i  Since  c.  TE  T ^ € R , i t i s d e f i n e d on a l l o f S  jL  H.  Hence  actually, TE  ±  =  T"E ,  =  T"E. l  i  and moreover TE. i  =  T'E.  B u t t h i s , t o g e t h e r w i t h t h e u n i q u e n e s s guarantee i n von Neumann' theorem, i m p l i e s t h a t T"  =  T'  T h i s p r o v e s theorem 30. JI. £ i E  Lemma. I f  :  ^ U }  {  F  j  :  are f a m i l i e s o f p r o j e c t i o n s such t h a t  59  and i j  E  =  F  F  j i> E  then a l s o ,  y T.j Proof. subset  (si'^cj/^  E  i j F  Given  =  I  xeH  * and  €>0  t h e r e must e x i s t a f i n i t e  such t h a t |y-x| < e / 2 ,  where  There i s a l s o a f i n i t e subset |y - Y ( J ^ ) €  F  such t h a t  jff's^  jyl  <  e/2  Thus |x - £ ( i e ^ j e / ) E ^ x l  = |x - £(J«UJ') P y | d  < |x-y| + |y - J ( j e $ ' ) P j y | < e/2 + e/2 = e Hence ^  F_. = -j  and t h e lemma i s p r o v e n .  I  60  32.  Lemma. ——•——  P R , where  in  R G  G  E . F „ be two r e s o l u t i o n s o f the i d e n t i t y a a  i s abelian. = E F ,  Q  then  Let  If  p i s a l s o a r e s o l u t i o n o f the i d e n t i t y i n R .  a  P Proof. If  a  G G, a b  Since  R  i s abelian,  =  E. F, E  =  E  a b a b  =  E  a  =  G, G b a  F  b b a a  G  Ev.F F,  F  a  a >  that G  a  <  V  Moreover, i f a g a i n ! ! G  b  x  "  G  a  - ^Vb  - a a> H  < "( a a  " a b) H  < f o r any F  x e H.  a _< b,  x I 1  E  and  e R ..  C b,  = so  G  Thus  E  P  E  F  x  P  x  a  G a  b  E  a  P  E  P  x  b  i n h e r i t s the r i g h t c o n t i n u i t y o f  properties (strong)G a ^ -oo  E a  a  lim  N( a b " b b> «  II(P - F )x|| + ||(E - E ) x | | ,  .  The  +  = a  0  61  and lim (strong)G a-* + 0 0  =  a  I  can be demonstrated i n s i m i l a r o r s i m p l e r  fashions.  T h i s p r o v e s 32.  33-  "Theorem. N  are unique  If R  i s a b e l i a n and  A,B € R , N  then there  .  R -operators  S, T, N  such t h a t  S 2 A + B T A N  Proof. F„ a  where R  ZD  i  and  i s abelian,  AB  Let  respectively. E(i).  - B  .  and  A  B  BA.  D  have s p e c t r a l r e s o l u t i o n s  E  ±  _  1  F(j) .  }  are integers  (negative  F  6  -  too!).  By lemma 31,  i  and  j  E ( i ) F ( j ) (A-B)  R  = E(i) F(j) A  -  E(i) F(j) B  = F(j) E(i) A  -  E(i) F(j) B  i s abelian) c  (since  and  P.^  E ( i )F(j) =1.  ^  Now f o r any  (since  E  Let E. -  j  and  N  F ( j ) (AE(i))  E ( i ) and  -  E ( i ) (BF(j))  F ( j ) are s p e c t r a l p r o j e c t i o n s )  since  62  = (AE(i)) F ( j ) (since  AE(1)  and  (A  (BF(J)) E ( i )  BF(j) e R , S  = AE(i) F ( j ) -  -  -  (a)  which i s abelian)  BE(i) F ( j )  - B) E ( i ) F ( j ) . c  S i n c e l i n e (a) i s an R  -operator,  E ( i ) F ( j ) (A-B) c (A-B) E ( I ) F ( j ) e Since the f a m i l y of  E(i) F(j)  R. S  i s c o u n t a b l e , and  theorem 30 a p p l i e s , so t h a t t h e r e e x i s t s a unique  R  i s abelian,  T e R^  such  that T => A-B. The a s s e r t i o n about The case o f  AB  and  (A+B) BA  f o l l o w s immediately. i s s i m i l a r , but i t requires  some a d d i t i o n a l remarks: E(i)  (since  E  &  -  F ( j ) E ( i ) AB  c  F ( j ) (AE(i)) B  i s the s p e c t r a l r e s o l u t i o n of =  (since  AE(i)  and  (since both  are i n  R, w h i c h i s a b e l i a n )  (AE(i)) (BF(j))  (b)  i s a s p e c t r a l p r o j e c t i o n of = BF(j)  A)  (AE(i)) F ( j ) B F(j)  c (since F ( j )  F ( j ) AB  B)  A (BF(j)) E ( i ) and  E ( i ) are i n  R,  which i s abelian)  =  AB E ( i ) F ( j ) . S  L i n e (b) i s a product o f E.(i) F ( j ) A B c  R  o p e r a t o r s , hence  ( A E ( i ) ) ( B F ( j ) ) » ABE(i) F ( j ) e  R. S  By  symmetry E ( i ) F ( j ) BA c ( B F ( j ) ) ( A E ( i ) ) = BAE(i) F ( J ) € R N Thus by 30 t h e r e e x i s t s unique N,N' in R such t h a t N' 2  N ^ AB However, s i n c e  R  S  BA.  i s abelian  ABE(i) F(J) = (AE(i))  (BF(j))  = ( B F ( J ) ) ( A E ( i ) ) U BAE(i) F ( J ) in  R.  Hence  S  N =.N'» .  T h i s completes the p r o o f of 33. Because the  S, T, N  o f theorem 33 are unique,  f o l l o w i n g d e f i n i t i o n s are p o s s i b l e .  34. S  Definition.  = A + B  I n 33  T = A - B  N = A  Note t h a t 33 implies' t h a t f o r  and  and  A ° B  =  B  o  A  A +* B  =  B  I  A  that  A,B  B.  o  €  R, W  the  64  A - A = 0. Moreover, i f C e R  W  a l s o , then  (A+B)C = AC + BC and AC + BC c (A°C) + (B.C) c (A.C) + (B.C) w h i l e on t h e o t h e r hand (A+B)C c (A+ B)C c (A+ B)«C. Since these extensions are unique (A+B)oC Thus  R  =  (A«C)  +  (BoC).  i s a commutative a l g e b r a under t h e o p e r a t i o n s  35.  I f A-B >_ 0  Corollary.  on  D(A-B), t h e n  +  A-B _> 0  also. Proof. D(A-B) =  C a r r y i n g on w i t h t h e n o t a t i o n o f 3 3 , by 30  £x : xeH,  ||(A-B)E(i) F ( j ) x | |  2  < *>}  and f o r x e D(A-B) (A-B)  x  =  ^  (A-B) E ( i ) F ( j ) x,  whence i t ' s o b v i o u s t h a t  36.  Corollary.  A-B > 0  implies  If A e R  and  A - B > 0.  B e R , then  and o  65  A  0  B  m  B o A  =  AB,  and so BA c  Proof.  AB  Let  €  R  W  N = A»B = BoA  .  Then  s  Since  B e R , D(BA)  so t h a t  BA  =  'D(A)  i s d e n s e l y d e f i n e d and t h e o p e r a t o r  Hence N = N* c (BA)* = A*B* = AB c N whence N = AB T h i s p r o v e s 36.  (BA)*  exists,  66 POSITIVE AND NEGATIVE" PART So OF? AN"R--OPERATOR  Let G . a rt  N  be an o p e r a t o r i n R^  with spectral  resolution  Then N = N[(I-G ) + G ] 0  Q  N(I-G )  4- N G  (I-G )N  4- GQN  =  [(I-G )  +  =  N(I-G )  2  Q  0  2  0  G  Q  Q  ]N  =  N  Thus N  0  -  (-NG ) G  Let N  +  = N(I-G )  + N , N  Then b o t h  N" = - N G  Q  Q  .  ^ N a r e p o s i t i v e , and by 26, b o t h a r e i n R . N  N  +  - N"  N = N  +  - N"  m  Hence  as w e l l as  37-  Theorem. I f N = A - B,  where  A,B  operator  P  are p o s i t i v e i n R , then there e x i s t s a unique p o s i t i v e N  in R  N  A = P + N  +  Proof.  such t h a t ,  I f A.B  B = P + N" have t h e r e s p e c t i v e  resolutions  E ,F ,  67  E ( i ) ==' E  j[  - E ^  i e ou  -  J e. w,  and F(j)  =  then ^  E(i)P(j) = I  ;  -J since  A . and  B  are p o s i t i v e .  Note f i r s t (A-B)  that  E(i)F(j) =  (A-B) E ( i ) F ( j )  = AE(i) F ( j ) - BE(i) F ( j ) , S where t h e l a t t e r (A-B)  i s in  R  .  Also  E ( i ) F ( j ) = (N+-N") E ( i ) F ( j ) = N E(i) +  F ( j ) - N"E(i) F ( j ) .  Thus N E(i)  F ( j ) - N~E(i) F ( j ) e  +  This  means t h a t  R . S  the s e l f - a d j o i n t operators  N E(i) +  P ( J ) N~E(i) P ( j ) r  c are defined  everywhere.  Hence t h e y  a r e bounded and b e l o n g  Thus AE(i)P(j)  - BE(i)F(j) = N E(i)F(j) +  where a l l t e r m s a r e i n AE(i)F(j)  S R .  - N E(i)P(j)= +  - N"E(i)P(j),  BE(i)F(j) -  N~E(i)F(j)  ( A - N ) E ( i ) F ( j ) ss - (B-N~) E ( i ) F ( j ) . +  to  R .  68  Thus A  N  +  =  B - N~  Now l e t B - N  P Then f o r any  (B-N") E ( i ) F ( j )  PE(i) F(J) = (since  G  (B-(-NG )) E ( i ) P ( j ) n  i s the s p e c t r a l r e s o l u t i o n o f  =  =  0  Q  (then  x € r n g P, r n g A, r n g B ) ,  Q  '((I-G ) 0  Q  Bx,x) + (G Ax,x) Q  G A c AG , e t c . ) Q  Q  =  ( ( I - G ) B x , ( I - G ) x ) + (G Ax,G x)  =  (B(I-G )x,  Q  Q  0  This l a s t i s non-negative. P = A - N  +  0  0  Q  Hence =  B - N"  > 0  B = N" + P  A = N" + P 4  This proves  37.  Q  ( I - G ) x ) + (AG x, G x ) .  and  as r e q u i r e d .  n  ( B ( I - G ) x , x ) + (AG x,x)  a  (hy 36  N~=-NG )  [B(I-G ) + AG ] E ( i ) F ( j )  Thus i f x e r n g E ( i ) F ( j )  (Px,x)  Nj and  0  69  MEASURE THEOREMS 38. Let  R  Theorem.  [A c o v e r i n g t h e o r e m f o r s e m i - f i n i t e  h e any v o n Neumann a l g e b r a .  measure on  R  Let  m  be a  ( d e f i n i t i o n s 7 and 8 ) , a n d l e t  P  projection i n  R .  semi-finite  E  Then t h e r e e x i s t s a f a m i l y  P  measures].  he a  ^ E j /£ R  non-zero P  such  that (i)  the zaE  (ii)  ^ . E  Proof.  are mutually  < 0=  ±  (iii)  E^  ±  for a l l  E  orthogonal  ±  = E  Let  X  he t h e s e t o f a l l s u b s e t s  I t R '  such  that (i)  ©-members a r e m u t u a l l y i f F 6 B , then  (ii) X  i spartially  orthogonal  0 < P _< E  and  mP  ordered by s e t i n c l u s i o n ,  linearly  o r d e r e d X - s u b s e t h a s a n u p p e r bound i n  union).  Hence b y Z o r n ' s Lemma, X Since  < »  P P jB c R , sup jB e R  X  and e v e r y (namely, i t s  has a maximal element  B.  P E - sup B e R , b y 2 7 .  and  Now E - sup B = 0, I  P f o r otherwise,  since  m  i s semi-finite,  there i s  P € R  such t h a t  0 < P.< E - sup JB and But  mP < <=°. this  That  i s , t h e X-member  © U  c o n t r a d i c t s the maximal!ty o f  B.  properly contains Thus  JB.  TO  sup'j| = E and  jB  furnishes  39.  a family  Theorem.  of the required  If  R  sort.  QED.  i s any v o n Neumann a l g e b r a and  P E  . E e R , where  then mE  Proof.  n  f mE  By lemma 21, E  = sup  .  [ E J  Now. { n+1  " n  E  E  ;  n  €  } p  Is a family  o f " m u t u a l l y o r t h o g o n a l R -members.  {E^  E = sup = E  so  -  that mE :  ^0 m E  1  =  +  Q  \  0 + +  0  mE  E  + sup  Q  +  \  ra  \  (  I  <Vl (nl  . n  +  1  - E  - n) E  E  +  m E  n  " n)  E  n+l " ^ n )  l i m (mE  l i m mE  E  n  -  mE ) Q  n  • n  Hence  e*j  F i n a l l y , since  m  i s monotone,  mE  40.  mE  Q  <  .  Corollary. E  and  mE  f  n  If  E  n  then, mE., 4. mE n  Proof.  If (0  E  n  4- E, t h e n  " n ) ' ^0  E  E  "  E  )  so t h a t m(E hy  39.  0  - E ) * n  m(E  Q  72  EXTENSION OF  m  TO  R  S P  qp 4.  Definition.  s  S  i s a positive R  -operator,  P  -r  ^ I E  i=l „  then define  I f  Y P  mS -=  a  j_  j_  m E  i=l (which i s n o n - n e g a t i v e and f i n i t e 2  is finite  (& ^0)  or i n f i n i t e according  as  ..mE  ±  ±  or infinite). S  i s summable i f mS  < ». SP  If  S  i s an a r b i t r a r y R S  where  S ,S S  mS  S ,S" +  i s summable.  i s integrable, define mS = m S  (this I s well defined  are  -operators.  i s i n t e g r a b l e i f a t l e a s t one o f S  then  - S", .  +  are p o s i t i v e R  If  S  -operator,  +  -  mS"  s i n c e one o f  i s summable i f  |mS|  mS ,mS" +  i s finite).  < », t h a t i s , i f b o t h  S ,S~ +  summable. qq Let  R  gp b e t h e s e t o f a l l summable R  -operators.  gp Remarks. support  has f i n i t e  An R  -operator  measure.  i s summable i f a n d o n l y i f i t s  73  If  S € R  I s integrable,  S P  P mS =  a^ mE  .  i  i=l Henceforth  R  i s assumed t o he an a b e l i a n von Neumann  algebra.  42.  Theorem.  If S € R , S P  then i t s i n t e g r a b i l i t y i s  SP Independent o f i t s r e p r e s e n t a t i o n i n R SP If S i s i n t e g r a b l e i n R , then i t s i n t e g r a l i s SP independent o f i t s r e p r e s e n t a t i o n i n R . P r o o f . I t i s enough t o p r o v e t h e second a s s e r t i o n f o r SP positive R -operators. qp  Let  S,T e R S  where  p S = T a  = i  E  such t h a t T >_ 0, T = £ b  i  Note f i r s t t h a t s i n c e  so t h a t  mS, mT  j  .  are f i n i t e or i n f i n i t e simultaneously.  the a s s e r t i o n f o r t h e case t h a t mS, mT  mS - mT  j  S = T,  qq  If  F  =  S £ R  are f i n i t e a  i i m E  - \\  15  J" ^ j  This proves  74  £ j V  (since  i ~ Z^j  E  =  1  = ; ij If  m E F j > 0, t h e n s  o  a  l  s  3 1 1 ( 1  R  i  s  m  E  i  P  abelian)  ( a . - b .) mE.F. " " E^Fj  i  mEj-Pj ^ ®>  i " V j - £ j *J  a  0, so t h a t  a  ±  = b ^. Thus i f  o  -  l)j =  0,  and hence mS  - mT  =  0  This proves 4 2 .  43.  Theorem. S+T  for a l l real  € R  I f S,T e R  aS e R  S S  S S  a . Moreover m(S+T) = mS+mT  The  , then  second a s s e r t i o n about  and one o f mS,mT  Proof.  maS = amS.  (S+T) h o l d s I f  S,T a r e p o s i t i v e  i s infinite.  The a s s e r t i o n s about  aS a r e o b v i o u s , t h e o t h e r s  tedious. If  S  and T  are represented as i n 4 2 , then  S+T  V  =  (a  i + b j  )  Xfy.  since °  ^ F. 3  =  I.  P Note t h a t  E^Fj e R , since (a^+bj) ^ 0  Now ^ 0,  R  i s abelian.  o n l y i f &^ £ 0  mE^ < » and mE. F < mE. < °=. i 3 ~ I b„ ^ 0 i m p l i e s t h a t mE. P. < ».  or  b j ^ 0.  If  then  Similarly  Therefore  ^ ( a j + b ^ O ) mEj^Pj <.» and  S+T e  R . SS  By d e f i n i t i o n m(S+T)  =  Y  (a^bj)  = £ i^ a  mE^F^  i j E, d ^ -Va  n E  F  +  b  •- I i i ( I ty a  mE  £, * J  +  -  \  =  mS + mT  a  I ^  i  +  2j  m  F  b  J  j  m  El)  F  J  T h i s p r o v e s 43.  44. operations  Theorem. (R  R  SS  i s abelian!).  " i s c l o s e d under t h e  R  s  lattice  76  qq  Proof.  where  F  For  S,T  in R  (Sl/T)  =  TF  (SHT)  =  T(I-F ) + SF  Q  recall  + S(I-F ) Q  0  ,  Q  i s the s p e c t r a l r e s o l u t i o n of  Q  that  a  S-T.  Obviously  a l l of  gq  the terms on t h e r i g h t a r e i n R SS belong to R . QED.  , hence by kj>, (SUT)  and  (SOT)  qp  45. then  S _< T  Theorem.  If  S  and  T  are i n t e g r a b l e i n  R  Implies mS < mT  I f 0 j : S _< T, t h e n ( u s i n g t h e r e p r e s e n t a t i o n s  Proof. i n 42)  Thus  mS = 00 If  implies that mS < 00  mT = «>.  t h e n as b e f o r e  mT - mS = )  ±3  (b .-a. )mE.F..  L  Now i f  mE^Fj ^ 0, b  since  S _< T.  j  Thus  i t must be t h a t - a  i  E F j £ 0, ±  3  1  1  0  and  >^ 0,  mT - mS _> 0  and so  mS _< mT. SP If  S  and  T  integrable, S < T  are a r b i t r a r y R  -operators  which are  77  implies  thai  s  + ^ +  T" <_ S",  T  whence from above mS so  < mT  +  mT"  +  < mS~,  that mS = mS  -•mS"  +  n  - mT~ = mT,  I f jfT ^ c R  Theorem. T  +  45.  T h i s proves  46.  < mT  and  S S  n  0  ^  then mT  n  0  i  SP Remark.  T h i s theorem i s not t r u e i n  T Then n  T  n  l0  i f ml =  n  =  iI  (uniform),  hence a l s o  T ^0. n  Let M(n) T  n  =  I  b  i n i n ' F  1=1  Por  .  Por l e t  n = 1, 2, ...  co.  Proof.  R  any r e a l number E (n,c)"- I  (b  c, l e t  i Q  > c) P  i n  But  mT  n  = »  for a l l  78  E(n,c)  i s thus the p r o j e c t i o n onto the subspace where  T  n  > c.  Note t h a t E(n,c) 2 E(n,c+e) for  e > 0.  Consequently  E(n,0) > E(n,c) and mE(n, c) < Moreover,  since  T ^  oo  .  ,  n  E(n,c) 2  E(n+l,c).  Let C(n) = max {Vj_  n  Then  C(n) J,  since  T n  i  : f i x e d nj .  •  The p i ^ c e de r e s i s t a n c e o f t h i s p r o o f i s the f a c t t h a t f o r  c >0 lim To prove t h i s ,  m E(n,c)  suppose lim  Since  n  =0.  contrarily  that  m E(n,c) > 0.  mE(n,c) < oo and  E(n,c) ^  , 4o a p p l i e s and  l i m mE(n,c) = m ( l i m [ s t r o n g ] E ( n , c ) ) Let E = i n f [E(n,c) : fixed c j . Then  E € R  P  by 2 6 , and by 2 2 E = l i m (strong) E ( n , c ) .  Therefore mE = m l i m (strong) E(n,c) = l i m m E(n,c) > 0, so t h a t  E ^ 0.  Thus there i s non-zero  x  in  rng E.  Now  x e rng E(n,c) for a l l  n € uo.  Thus ( T x , x ) > c||xj| > 0 2  n  f o r a l l n, c o n t r a d i c t i n g ; the -fact ;that  T  R  f 0.  Hence  l i m m E(n,c) = 0 for  c > 0. To complete the p r o o f , f o r a l l non-negative i n t e g e r s E ( 0 0 ) _> E(n,c) 5  for  c > 0. T  n  =  E(0 0)  =  E(n,c) T  <  C(n) E(n,c) + c ( E ( 0 0 ) — E ( n , c ) )  <  G(0) E(n,c) + cE(0,0) .  Now,  then choose  3  n  n  + (E(0,0) - E ( n , c ) ) T  n  5  given  N  T  e > 0, choose  0 < c <  — — 2mE(0j0)  so t h a t  n > N  c ,  implies  so t h a t  [*]  80  mE(n,c) < e / 2 C ( 0 ) . Then from [*], n > N m T  implies  1  n  € / / 2  +  G / / 2  =  e  »  that i s , l i m mT By  =  n  0,  4 5 , t h i s convergence i s monotone. I n the next s e c t i o n , m  f o l l o w i n g theorem h i n t s how 47. s  n -  0  a  n  Theorem.  If  QED.  w i l l he extended t o  R .  The  t h i s w i l l he done. R  i s abelian, S e R , SP  £s j n  c  R , SP  d  s  f  n  S,  then mS  Proof;  Since 0  for a l l  mS.  n  < s  < S  n  n,  0  < mS„ < mS„,, < mS — n — n+i —  by 4 5 , and l i m mS  n  _< mS  Thus i f  l i m mS^ = °=, mS = *> . n For the converse o f t h i s , l e t E(n,c) 3  r e s p e c t i v e p r o j e c t i o n s on which  S  n  and  S  and  E(c)  be the  are g r e a t e r than  81  c >_ 0.  Note t h a t E(n,c) If  of  S  E(n+l,c) < E ( c )  c < max  (a)  S, the l a r g e s t o f the non-zero c o e f f i c i e n t s  (assume S^O!)  and ||x|| = 1,  x e rng E ( c ) then (Sx,x) > c and  since s  rr  n  .s,  there e x i s t s an i n t e g e r  Q(c,x)  such t h a t  n > Q  implies  (Sx,x) > (S x,x) > c n  so t h a t x e rng E(n,c) for  n _> Q, t h a t i s , u s i n g  (a)  E(n,c) f E ( c ) . By lemma 3 9 mE(n,c) T.mE(c) where  c < max Now  (b)  S. c _> 0  i f S  > E(n,c) S  n  n  > c E(n,c).  Thus mS for a l l  c > 0.  n  2  c  mE(n,'c)  (c)  82  If  0 < c < min S  (d)  then E(c) = E(0) and  so, i f mS = « and . mE(c) =' a ,  s u b j e c t t o ( d ) , then by (b) l i m mE(n, c) =  ooj  and by (c) lim m S = n Thus  oo.  mS = oo i f and o n l y i f l i m mS  theorem i s proven i n t h i s  n  = oo, and the  case.  The above a l s o shows t h a t mS < ce  lim m S  are s i m u l t a n e o u s l y t r u e .  Since n  S,  also  s - s ^o n  and by lemma 46 m(S-S ) ^ 0 . n  But m(S-S ) n  <  oo  I n t h i s case a l l the o p e r a t o r s i n v o l v e d  are i n R ^ ° .  s t  n  = mS  -  mS  n  83  by 4 3 .  Hence mS f  mS.  n  op  48. which  Lemma.  If  S  is a positive R  mS = + « , t h e n t h e r e e x i s t s S  n i  S  >  Proof.  S  n  £S f  '  f  m S  n  t  -operator for  c R  such t h a t  »  Let n  3  -I  a  i  i  E  1=1  Then a  ±  mS = »  i f and o n l y i f mE^ = »  f o r some  such t h a t  ^ 0 By  38  E. , J ( j where t h e  e J  )  are mutually  F  ,  j  orthogonal  and a r e o f f i n i t e measure  Choose a n e s t o f f i n i t e J - s u b s e t s P ( n ) : c P(n+1)  P(n)  for  i  n € ai  }  J  c  such t h a t Y (jeP(n)) m P j >-S i  .  a  Then t h e sequence S  [_^ j >  n "  n  a  i I(J  has t h e r e q u i r e d p r o p e r t i e s . T h i s p r o v e s 48.  e P  ( )) n  F  J>  8k  k9.  Independence Theorem.  If  £s j  and  £T | n  are  SP sequences i n  R  such that S t S n  then  T < S  T  n  f  T, *  implies lim mT  < lim mS„ . n — n  By 20  Proof.  (  ns j  T m  m  v  t (T n s) ,  n<  n  v  m  '  and since S > T > T  ,  n  then T H S = T m m so that (T  m  s) f  n  n  T.  m  and by 4? m  ( m T  n  S  n)  ?n  Thus f o r a r b i t r a r y R(m,<37 )  m T  n •  m e co  such that i f n > N Dtf < M(T  m  ns ) . n  But  (T v  ns j <s , n  m  n' —  n  3  so  3?7< m(T  m  0 S ) < mS n  n  and Df?'< mT , m  there e x i s t s  85  for  n > If. Thus Off  < l i m mS  f o r a l l Off <. mT ,  n  hence  m  mT m —< l i m mS„n . Since  m  i s arbitrary l i m mT m —< l i m mS n . T h i s proves 4 9 .  Remark.  I f , as above,  s  n  f  S  T  T.  then l i m mS„n = l i m mT n.  )  s  86 EXTENDING  m  R  TO  S  CJ  50.  Definition.  be a p o s i t i v e R - o p e r a t o r ,  Let T  -^T^J c R  By 2 4 t h e r e I s a sequence  S P  such t h a t  n Define mT a' l i m mT . (Theorem 49 shows t h a t sequence  £T j  mT  does n o t depend on t h e c h o i c e o f  Theorem 47 shows t h a t t h i s i s a c o n s i s t e n t  .  e x t e n s i o n o f d e f i n i t i o n s a l r e a d y made.) T  i s summable i f mT < <» .  51.  Lemma.  i s a p o s i t i v e R - o p e r a t o r f o r which  If S  mS3 - +<*>, t h e n t h e r e e x i s t s n —  n'  « R  [s ] n  such t h a t n  3  T  nJ  —  R  such  that n  3  and by d e f i n i t i o n mT Either T  ( c R  i R . S S  N  t mS ==  n  sequence  Then  oo  .  and t h e theorem i s p r o v e n , o r t h e r e T  n  £ R  y£Lv c R \  S S  By 48 t h e r e e x i s t s a  f o r a l l n _> N.  such t h a t < T  N  T h i s p r o v e s 51.  < S,  S f , n  exists  mS t n  ».  s 5«  Corollary.  2  there e x i s t s  If T  £" j- £ 0 < T — n —< T T  r S S  s  u  c  that  n  n  l i m mT  n  i s a positive  R  operator,  then  Tn t  = mT, g  55•  Theorem.  If  S  and  T  are  R  operators f o r which  0 < T < S, then  mT _< mS.  Hence i f S  i s summable, so i s  Proof. in  -R j  T.  By 24 t h e r e e x i s t  ^"s ^ n  and ^ * T j , sequences n  such t h a t Sn t S  Tn f T  Theorem 53 t h u s f o l l o w s i m m e d i a t e l y  54.  Theorem.  G m fH s B  X =  [K  i s a p o s i t i v e R - o p e r a t o r and  H € R , 0 _< H _< TJ S  (n : H  =  If T  G  R , S P  : H e R , SS  from 49.  0 ^ H < T j 0 _< H < T  then mT =  sup  £mH : H e  G]  =  sup  £mH : H e  BJ  -  sup £mH : H €•  ,  j  88  Proof.  By 5 2 , x ^ 0.  X  c. 8 c  Obviously  G.  Thus by 53 mT 2 sup £ m H : H e  GJ  2 sup £ m H : H € Bj . 2 sup I mH : H e 2 l i m mT c  = mT,  SS  7  c R  where  n  Xj  i s t h e sequence whose e x i s t e n c e i s guaranteed  by 5 2 . T h i s p r o v e s 54.  s  The e x t e n s i o n o f  m  to R -operators of a r b i t r a r y sign  i s f o r m a l l y t h e same as t h e analogous e x t e n s i o n i n d e f i n i t i o n 41.  Without f u r t h e r ado t h e terms " i n t e g r a b l e " and "summable" S w i l l be a p p l i e d t o R - o p e r a t o r s . The f o l l o w i n g i s a lemma toward a p r o o f t h a t m i s l i n e a r S on t h e summable R - o p e r a t o r s . g  55. (a)  Lemma. L e t A and m(A+B) = mA + mB  (b)  I f one o f  A,B  i n t e g r a b l e and Hence i f A (A-B) .  and  B  B  be p o s i t i v e R - o p e r a t o r s .  i s summable t h e n  (A-B) I s  m(A-B) = mA - mB. a r e summable, so a r e  (A+B)  and  89  Proof. l ii> k  f n? B  i  n  Since SP  R  are positive,  there exist"sequences  such t h a t  A r n  A,B  B TB  A  n  and mA = l i m mA„ n Now  mB = l i m mB . n  (Aj, + B ) t (A + B ) , n  and s i n c e  m(A+B)  i s independent  a s s e r t i o n (a) now f o l l o w s m(A+B)  o f t h e sequence  (A +B ), the n  n  easily:  =  l i m m(A + B )  =  l i m (mA + mB )  n  n  n  n  (by 45) =  I i i  =  mA  4-  + l i m mB  n  mB,  so t h a t (a) i s p r o v e n . To prove  ( h ) , note t h a t by 25  A = (A-B) for  +  4- P  B =a (A-B) ~ 4- P  > 0  B > (A-B)" > 0  P > Oj whence A _> ( A - B )  +  A > P > 0.  B > P VO,  so t h a t by 53, s i n c e one o f A,B  i s summable, so i s P  i s one o f  (A-B) i s i n t e g r a b l e .  ( A - B ) , (A-B)". +  Thus  arid so  90  Now mA  -  mB  a  m[(A-B) +P] -m[(A-B)~+P] +  •a' m(A-B) +mP - m(A-B)"-mP +  (by part (a)) =  m(A-B) - m(A-B)~  =  m(A-B).  +  This proves 5 5 .  56. (S+T)  Theorem. I f S  and T  are summable operators, then  i s summable and m(S+T) = mS + mT  Proof.  If S  and T  are summable, then so are  S , S", T , T" . +  +  Since these are postive S  +  + T  S" + T~  +  are summable by 5 5 (a). Thus  (S+T) can be expressed as the  difference of two p o s i t i v e summable operators: S+T  By 5 5 (*>), + S  =  ( S - S") + ( T - T~)  a  ( S + T ) - (S~ + T")'.  T  m(S+T)  +  +  +  i  +  thus also summable and  s  a  m(S + T ) - m(S" + T") +  +  91  = mS  + mT  +  - (mS" +  +  mT")  (by 54 (a)') = (mS  +  - mS") + (mT  -  +  mT")  = mS + mT. This proves  56. g  57. and  a  Theorem.  If  S  i s an i n t e g r a b l e R - o p e r a t o r  i s any r e a l number, t h e n  m(aS)  •58.  =  Theorem.  aS  i s a l s o i n t e g r a b l e and  a mS  If  S  and  T  a r e summable, and  S >^ / T  then mS _> mT Proof.  The s t a n d a r d argument.  Since  S-T _> 0,  m(S-T) j> 0, hence mS s> m(S-T) + mT _> mT.  59. If  Lebesgue's Monotone Convergence Theorem. {T^  i s a sequence o f p o s i t i v e summable  such t h a t  T t  T,  n  then mT  s=  l i m mT . n  R -operators S  Proof.  F o r each  n  there e x i s t s  £S^j  c  R  S S  such t h a t n  m  n  and lir»  mS ™ = mT < *>. n n  Let U  where t h e 44.  sup  m  -  S U  P c^q*  :  * < j> m  i s t a k e n i n t h e sense, o f 1 J . Then  U e R n  SS  Moreover n  '  for  V l  -  S U  Pf q S  >_ sup £s (since  > S -  m+1  m q  :  * <  m + 1  J  : q _< m+lj  ....  2  sup i^Sq  =  U . m  111  : q <mj  .  Now S q„ for  q_< m.  m  —<; U m —< T m m  Hence ( s t r o n g l i m i t s )  lirrim Sn„  m  = T n —< l i m U m .—< l i m T m = T  so t h a t T n —< l i m U m —< T f o i * a l l n,  or  (*) v7  93  T = l i m Tn —< l i m U_m — < T so t h a t T = lim U m T. S i n c e n", m €  Um t  and hence  mT = l i m mU Now hy  q < m.  S S  (**)  R  (*) m S„ q  for  R  < mU  m  —  < mT m — m  Hence m  q  m —  q—  n  and mT  q  —  < l i m mU  < l i m mT n — n  whence mT = l i m mU n = l i m mT n hy ( * * ) . Thus mT = l i m mT . n T h i s p r o v e s 59 • The f o l l o w i n g theorem i s an a n a l o g o f t h e d e f i n i t i o n f ( x ) dx = l i m / f ( x ) dx f o r t h e Riemann i n t e g r a l .  60.  G_  Theorem.  If T  i s any i n t e g r a b l e R - o p e r a t o r and P i s any r e s o l u t i o n o f t h e i n d e n t i t y i n R , t h e n  94  mT = l i m ( a  =>) mT G SI  Proof.  F i r s t o b s e r v e t h a t i f T > 0, t h e n t h e l i m i t must  e x i s t and mT > l i m ( a -* °°) mT  2 Hm If  T > 0  mT  there e x i s t s  G  ^T ^ R  Now  T nG n e R  o p  c R  such t h a t T-n f T  t mT i  f o r a l l n. n  Now f o r any  n  Q  n  0 —< Tn —< T mT  G  Moreover  n  x € H  !! ( T G n  n  - T)x||  < ll(T G n  - TG )x!l + ||(TG - T)x|l  n  n  < l l f f j l II ( T =  II(T  n  n  - T)x|| + llTll |](I •- G ) x | n  - T)x|||.+ ||T|| ||(I - G ) x | j ,  n  n  whence T n Gn T  T, 3  since T„n t T Since  mT  G nt  l  .  i s i n d e p e n d e n t o f t h e sequence mT G T n  n  mT,  95  , Wow  the fact mT  and  n  that  G_ < mT G„ n — n  (*)• p r o v e r . t h e t h e o r e m f o r T _> 0. q For  arbitrary  mT  =  mT  -  lim  +  integrable  T  i n R ,  - mT" (a-^oo)  mT G  - l i m (a->•<») mT~G  +  Q  cl  lim  =  l i m ( a ^ » ) ( m ( T - T")  (a-vco)  (mT G  cl  =  +  cL +  - mT'Gj St  G j Cb  \  =  l i m (a->») mT G  This proves  cl  60.  96  MEASURE AND POSITIVE  If  then  N S NE„ e R a  R -OPERATORS W  i s a p o s i t i v e R -operator with resolution N  for a l l  .E ,  a. Moreover  NE„ < NE. a — h for  a < b, so t h a t  mNE„  i s an i n c r e a s i n g f u n c t i o n o f  a.  This  j u s t i f i e s the f o l l o w i n g .  61.  Definition.  If N  i spositive i n  R, W  mN = l i r a (a-*®) mNE N  i s "summable i f mN < <*>.  62.  Theorem. • r  If N  i s a p o s i t i v e R - o p e r a t o r and  i s any r e s o l u t i o n o f t h e i d e n t i t y I n mN = l i m (a-»») mN G  Proof. — — show t h a t  By 36,  llm(a->co)  NG  e R  a Q  N  cl  P R , then .  for a l l  a.  The p r o o f must  mNG  .a e x i s t s and e q u a l s Let  mN.  E„ be t h e r e s o l u t i o n o f a [ H ( a ) : — < c < »J  N  c  be t h e r e s o l u t i o n d f : the R ^ - o p e r a t o r Then NG  a  H (a) e R c  S  NG . &  and l e t  G a  for a i l  a  and c, and f o r f i x e d  a, mNG  H (a)  i s an  i n c r e a s i n g f u n c t i o n o f c, mNG  = l i m ( c ^ » ) mNG  ct  ct  H fa) C  so t h a t mNG  > m NG„ H (a)  <3» —  (a)  C  ct  .  f o r a l l c. Furthermore, f o r a l l h<; and c, NG H„(a) E, e R a c ' b S  v  and  NG  H ( a ) > NG H j a ) E ^  a  c  a  so t h a t mNG  H ( a ) > mNG H ( a ) B^.'  a  c  a  c  Combining t h i s w i t h (a) mNG  > mNG H ( a ) E  a  for a l l b  a  c  (b)  f e  and c.  Now NG NE  fe  G  H fa) E, = NE, c b b  a  x /  G H„(a), a c ' v  /  e R , and by 60 S  a  mNE^  G  a  = l i m (c->») mNE^ G  a  H (a) c  Hence by (b) mNG  a  ci  > mNE, G D  (c)  ct  for a l l b. To o b t a i n a f u r t h e r i n e q u a l i t y on mNG . note t h a t ct  m  in R  for a l l  for a l l  a  b  V*' and  E  b  <  c.  ^b  A g a i n by 60, s i n c e NG  a  c  a  _< lim(b-^») mNE^ c.  c  E^  = mN  Thus  mNG  Q  for a l l  H (a) e R  c, raNG H ( a ) = lim(b-»») mNG H ^ a )  for a l l  &  = llra(c^oo)  mNG  H„ (a)  < mN  (d)  a. I t remains t o apply 60 once more:  mN  =» l i m ( b ^ e o ) mNE^  = l i m ( b l i m ' a - ^ o o )  T h i s means t h a t g i v e n 377 < mN, mN > mNE^, Por  there e x i s t  G,  a > a' mNE, . G„ > mNE, , G , , b a — b a' ' . 1  1  o  and by (c) mNG„ > mNE, . G a — D a  .  1  Thus f o r  a > a', (e) becomes  mN _> mNG Since 077  >_  m W E  h « a - "^b' G  G  a'  y  ^  *  I s a r b i t r a r y , t h i s Implies that lim  (a-*  co)  mNG ct  e x i s t s , and mN  = l i m (a-*<») mNG_  . ct  T h i s proves 62.  G  a',b»  &  .  such t h a t  (e)  > ^7/  &  mNE^  99  ^3.  Theorem.  If A  and  B  are p o s i t i v e R  -operators  m(A + B) = mA + mB  Proof. of  A  and  B.  Let  E , F a  he t h e r e s p e c t i v e s p e c t r a l r e s o l u t i o n s  &  Then G  = E F a a  a a  P i s a r e s o l u t i o n of the i d e n t i t y i n R  (hy 32).  Now A + B  so t h a t (A '+ B)  3  A + B  r> (A + B) G = A G + BG . a "— a a a  G-  0  a  Since the right-hand side i s i n R , in in  H R  f o r a l l a, hence (A + B) G S  a  AG^ + BG a  and m(A + B) G  Letting  i t i s d e f i n e d everywhere  a  tend t o  00  = mAG  + mBG  .  and a p p l y i n g 62 y i e l d s  m(A + B) = mA + mB . T h i s p r o v e s 63.  100  MEASURE AID R -0PERATORS OP ARBITRARY SIGN N  Definition. summable, t h e n  H  If N c R  N  and one o f  N , N"  is  B  are p o s i t i v e R^-operators,  +  i s i n t e g r a b l e and  mN « mN " - mST 4  ®  summable i f |mN|  i s  ^5.  Theorem.  If A  one o f which i s summable, t h e n  < »  and  (A - B)  i s i n t e g r a b l e and  m(A - B) = mA - mB  Proof.  N ' a A - B, then by' JJ t h e r e i s a p o s i t i v e  Let  N R -operator  P  x  By  such t h a t A = N + P  63,  mA = mN mB And  B = N" + P .  +  since  a  + mP  ml" +  N , N", p  mP.  are p o s i t i v e  +  raA _>  +  m  >.  P  mA > mN so t h a t , s i n c e one o f  mB > mP > 0  0  +  > 0  A,B  mB >m~ I s summable,  I s summable. Thus  N mN  i s i n t e g r a b l e and a  mN  +  - mN"  ,> 0, P  and one. o f  N ,N" +  = mlf " + mP - mN~ - mP 1  (since  mP  is  finite) = ra(N + P) - m(N~ + P) +  (by  63) =a  mA - mB .  T h i s p r o v e s 65.  102  ABSOLUTE CONTINUITY L e t now Let  m  and 8) such t h a t (definition 9).  R  be an a b e l i a n von Neumann a l g e b r a .  and  be s e r a i - f i n i t e measures ( d e f i n i t i o n s 7  n  n  i s a b s o l u t e l y continuous  with respect to  The remainder o f t h i s opus w i l l be taken up w i t h  p r o v i n g a Radon-Nikodym theorem f o r t h i s s i t u a t i o n . N t h e r e e x i s t s an o p e r a t o r  N  In  R  nT = m(N for a l l T  P e G(a)  In  exists  R. N  P o r any r e a l number  67.  Definition.  Let  whenever  0 < P < E  Lemma. P  If  F  E  E e G(a) . and  P e  E  i s a-good i f  R. P  i s a non-zero G(a)-member, t h e n t h e r e  such t h a t 0  and  « T)  Definition.  F € R  Approximately:  such t h a t  66.  68.  m  < P < E  i s a-good.  Proof.  Let  X  be t h e s e t o f a l l f a m i l i e s  that (i)  B-members a r e m u t u a l l y  orthogonal  IB er R',P  such  (ii)  i f K e ft, t h e n 0 < K < E  and a i K < nK. X  i s p a r t i a l l y o r d e r e d by s e t i n c l u s i o n .  every nest i n X  has an upper bound i n X.  Thus  Obviously X  Contains  P  a maximal element ft c R . Now a m sup JB=  a m ^ (Keft) K P  ( s i n c e j|-members a r e m u t u a l l y o r t h o g o n a l and sup" B e R  <  = n sup ft that i s , a m sup B <  n sup ,B  Now sup ft < since  E,  E e G(a), therefore, sup B  < E.  The non-zero R -member F = E - sup ft  by 27)  104  has t h e d e s i r e d p r o p e r t i e s . 0 < K <: F  F o r i f KcRP j  and  a mK < nK,  then £ X and  B.  c o n t r a d i c t i n g the maximality o f T h i s p r o v e s 68. The s e t  G(a)  i s non-empty f o r a l l r e a l  at l e a s t the zero p r o j e c t i o n ) .  a  ( I t contains  Thus t h e f o l l o w i n g d e f i n t i o n i s  justified. 69.  Definition.  (& c R  i s a maximal f a m i l y o f m u t u a l l y  o r t h o g o n a l a-good p r o j e c t i o n s .  70.  71.  Lemma.  sup  e G(a)  i s a-good.  105  Proof.  Let  P be an R -member such t h a t  P  0 < F < sup^ . 0  T/O show  F s G(a), amP  (where  (since  FE  =  a m (P s u p ^  »  a  =  a m  m  Y  (F  (  E  ^ ( ? )  ( ^f) E  Y  i s a projection^ since  PE _< E  )  0  F  Y ( C ? ) am FE  2  Y  and  i s abelian)  E €  E =  )  E  R  =  E  ( <f) Ee  ^  n  i s a-good) n  Y  (  E E  <?)  F  E  = n (F s u p ^ ) =» nF Thus  72. Since  sup (@  i s a-good.  Definition. G(a)  By 27, 70 and 71, E  E  = sup^° .  a  i s n e v e r empty, E e R, E  e G ( a ) , and  P  Q  CU  CL  Lemma.  If  F € R  and  0 < F < I-E  —  then  E  i s a-good. Cv  P  73.  exists f o r a l l real  F £ G(a'), t h a t i s , amP < nF„  a o  a.  106  Suppose on t h e c o n t r a r y t h a t . F € G ( a ) . By 68,  Proof. P  there I s  F  1  e R  such t h a t  0 < F  F'  i s a-good and  < F.  T  Now  0 < F» —< I - Ea o  so t h a t  F» i s o r t h o g o n a l t o every  E  in^ * . 2  T h i s means t h a t  i s a f a m i l y o f m u t u a l l y o r t h o g o n a l a-good elements w h i c h p r o p e r l y contains and  t h e maximal f a m i l y ( @  , a contradiction.  Thus  F' / G(a)  73 i s proven.  74.  Summary 1  of the p r o p e r t i e s o f the p r o j e c t i o n  1  F o r every r e a l number . a  there e x i s t s a p r o j e c t i o n  such t h a t (I) (ii)  E  e R  a  If  P  F e R  0 < F < E .  and  P  *~~  amF 2 (iii)  If  F e R  n  and  P  —*~  CL  F  0 < F < I-E -  amF < nF (iv)  Proof Since  R  E  CL  P i s t h e u n i q u e such R - p r o j e c t i o n .  of ( i v ) : Let  E€R  P  s a t i s f y ( i i ) and ( i i i )  i s abelian, P = E(I-E ) a  Q = E (I-E) a  E . CL  E  Q  CL  107 P  are p r o j e c t i o n s i n R . I f E / E . a t l e a s t one o f cl  P,Q  is  non-zero. If  P ^ 0, t h e n s i n c e  P _<'E, ( I i ) i m p l i e s t h a t  amP 2 nP. But s i n c e  P < I-E . ( i i i ) i m p l i e s t h a t amP < nP A simitar contradiction arises i f  Q £ 0.  T h i s proves  (iv).  The n e x t few lemmas w i l l show t h a t [ \  -  1  <- j  <a  P  i s almost a r e s o l u t i o n o f the i d e n t i t y i n R . 75.  Lemma. 1  If  a < h, t h e n  Proof.  E  • "  E  o  i s a monotone i n c r e a s i n g f u n c t i o n o f  CL  < E. .  Since  R  i s ahelian  Va'  =  W  so i t .remains t o show E.E^ . E.. a D a Now P =E_ - E J ^ - E j l - E ^ a a  > 0  I f • P > 0, t h e n 0 < P —< Ea• and 0  <  p  <  x  ~\>  a:  108  so t h a t hy (11) and ( i l l ) o f 74 araP _> nP  bmP  < nP  or bmP < nP _< amP, w h i c h i s a c o n t r a d i c t i o n s i n c e a _< h.  76.  P = 0  and 75 i s p r o v e n .  Lemma. l i m ( s t r o n g ) (a-* +«) E  Proof.  =1  Let  E = sup [E t h e n a l s o by  Thus  : a real]  75, E =» sup |Ep : pcuoj  75,  A g a i n by  so t h a t by 22, E = l i m ( s t r o n g ) EL.  ir and  similarly E = l i m (strong)  where  £ (p)J a  1 8  a n v  V^p)  r e a l sequence such t h a t  E =• l i m ( s t r o n g )  (a-^eo)  a(p)"t».  Thus  E Cb  The second p a r t o f t h e p r o o f i s t o show t h a t If  I-E £ 0, t h e n , s i n c e  n  E = I.  i s a s e m i - f i n i t e measure,  109  there  exists  P € R  P  such  0 < P<  that  I-E  nP  <  eo.  Now  0 < P < I-E  < —  —  for  a l l real  a.  T h u s hy  I-E„ a  ( i i i ) of  a mP for a l l real  a.  mP  n  => 0  implies  nP  Thus the  =» 0 .  assumption  0.  >  continuous with respect  nP I-E  to  m,  so  that  T h i s means t h a t  =0  nP  proves  nP  i s absolutely  =* 0  nP  Thus  mP But  <  74  ^ 0  has  >  0. lead  to a contradiction.  This  76.  77.  Lemma. lim  Proof.  E_  i s strongly  right  continuous:  cl  11  (strong)  (b.-*a+) E.  This proof  i s exactly  =  E  like  that of  76.  Let E Then, s i n c e  E^  =5  £a(p)£  [\  : b  > aj €  R .' P  i s a monotone f u n c t i o n o f E >  where  Inf  inf  i s any  f E  a  real  (  p  )  :  b,  peu)|  sequence such t h a t  a(p)^a.  By  22  110  E  a(p)  ^  E  f o r a l l such sequences, hence E = l i m ( s t r o n g ) (b-> a+) E^ Obviously  E > E . — a  If  P = E-E J ft  0,  then 0 < P < I-E — a o  and 0 < for  b > a.  p  <  \  Hence by 7 4 amP < nP  and bmP 2. nP for  b > a, so t h a t amP 2 nP  also.  This i s a contradiction. Hence If  P = E-E  ^E  a  o  =0  and 77 i s p r o v e n .  : - oo < a < e o j i s a r e s o l u t i o n o f t h e i d e n t i t y  then l i m (strong)  (a-*-co)  E  =0. ct  As t h i n g s now s t a n d , t h i s i s n o t t r u e : Then  amE > nE  l e t E e G(a)  f o r a < 0.  Ill  or 0  amE > nE. _> 0  whence amE = nE = 0 , or,  since  a < 0, mE == nE =* 0. Thus hy 77 lim  where  F  ( s t r o n g ) (a-> -®) E  i s a p r o j e c t i o n o f m-measure z e r o . The troublesome  by d e f i n i n g ." " 1 liri  = P  111  E  = 0  p o s s i b i l i t y that  f o r a < 0.  Fj^b can be e l i m i n a t e d  This preserves the r i g h t  cont-  ct  i n u i t y and m o n o t o n i c i t y o f  E.  So now  a  ct  1 11  E  ct  i s a resolution of  the i d e n t i t y . N  78. w h i c h has E  79.  a  D e f i n i t i o n . N i s the unique p o s i t i v e R -operator as a s p e c t r a l r e s o l u t i o n ,  Theorem. nE =  r  If E € R  P  and  mE < »  3  then  a d(mE E ) &  Jo  a,  Proof. •—  Since  mEE„ a  i s a f i n i t e monotone f u n c t i o n o f  s(Q) = / a d(mEE ) a  JQ  e x i s t s and i s f i n i t e f o r a l l i n t e g e r s  Q > 0.  Moreover, s i n c e  112  s(Q) CO  1S  oo  a d(mEE ) a  '0 exists,  ^0  and, g i v e n  such t h a t  0 there e x i s t s a  Q  sueh t h a t CD _  0 P a r t i t i o n the i n t e r v a l  0 = a  [0,Q]  < a^ < ... < a  Q  p  =» Q.  Let E(i) = E a  i  - E i - l  a  and form the c o r r e s p o n d i n g upper and lower sums: P U = Y i=l  a  i  a  i-l  m  E  E  (l)  P L  =  Y, i=l  m  E  E  ( ) i  Since s(Q) = sup [lj the  = i n f £uj ,  p a r t i t i o n can he made so t h a t oo  077 <  L _< S(Q) <u <  y  113  Now a ^  mEE(i) < n E E ( i ) _< a, mEE(I) ±  Thus  D?y< L  = <  Y  =  n E(E  (since  nE  Q  i _ i  a  m  E  (i)  E  nEE(i)  1  n E E  -  Q  E ) p  Q  = 0) _<  ^  a  mEE(i)  ±  CO  That i s , g i v e n  Q/J'<  07/  j  there e x i s t s  CO  < nEE . < Q  Since EE hy  Q  f E,  39  n E E t nE, Q  so t h a t 00  nE =  /  a d(nEE )  T h i s p r o v e s 79.  a  Q  such t h a t  114  To g e t on w i t h t h e t r u e b u s i n e s s , r e c a l l t h a t p o s i t i v e R -operator with spectral r e s o l u t i o n  E  o  N  i s the  ( d e f i n t i o n 78),  Let E(a,b) = E - E b  for  &  a < b.  Then N(a,b) = NE(a,b) e R  S  N T e R  and f o r any  T(a,b) = T • E(a,b) = TE(a,b) e R by  N  36.  80.  Lemma•  If  T  i s a p o s i t i v e R^-operator,  then f o r  a < b amT(a,b) _< mT  P r obo —f .  Let  o H(a b) _< bmT(a,b) . i)  T  have t h e r e s o l u t i o n  Fc  p i n R ,• t h e n  T c = TF„ c S i s a p o s i t i v e R - o p e r a t o r f o r a l l c. T  0  N(a,b) = T N(a,b) = N(a,b)T  I f now (T  x € H c  - ( N  a j b  =  (T  1/? '  Hence  and  c  )  )  X j X  1/2  (  S e x i s t s i n R , since (T  T l / 2  c  € R . S  i s fixed  1/2  H  (  a  }  X ) X )  q i s postive i n R ) H ( b ) x,  (N(a,b) T  a >  1 / 2 c  x,  T/^x) T ^ x )  3  Now a(E(a,b) T ^ x , < (N(a,h) T  TV^)  l / 2 c  < b(E(a,b). T_  x,  l/2  T  l / 2 c  x, T  l / 2  x)  x)  Thus aT (a,h) < T c  in  . N ( a , b ) < b' T ( a , b )  c  c  R. S  If  a < b < 0, t h e n a l l t h e above terms a r e z e r o ,  If  a _< 0 < b , t h e n 0  In R . S  < T  • N(a,b) < b  T„(a,b)  Hence by 53 mT  o N(a,b) < mbT (a,b) = b m T ( a b )  c  c  c  5  If 0 < a < b 0  < a T„(a,b) < T  0  N(a b) <  T„(a,b)  5  S in R  and a g a i n by 5~}  3  maT (a,b) _< mT c  Hence, y -. • •  -_ ;  c  ,••  c  and  h o l d s I n t h e l i m i t as  c  ,- ,.  amT (a,b) < mT holds f o r a l l  o N(a,b) < mbT (a,b) . \vi,Lcy.\:-\r  • N(a,b) <. bmT a < b.  ••-  (a,b).  In particular  this  c-*•+=>. Thus by 62, s i n c e  amT (a,b) _< mT T h i s p r o v e s 80.  o N(a,b) _< bmT(a,b) .  inequality T  = TP ,  8l.  Lemma.  If T  i s a p o s i t i v e R^-operator, then f o r  a <b amT(a,b) _< nT(a,b) < bmT(a,b)  Proof. the number  Let T  he as i n t h e p r e c e e d i n g lemma. F i x  c. -^S ^ G R  Wow t h e r e e x i s t s a sequence p > °  S  S  p  p * T (a,b) c  and by d e f i n i t i o n nS .f  mT  n  (a,b)  nS„ f  nT (a,b)  Let q  - -I 'P li=l  S  a  i  i  G  be an a r b i t r a r y member o f t h i s sequence. S  p  .« S E(a,to) p  and bmS  p  == b m E(a,b) ^  . =» •a  — >  b m b  j[_  a  i  a  a  i  a £\ a m S P  n  G  i  raG  E  G  ±  i E(a,b)  G  i  E(a,b)  ( > ) a  b  n  s  a^ mG^ E(a,b)  p  Then  S P  such t h a t  117  so  that amS„ P  < nS„ < bmS^ . P— P  P a s s i n g t o the l i m i t i n  p,  a m T (a,b) < n T„(a,b) < h mT for a l l  c  and  a < b.  fa,b)  Employing 62 again and l e t t i n g e->+»  a m T(a,b) < n T(a,b) _< h m T(a,b) for  a < b. T h i s proves 8 l .  N  n  82.  Lemma. mT  for  If  T  R  o p e r a t o r , then  «. N(a,b) = nT(a,b)  a < b. Proof.  Let  P  denote a p a r t i t i o n o f the i n t e r v a l <a.  P : a => a. <a, < o l Por  i s a positive  p-1  = b  iep l e t E  ±  =»E(a ,  Prom 80 and 8 l  for a l l iep.  i  a  i + 1  )  N, = W(a.  [a,b]  118  Examining upper and lower sums, L(P) = V T.ep  < \ Now i f  i mTEj mTE a, ±  i  ^  m T * U  ^  n TE  a  ±  i i  m T • N(a,b)  =  ±  m T E  +  •»  n T(a,b)  i = < )u  p  m T(a,b) < «, then U(P) - L ( P ) = ( a  ±  +  1  - a ) m ±  _< (max P) J\ m T E  (where  max P =3 max  "  a  TE  ±  1  i  ^  :  e  p  J )  = (max P) m T(a,b) Thus (max P) mT(a,b) > U(P) ~ L ( P ) > jmT • N(a,b) - nT(a,b)| If  mT(a,b) = 0, the lemma i s proven.  given  e > 0, t h e r e e x i s t s a p a r t i t i o n max P < e/mT(a,b)  so t h a t |mT  • N(a,b) - nT(a,b)| < €  I f mT(a,b) > 0, then, P  such t h a t  119  for arbitrary  e > 0.  Hence  o N(a,b) = nT(a,b)  mT  I n the case t h a t mT(a,b) = +», the a s s e r t i o n i s obvious from 80 and 8 l . T h i s proves 82.  83.  Theorem.  If  T  i s a p o s i t i v e R^-operator  mT. • N = nT  Proof.  Prom 82,  mT  • N(0,a) = nT(0,a)  or m(T By 62, l e t t i n g  84.  o  N)E  ,= nT • E„  n  cl  cl  a->+ «>, 83 i s proven.  Theorem.  If  T e R  K  i s n - i n t e g r a b l e , then  i s m- i n t e g r a b l e and mT  Proof.  If  T  nT = n T  • "N = nT  i s n-integrable,  then  - nT"  +  where a t l e a s t one o f nT  +  (by 83) I s f i n i t e .  = mT  +  0  ET  nT" = mT"  0  ,  N  T • N  120  How (T • H)  ( T - T") • N +  s=  = (T where one o f T oN  (T  +  • N ) - (T" > N)  +  • N ) , '(T" o H)  i s m-integrable and m(T •• N)  =a  m(T  = nT  +  +  i s m-summable.  T h e r e f o r e , hy 6 5 ,  ;  • N ) - m(T"  N)  - nT" = nT  T h i s p r o v e s 84. Theorem 84 i s t h e p r o m i s e d Radon-Nikodym theorem.  

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