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Chapter 6 Decomposition of a Hubert Space into a Direct Integral
CHAPTER 6 .
1.
D E C O M P O S I T I O N OF A H I L B E R T S P A C E INTO A DIRECT INTEGRAL
Posing the problem.
A s i n c h a p t e r s 1, 2 , and 3 , given:
1' a compact m e t r i s a b l e s p a c e Z; 2' a p o s i t i v e measure v on 3' a v-measurable over Z ,
z
of s u p p o r t Z;
f i e l d 5 + f f ( < ) of non-zero H i l b e r t s p a c e s
w e can c o n s t r u c t c a n o n i c a l l y : lo t h e s e p a r a b l e H i l b e r t space ff
=
2' t h e a b e l i a n von Neumann a l g e b r a tors;
I'B
ff ( < ) d V ( < )
z
;
of d i a g o n a l i s a b l e opera-
3' more p r e c i s e l y , t h e C*-algebra Y of c o n t i n u o u s l y d i a g o n a l i sable o p e r a t o r s , whose weak c l o s u r e i s
z.
W e a r e now going t o show t h a t t h e o r d e r of t h e s e c o n s t r u c t i o n s can be r e v e r s e d , i n an e s s e n t i a l l y unique way. T h i s w i l l g r e a t l y add t o t h e importance of t h e p r e c e d i n g c h a p t e r s .
2.
Existence theorems.
THEOREM 1. Let ff be a separable complex H i l b e r t space, y an abelian c*-algebra of operators i n H, z t h e spectrum of y, and v a b a s i c measure on Z . Suppose t h a t I i s in t h e weak closure of y. Then, there e x i s t s a v-measurable f i e l d 5 + H ( < ) of non-zero coq?lex lif'lbert spaces over Z , and an isomorphism of H onto
H ( < ) d v ( < ) ,which transforms t h e Gelfand isomorphism into the canonical isomorphism of & ( Z ) onto the algebra of continuously diagonalisable operators. Proof. ( i ) o o W e w i l l d e n o t e by f -+ T t h e weakly c o n t i n u o u s i s o morphism of L C ( Z , V) o n t o t h e weak cltfssure Z of y which e x t e n d s t h e Gelfand isomorphism ( p a r t I , c h a p t e r 7 , p r o p o s i t i o n 1 ) ; t h i s weak c l o s u r e i s moreover yr' s i n c e i t c o n t a i n s I ( p a r t I , c h a p t e r 233
234
PART 11, CHAPTER 6
3 , theorem 2 ) . L e t ( ~ 1~ , 2 ... , ) be a dense sequence i n ff. Adding t o t h e x i ' s t h e i r l i n e a r combinations, w i t h r a t i o n a l comp l e x c o e f f i c i e n t s , w e can suppose t h a t t h e x i ' s form a l i n e a r subspace ff' o f ff o v e r t h e f i e l d of r a t i o n a l complex numbers.
For x, y E f f , l e t hx,y be t h e Radon-Nikodym d e r i v a t i v e of t h e s p e c t r a l measure v ~ w,i t h ~ r e s p e c t t o v . By t h e formulas of p a r t I , c h a p t e r 7 , s e c t i o n 1, and t h e c o u n t a b i l i t y of f f ' , t h e r e e x i s t s a v - n e g l i g i b l e subset N of Z such t h a t , f o r 5 J N , t h e f u n c t i o n (x, y ) + hx,8(5) i s , on ff', a p o s i t i v e h e r m i t i a n sesq u i l i n e a r form. L e t ( 5 ) be t h e H i l b e r t space o b t a i n e d from f f ' by p a s s i n g t o t h e q u o t i e n t and completing, f f ' b e i n g endowed w i t h t h i s s e s q u i l i n e a r form, and l e t be t h e c a n o n i c a l mapping of ff' i n t o t i ( < ) . L e t N l c Z \ N be t h e s e t of t h e < c Z \ N such t h a t f f ( < ) = 0 . For < E Z \ N , t h e c o n d i t i o n cN1 i s e q u i v a l e n t t o t h e c o n d i t i o n " hxi,xj(<) = 0 f o r e v e r y i and e v e r y i"; w e t h e r e f o r e see t h a t N 1 i s V-measurable; l e t f be i t s c h a r a c t e r i s t i c funct i o n ; w e have
I$(c)
<
J
( ~ y i l ~=p(~yiI3:i) i) = f ( ~ ) d v , ~ , = ~ Jf(5)h,i,xi(5)dv(c) ~ ( ~ )
=o,
hence T p i = 0 f o r e v e r y i, 0 ; w e t h u s see t h a t N 1 i s V-negli'gible. Choosing new ' f ~ ~ ~ ~ s Tafrib(i ;t r) a r i l y on N u N l , which i s v - n e g l i g i b l e , w e can a r r a n g e t h a t ff(5) # 0 f o r every
< E z.
(ii)W e now endow t h e f f ( 5 ) ' s w i t h a measurable f i e l d s t r u c . number t u r e . For NUN^, p u t xi(<) = @ ( < ) x iThe
depends measurably on 5; and, f o r e v e r y 5 4 N U N l , t h e X i ( <) ' s form a t o t a l s e t i n f f ( 5 ) There t h u s e x i s t s on t h e f f ( 5 )' s e x a c t l y one measurable f i e l d s t r u c t u r e such t h a t t h e f i e l d s 5 + X i ( < ) a r e measurable v e c t o r f i e l d s ( c h a p t e r 1, p r o p o s i t i o n 4).
.
(iii)W e a r e going t o d e f i n e an isomorphism of ff o n t o
W e have
REDUCTION O F VON NEUMA” ALGEBRAS
235
n
i,j=1
T h i s shows a t once t h a t t h e v e c t o r f i e l d
i s s q u a r e - i n t e g r a b l e , and consequently t h a t t h i s f i e l d o n l y depends ( u p t o n e g l i g i b l e s e t s ) on t h e v e c t o r x and n o t on i t s r e p r e s e n t a t i o n i n t h e form
n
1T
x
.t’=1 f i i’
mapping UO which, t o t h e element
n
a r e dense i n
J
x
and f i n a l l y t h a t t h e
of ff, assigns t h e f i e l d
ff ( < ) d V ( < ) ( c h a p t e r 1, p r o p o s i t i o n
t h e v e c t o r s of t h e form
n
1T
.x.
i = 1 fz e x t e n d s t o an isomorphism U of
7-,
a r e dense i n
ff o n t o
co
le
7).
ff.
Moreover,
Hence UO
ff ( 5 )dV ( 5 ) .
A f u n c t i o n f E L c ( Z , V ) d e f i n e s on t h e one hand an o p e r a t o r T f i n f f , and on t h e o t h e r hand a d i a g o n a l i s a b l e o p e r a t o r T ’
f
le
in
f f ( r ; ) d v ( < ) . With t h e above n o t a t i o n , w e have n
Moreover, T’UOx i s t h e v e c t o r f i e l d
f
n
W e t h u s see t h a t TF o x = U O T p , and hence t h a t T ’
f
= UT U
f
-1
.
0
2 36
PART 11, CHAPTER 6
z
THEOREM 2 . Let H be a separable comptex HiZbert space, and a n a b e l i a n v o n Neumann aZgebra in H. Then, there e x i s t a compact metrisabte space Z , a p o s i t i v e measure v on z w i t h support Z , a v-measurabte f i e t d < + H(<) of non-zero compZex H i t b e r t spaces
over
Z,
and an isomorphism of H onto
r
H ( < ) d v ( < )which trans-
forms Z into the algebra of diagonatisable operators.
z,
weakly dense i n z, Proof. Let Y be a sub-C*-algebra of whose spectrum Z i s compact m e t r i s a b l e and c a r r i e s a b a s i c measure v [ p a r t I , c h a p t e r 7 , p r o p o s i t i o n 41. There e x i s t (theorem 1) a v-measurable f i e l d + H ( < ) of non-zero
<
H i l b e r t spaces over Z and an isomorphism of
f~' onto
je
H(<)dV(<)
which transforms Y i n t o t h e a l g e b r a of continuously d i a g o n a l i s a b l e o p e r a t o r s , and hence z i n t o t h e a l g e b r a of d i a g o n a l i s a b l e 0 o p e r a t o r s [chapter 2 , p r o p o s i t i o n 7 , ( i )
3.
Let H be a separabte complex H i t b e r t space, and A COROLLARY. a von Neumann atgebra in H. There ex-ist a compact metrisabte space Z , a p o s i t i v e measure v on z of support Z , a v-measurabte f i e l d 5 + H(<) of non-zero comptex H i l b e r t spaces over Z , a vmeasurable f i e t d < + A ( < ) of f a c t o r s i n t h e H(<) 's, and an isomorphism of H onto
rA(i)dv(i). Proof.
5
+
Is
H(<)$v(~) which transforms A into
Apply theorem 2 t o t h e c e n t r e
H ( < ) , and
an isomorphism U of
ff
2
of
A.
W e obtain Z , V,
o n t o r H ( < ) d V ( I ; ) which J
transforms z i n t o t h e a l q e b r a of d i a g o n a l i s a b l e o p e r a t o r s . We have c A c 2 ' , hence U A u - l i s decomposable ( c h a p t e r 3 , theorem 2). Hence t h e r e e x i s t s a v-measurable f i e l d 5 + A ( < ) of von
z
Neumann a l g e b r a s i n t h e ff(5) ' s such t h a t U A U - l A s t h e c e n t r e of UAU-'
=
Is
A ( < ) d v ( <.)
i s t h e a l g e b r a of d i a g o n a l i s a b l e operat o r s , t h e A ( < ) ' s a r e f a c t o r s almost everywhere ( c h a p t e r 3, theorem 3 ) . 0
The c o r o l l a r y t o some e x t e n t reduces t h e study of von Neumann a l g e b r a s t o t h a t of f a c t o r s ; t h i s was one of t h e p r i n c i p a l g o a l s of " r e d u c t i o n theory." N e v e r t h e l e s s , we saw i n p a r t I t h a t one can study g e n e r a l von Neumann a l g e b r a s d i r e c t l y by methods which comprise t h e " g l o b a l thoery
.
'I
References [194], [205],
:
[28], [206].
[49],
[8O],
[loo],
[117],
[145],
[193],
237
REDUCTION O F VON NEUMANN ALGEBRAS
3.
Uniqueness t h e o r e m s .
Given ff and Y, w e wish t o show t h a t Z , V and t h e f i e l d f f ( < ) a r e e s s e n t i a l l y unique. W e have a l r e a d y remarked ( c h a p t e r 2 , s e c t i o n 4 ) t h a t Z may be c a n o n i c a l l y i d e n t i f i e d w i t h t h e spectrum of Y i n such a way t h a t t h e c a n o n i c a l isomorphism o n t o Y may be i d e n t i f i e d w i t h t h e Gelfand isomorphism. of L,(Z) T h i s e s t a b l i s h e s t h e uniqueness of Z ( u p t o homeomorphism), and a l l o w s us t o s t a t e t h e uniqueness theorem i n t h e f o l l o w i n g way:
5
+
Let z be a ZocaZZy compact space, countabte a t THEOREM 3 . i n f i n i t y . Let v be a p o s i t i v e measure on z o f support Z, < + ff(5) a v-measurable f i e t d of non-zero HiZbert spaces over
ff
='rff
Z,
( 5 ) d v ( < ) , Y t h e algebra of continuousty diagonatisabZe
s $
operators in ff, and f + Tf the canonicat isomorphism of &(z) onto Y. Define, anatogouszy, v l , < + tl,(s), HL, Y,, + TI Let u be an isomorphism of ff onto ff, transformzng T z n t o for every f E ~ ( z ) . Then, v and vl are equioalent, an there e x i s t s , a f t e r necessary modification of t h e ff(5)'s and the ff (5)'s on n e g t i g i b l e s e t s , an isomorphism 5 + v(<) of t h e f i e i d ( f f ( 5 ) ) onto t h e f i e l d ( f f l ( < ) ) , such t h a t u = WV, where v is the
6
isomorphism
I
I
8
8
v(<)dv(<)of
t h e canonical isomorphism
ff,(C)dv(<), and where w is
( 5 )dv ( 5 ) onto H,.
Proof. I d e n t i f y Z w i t h t h e spectrum of Y and of Y 1 i n such a are way t h a t , f o r e v e r y f u n c t i o n f E L,(Z) , T and T1 = UTP-' t h e elements of Y and Y, corresponding t o f u n d l r t h e Gelfand - vux u y , hence isomorphisms. Then, i f x , y E ff, w e have v t h e b a s i c measures d e f i n e d on Z by Y and %'are t h & s a m e . Hence
f
v and v l a r e e q u i v a l e n t .
Let
q
=
[ g f f L ( i ; ) d v ( <, )W be t h e canoni-
9
c a l isomorphism of 'fl' o n t o ff,, and t h e image o f Yl under W-l., which i s t h e a l g e b r a of c o n t i n u o u s l y diagonalisabble o p e r a t o r s i n q. Every f u n c t i o n f E L,(Z) d e f i n e s an o p e r a t o r T i n 77, and w e have f T
N
=
f
.
(W-1 U)T (W-1u) -1
f
Hence W - I U = V i s a decomposable l i n e a r mapping of ff i n t o q ( c h a p t e r 2 , theorem l ) , from which i t f o l l o w s t h a t t h e r e e x i s t s a measurable f i e l d + V(<) of l i n e a r mappings o f ff(5) i n t o
<
ff,(c),
with V
of ff o n t o
q,
F i n a l l y , s i n c e V i s an isomorphism
=
w e have
v*v
=
If/,
w*
=
I
T7r
PART 11, CHAPTER 6
238
hence
almost everywhere, i . e . V ( < ) i s almost everywhere an isomorphism of H ( < ) onto f f l ( < ) . 0 THEOREM 4. Z,
measure on
r
Let Z be a Bore1 space, v a standard p o s i t i v e E = ( f f ( < ) ) a v-measurable f i e l d of non-zero com-
p l e x Hilbert spaces over
ff
Z,
=
f f ( < ) d v ( < ) ,and Z the algebra
Define zlJ v1, El = ( f f l ( c l ) ) , ffl, and Zl analogously. Let u be an isomorphism o f ff onto ffl transforming 2 i n t o Z., Then there e x i s t :
of diagonalisable operators i n ff.
z, and a vl-negligible Bore1 in z1; 2' a Bore2 isomzrphism q of Z \ N onto Z 1 \ N 1 which transforms v i n t o a measure vl equivalent t o vl; 3 O an q-isomorphism (~(5))of E I z \ N onto E ~ / Z ~ \ Nwhich ~ , 1 ' a v-negligible Bore1 s e t N i n
set
N~
v of ff wv, where w is
I
~
8
N
1 4 , ( < ~ ) & ~ ( i
defines an isomorphism
onto ti,
a way t h a t u onto H,.
the canoni&Z isomorphism of
=
=
n,
Proof. We can suppose t h a t Z and Z1 a r e compact m e t r i s a b l e . The isomorphism U d e f i n e s an isomorphism of 2 onto and t h e r e f o r e an isomorghism @ of L :(Z, v) onto LF(Z1, Vl). Hence t h e r e e x i s t N, N l , v l , q , with p r o p e r t i e s 1 ' and 2' of t h e theorem, q a l s o d e f i n i n g t h e isomorphism @ (appendix I V ) .
z,,
For f. Lod)(Z, v ) [ r e s p . f l E LF(Z1, v,)], denote by Tf ( r e s p . ) t h e d i a g o n a l i s a b l e o p e r a t o r of ff ( r e s p . defin d b ( & & s p .fl). If f and f1 correspond under @, Tf and WTflW respond under U, i . e .
Tl)
or
-1 -1 -1 T W U = W UT or
fl
-1 T V=VT fl
putting
f'
f'
239
REDUCTION OF VON NEUMANN ALGEBRAS
I t a l l , t h e r e f o r e , comes down t o showing t h a t V i s "decomposable" i n a g e n e r a l i s e d s e n s e . W e c o u l d have p r e s e n t e d t h i s g e n e r a l i s a t i o n i n c h a p t e r 2 , b u t it would have been cumbersome. W e w i l l b r i e f l y i n d i c a t e how t h e arguments o f c h a p t e r 2 may be extended, l e a v i n g t h e d e t a i l s t o t h e r e a d e r , which s h o u l d n o t , however, c a u s e any d i f f i c u l t y . L e t (xl, x 2 , ) be a fundamental sequence of measurable v e c t o r f i e l d s o v e r Z \ N such t h a t xi E ff; w e -1 can p u t yi = V x i E gl. Making u s e of t h e e q u a l i t y TflV = VTf, w e show, j u s t a s f o r theorem 1 of c h a p t e r 2 , t h a t
...
almost everywhere on Z \ N , f o r any r a t i o n a l complex numbers p l , p 2 , ..., .p, Hence t h e r e e x i s t s a continuous l i n e a r mapping V ( 5 ) of f f ( 5 ) i n t o f f l ( n ( < ) ) , f o r 5 E Z \ N , such t h a t V(C) 5 1 and
11
11
V ( < ) Z i ( 5 )= y i ( l l ( < ) ) f o r e v e r y i, almost everywhere. G e n e r a l i s i n g p r o p o s i t i o n 1 of c h a p t e r 2 , w e conclude from lzhis t h a t t h e V ( 5 ) ' s t r a n s f o r m e v e r y measurable v e c t o r f i e l d o v e r Z \ N i n t o a measurable v e c t o r f i e l d over Z1\N1. Arguing s i m i l a r l y w i t h V - l , t h e r e e x i s t s , f o r 51 E Z 1 \ N 1 , a continuous l i n e a r mapping V ' (51) of ffi(5,) i n t o ff ) , such V' (C1) I 1, and V ' ( < l ) y i ( < l ) = X ~ ( U - ~ ( < ~f )o )r e v e r y i, that almost everywhere; and t h e V ' ( C 1 ) ' s t r a n s f o r m e v e r y measurable v e c t o r f i e l d over z1\ N 1 i n t o a measurable v e c t o r f i e l d o v e r Z\N. Consequently, V ( < ) and V ' ( n ( < ) ) a r e , almost everywhere, i n v e r s e isomorphisms of f f ( 5 ) o n t o f f ' ( q ( < ) ) and of ff' (n(<)) o n t o f f ( 5 ) . A f t e r m o d i f i c a t i o n on a n e g l i g i b l e s e t , ( V ( < ) ) C ~ ~i,s~ an The c o r r e s p o n d i n g i s o q-isomorphism of E 1'2 \N o n t o E l 121\ N 1 . morphism of ff o n t o ff1 a c t s on elements of t h e form Tf"i i n t h e same way a s V , and i s t h e r e f o r e e q u a l t o V. 0
1)
(nd1(cL)
1)
Reference : [ 8 0 ] .
Exercises.
1.
Let
5
+
r
f f ( 5 ) be a v-measurable f i e l d of com-
p l e x H i l b e r t s p a c e s o v e r Z , ff
=
ff ( c ) d v ( < ) , z t h e a l g e b r a of
d i a g o n a l i s a b l e o p e r a t o r s , and A a f a c t o r c o n t a i n e d i n Z ' . Suppose t h a t i s a maximal a b e l i a n von Neumann s u b a l g e b r a i n A ' .
z
a. I f A i s d i s c r e t e , and i f f f ( 5 ) P 0 almost everywhere, t h e r e e x i s t : lo a s e p a r a b l e H i l b e r t s p a c e K O ; 2 O a l m o s t everywhere on Z , an isomorphism U ( < ) o f f f ( < ) o n t o K O ; 3' f o r e v e r y T E A , a unique o p e r a t o r T1 E !-(KO) such t h a t
240
PART 11, CHAPTER 6
(Use c h a p t e r 2 , p r o p o s i t i o n 8 and e x e r c i s e 1, and theorem 3 of t h e p r e s e n t c h a p t e r ) . A s T runs through A , T1 runs through L(K0).
b.
I n t h e g e n e r a l c a s e , f o r every T =
'
re
J
T(<)dV(<)E
A,
T
#
0
i m p l i e s T ( < ) # 0 almost everywhere. [Let Y be t h e measurable s u b s e t of Z c o n s i s t i n g of t h e 5 such t h a t T ( < ) = 0 . Let E be t h e d i a g o n a l i s a b l e p r o j e c t i o n corresponding t o Y . W e have TE = 0 , hence E = 01 [l6], [57], [58]. Problem:
What r e l a t i o n s e x i s t between t h e T ( < ) ' s ?
"51,
[265]. 2. With t h e hypotheses of theorem 2 , show t h a t one can, with t h e n o t a t i o n of t h a t theorem, t a k e f o r Z a compact i n t e r v a l of the r e a l l i n e . (Thanks t o p a r t I , c h a p t e r 7, e x e r c i s e 3 f, take Y t o be generated by a s i n g l e h e r m i t i a n element. Then t h e spectrum of Y i s a compact s u b s e t of t h e r e a l l i n e ) [ 8 O ] .
1.
D E C O M P O S I T I O N OF A H I L B E R T S P A C E INTO A DIRECT INTEGRAL
Posing the problem.
A s i n c h a p t e r s 1, 2 , and 3 , given:
1' a compact m e t r i s a b l e s p a c e Z; 2' a p o s i t i v e measure v on 3' a v-measurable over Z ,
z
of s u p p o r t Z;
f i e l d 5 + f f ( < ) of non-zero H i l b e r t s p a c e s
w e can c o n s t r u c t c a n o n i c a l l y : lo t h e s e p a r a b l e H i l b e r t space ff
=
2' t h e a b e l i a n von Neumann a l g e b r a tors;
I'B
ff ( < ) d V ( < )
z
;
of d i a g o n a l i s a b l e opera-
3' more p r e c i s e l y , t h e C*-algebra Y of c o n t i n u o u s l y d i a g o n a l i sable o p e r a t o r s , whose weak c l o s u r e i s
z.
W e a r e now going t o show t h a t t h e o r d e r of t h e s e c o n s t r u c t i o n s can be r e v e r s e d , i n an e s s e n t i a l l y unique way. T h i s w i l l g r e a t l y add t o t h e importance of t h e p r e c e d i n g c h a p t e r s .
2.
Existence theorems.
THEOREM 1. Let ff be a separable complex H i l b e r t space, y an abelian c*-algebra of operators i n H, z t h e spectrum of y, and v a b a s i c measure on Z . Suppose t h a t I i s in t h e weak closure of y. Then, there e x i s t s a v-measurable f i e l d 5 + H ( < ) of non-zero coq?lex lif'lbert spaces over Z , and an isomorphism of H onto
H ( < ) d v ( < ) ,which transforms t h e Gelfand isomorphism into the canonical isomorphism of & ( Z ) onto the algebra of continuously diagonalisable operators. Proof. ( i ) o o W e w i l l d e n o t e by f -+ T t h e weakly c o n t i n u o u s i s o morphism of L C ( Z , V) o n t o t h e weak cltfssure Z of y which e x t e n d s t h e Gelfand isomorphism ( p a r t I , c h a p t e r 7 , p r o p o s i t i o n 1 ) ; t h i s weak c l o s u r e i s moreover yr' s i n c e i t c o n t a i n s I ( p a r t I , c h a p t e r 233
234
PART 11, CHAPTER 6
3 , theorem 2 ) . L e t ( ~ 1~ , 2 ... , ) be a dense sequence i n ff. Adding t o t h e x i ' s t h e i r l i n e a r combinations, w i t h r a t i o n a l comp l e x c o e f f i c i e n t s , w e can suppose t h a t t h e x i ' s form a l i n e a r subspace ff' o f ff o v e r t h e f i e l d of r a t i o n a l complex numbers.
For x, y E f f , l e t hx,y be t h e Radon-Nikodym d e r i v a t i v e of t h e s p e c t r a l measure v ~ w,i t h ~ r e s p e c t t o v . By t h e formulas of p a r t I , c h a p t e r 7 , s e c t i o n 1, and t h e c o u n t a b i l i t y of f f ' , t h e r e e x i s t s a v - n e g l i g i b l e subset N of Z such t h a t , f o r 5 J N , t h e f u n c t i o n (x, y ) + hx,8(5) i s , on ff', a p o s i t i v e h e r m i t i a n sesq u i l i n e a r form. L e t ( 5 ) be t h e H i l b e r t space o b t a i n e d from f f ' by p a s s i n g t o t h e q u o t i e n t and completing, f f ' b e i n g endowed w i t h t h i s s e s q u i l i n e a r form, and l e t be t h e c a n o n i c a l mapping of ff' i n t o t i ( < ) . L e t N l c Z \ N be t h e s e t of t h e < c Z \ N such t h a t f f ( < ) = 0 . For < E Z \ N , t h e c o n d i t i o n cN1 i s e q u i v a l e n t t o t h e c o n d i t i o n " hxi,xj(<) = 0 f o r e v e r y i and e v e r y i"; w e t h e r e f o r e see t h a t N 1 i s V-measurable; l e t f be i t s c h a r a c t e r i s t i c funct i o n ; w e have
I$(c)
<
J
( ~ y i l ~=p(~yiI3:i) i) = f ( ~ ) d v , ~ , = ~ Jf(5)h,i,xi(5)dv(c) ~ ( ~ )
=o,
hence T p i = 0 f o r e v e r y i, 0 ; w e t h u s see t h a t N 1 i s V-negli'gible. Choosing new ' f ~ ~ ~ ~ s Tafrib(i ;t r) a r i l y on N u N l , which i s v - n e g l i g i b l e , w e can a r r a n g e t h a t ff(5) # 0 f o r every
< E z.
(ii)W e now endow t h e f f ( 5 ) ' s w i t h a measurable f i e l d s t r u c . number t u r e . For NUN^, p u t xi(<) = @ ( < ) x iThe
depends measurably on 5; and, f o r e v e r y 5 4 N U N l , t h e X i ( <) ' s form a t o t a l s e t i n f f ( 5 ) There t h u s e x i s t s on t h e f f ( 5 )' s e x a c t l y one measurable f i e l d s t r u c t u r e such t h a t t h e f i e l d s 5 + X i ( < ) a r e measurable v e c t o r f i e l d s ( c h a p t e r 1, p r o p o s i t i o n 4).
.
(iii)W e a r e going t o d e f i n e an isomorphism of ff o n t o
W e have
REDUCTION O F VON NEUMA” ALGEBRAS
235
n
i,j=1
T h i s shows a t once t h a t t h e v e c t o r f i e l d
i s s q u a r e - i n t e g r a b l e , and consequently t h a t t h i s f i e l d o n l y depends ( u p t o n e g l i g i b l e s e t s ) on t h e v e c t o r x and n o t on i t s r e p r e s e n t a t i o n i n t h e form
n
1T
x
.t’=1 f i i’
mapping UO which, t o t h e element
n
a r e dense i n
J
x
and f i n a l l y t h a t t h e
of ff, assigns t h e f i e l d
ff ( < ) d V ( < ) ( c h a p t e r 1, p r o p o s i t i o n
t h e v e c t o r s of t h e form
n
1T
.x.
i = 1 fz e x t e n d s t o an isomorphism U of
7-,
a r e dense i n
ff o n t o
co
le
7).
ff.
Moreover,
Hence UO
ff ( 5 )dV ( 5 ) .
A f u n c t i o n f E L c ( Z , V ) d e f i n e s on t h e one hand an o p e r a t o r T f i n f f , and on t h e o t h e r hand a d i a g o n a l i s a b l e o p e r a t o r T ’
f
le
in
f f ( r ; ) d v ( < ) . With t h e above n o t a t i o n , w e have n
Moreover, T’UOx i s t h e v e c t o r f i e l d
f
n
W e t h u s see t h a t TF o x = U O T p , and hence t h a t T ’
f
= UT U
f
-1
.
0
2 36
PART 11, CHAPTER 6
z
THEOREM 2 . Let H be a separable comptex HiZbert space, and a n a b e l i a n v o n Neumann aZgebra in H. Then, there e x i s t a compact metrisabte space Z , a p o s i t i v e measure v on z w i t h support Z , a v-measurabte f i e t d < + H(<) of non-zero compZex H i t b e r t spaces
over
Z,
and an isomorphism of H onto
r
H ( < ) d v ( < )which trans-
forms Z into the algebra of diagonatisable operators.
z,
weakly dense i n z, Proof. Let Y be a sub-C*-algebra of whose spectrum Z i s compact m e t r i s a b l e and c a r r i e s a b a s i c measure v [ p a r t I , c h a p t e r 7 , p r o p o s i t i o n 41. There e x i s t (theorem 1) a v-measurable f i e l d + H ( < ) of non-zero
<
H i l b e r t spaces over Z and an isomorphism of
f~' onto
je
H(<)dV(<)
which transforms Y i n t o t h e a l g e b r a of continuously d i a g o n a l i s a b l e o p e r a t o r s , and hence z i n t o t h e a l g e b r a of d i a g o n a l i s a b l e 0 o p e r a t o r s [chapter 2 , p r o p o s i t i o n 7 , ( i )
3.
Let H be a separabte complex H i t b e r t space, and A COROLLARY. a von Neumann atgebra in H. There ex-ist a compact metrisabte space Z , a p o s i t i v e measure v on z of support Z , a v-measurabte f i e l d 5 + H(<) of non-zero comptex H i l b e r t spaces over Z , a vmeasurable f i e t d < + A ( < ) of f a c t o r s i n t h e H(<) 's, and an isomorphism of H onto
rA(i)dv(i). Proof.
5
+
Is
H(<)$v(~) which transforms A into
Apply theorem 2 t o t h e c e n t r e
H ( < ) , and
an isomorphism U of
ff
2
of
A.
W e obtain Z , V,
o n t o r H ( < ) d V ( I ; ) which J
transforms z i n t o t h e a l q e b r a of d i a g o n a l i s a b l e o p e r a t o r s . We have c A c 2 ' , hence U A u - l i s decomposable ( c h a p t e r 3 , theorem 2). Hence t h e r e e x i s t s a v-measurable f i e l d 5 + A ( < ) of von
z
Neumann a l g e b r a s i n t h e ff(5) ' s such t h a t U A U - l A s t h e c e n t r e of UAU-'
=
Is
A ( < ) d v ( <.)
i s t h e a l g e b r a of d i a g o n a l i s a b l e operat o r s , t h e A ( < ) ' s a r e f a c t o r s almost everywhere ( c h a p t e r 3, theorem 3 ) . 0
The c o r o l l a r y t o some e x t e n t reduces t h e study of von Neumann a l g e b r a s t o t h a t of f a c t o r s ; t h i s was one of t h e p r i n c i p a l g o a l s of " r e d u c t i o n theory." N e v e r t h e l e s s , we saw i n p a r t I t h a t one can study g e n e r a l von Neumann a l g e b r a s d i r e c t l y by methods which comprise t h e " g l o b a l thoery
.
'I
References [194], [205],
:
[28], [206].
[49],
[8O],
[loo],
[117],
[145],
[193],
237
REDUCTION O F VON NEUMANN ALGEBRAS
3.
Uniqueness t h e o r e m s .
Given ff and Y, w e wish t o show t h a t Z , V and t h e f i e l d f f ( < ) a r e e s s e n t i a l l y unique. W e have a l r e a d y remarked ( c h a p t e r 2 , s e c t i o n 4 ) t h a t Z may be c a n o n i c a l l y i d e n t i f i e d w i t h t h e spectrum of Y i n such a way t h a t t h e c a n o n i c a l isomorphism o n t o Y may be i d e n t i f i e d w i t h t h e Gelfand isomorphism. of L,(Z) T h i s e s t a b l i s h e s t h e uniqueness of Z ( u p t o homeomorphism), and a l l o w s us t o s t a t e t h e uniqueness theorem i n t h e f o l l o w i n g way:
5
+
Let z be a ZocaZZy compact space, countabte a t THEOREM 3 . i n f i n i t y . Let v be a p o s i t i v e measure on z o f support Z, < + ff(5) a v-measurable f i e t d of non-zero HiZbert spaces over
ff
='rff
Z,
( 5 ) d v ( < ) , Y t h e algebra of continuousty diagonatisabZe
s $
operators in ff, and f + Tf the canonicat isomorphism of &(z) onto Y. Define, anatogouszy, v l , < + tl,(s), HL, Y,, + TI Let u be an isomorphism of ff onto ff, transformzng T z n t o for every f E ~ ( z ) . Then, v and vl are equioalent, an there e x i s t s , a f t e r necessary modification of t h e ff(5)'s and the ff (5)'s on n e g t i g i b l e s e t s , an isomorphism 5 + v(<) of t h e f i e i d ( f f ( 5 ) ) onto t h e f i e l d ( f f l ( < ) ) , such t h a t u = WV, where v is the
6
isomorphism
I
I
8
8
v(<)dv(<)of
t h e canonical isomorphism
ff,(C)dv(<), and where w is
( 5 )dv ( 5 ) onto H,.
Proof. I d e n t i f y Z w i t h t h e spectrum of Y and of Y 1 i n such a are way t h a t , f o r e v e r y f u n c t i o n f E L,(Z) , T and T1 = UTP-' t h e elements of Y and Y, corresponding t o f u n d l r t h e Gelfand - vux u y , hence isomorphisms. Then, i f x , y E ff, w e have v t h e b a s i c measures d e f i n e d on Z by Y and %'are t h & s a m e . Hence
f
v and v l a r e e q u i v a l e n t .
Let
q
=
[ g f f L ( i ; ) d v ( <, )W be t h e canoni-
9
c a l isomorphism of 'fl' o n t o ff,, and t h e image o f Yl under W-l., which i s t h e a l g e b r a of c o n t i n u o u s l y diagonalisabble o p e r a t o r s i n q. Every f u n c t i o n f E L,(Z) d e f i n e s an o p e r a t o r T i n 77, and w e have f T
N
=
f
.
(W-1 U)T (W-1u) -1
f
Hence W - I U = V i s a decomposable l i n e a r mapping of ff i n t o q ( c h a p t e r 2 , theorem l ) , from which i t f o l l o w s t h a t t h e r e e x i s t s a measurable f i e l d + V(<) of l i n e a r mappings o f ff(5) i n t o
<
ff,(c),
with V
of ff o n t o
q,
F i n a l l y , s i n c e V i s an isomorphism
=
w e have
v*v
=
If/,
w*
=
I
T7r
PART 11, CHAPTER 6
238
hence
almost everywhere, i . e . V ( < ) i s almost everywhere an isomorphism of H ( < ) onto f f l ( < ) . 0 THEOREM 4. Z,
measure on
r
Let Z be a Bore1 space, v a standard p o s i t i v e E = ( f f ( < ) ) a v-measurable f i e l d of non-zero com-
p l e x Hilbert spaces over
ff
Z,
=
f f ( < ) d v ( < ) ,and Z the algebra
Define zlJ v1, El = ( f f l ( c l ) ) , ffl, and Zl analogously. Let u be an isomorphism o f ff onto ffl transforming 2 i n t o Z., Then there e x i s t :
of diagonalisable operators i n ff.
z, and a vl-negligible Bore1 in z1; 2' a Bore2 isomzrphism q of Z \ N onto Z 1 \ N 1 which transforms v i n t o a measure vl equivalent t o vl; 3 O an q-isomorphism (~(5))of E I z \ N onto E ~ / Z ~ \ Nwhich ~ , 1 ' a v-negligible Bore1 s e t N i n
set
N~
v of ff wv, where w is
I
~
8
N
1 4 , ( < ~ ) & ~ ( i
defines an isomorphism
onto ti,
a way t h a t u onto H,.
the canoni&Z isomorphism of
=
=
n,
Proof. We can suppose t h a t Z and Z1 a r e compact m e t r i s a b l e . The isomorphism U d e f i n e s an isomorphism of 2 onto and t h e r e f o r e an isomorghism @ of L :(Z, v) onto LF(Z1, Vl). Hence t h e r e e x i s t N, N l , v l , q , with p r o p e r t i e s 1 ' and 2' of t h e theorem, q a l s o d e f i n i n g t h e isomorphism @ (appendix I V ) .
z,,
For f. Lod)(Z, v ) [ r e s p . f l E LF(Z1, v,)], denote by Tf ( r e s p . ) t h e d i a g o n a l i s a b l e o p e r a t o r of ff ( r e s p . defin d b ( & & s p .fl). If f and f1 correspond under @, Tf and WTflW respond under U, i . e .
Tl)
or
-1 -1 -1 T W U = W UT or
fl
-1 T V=VT fl
putting
f'
f'
239
REDUCTION OF VON NEUMANN ALGEBRAS
I t a l l , t h e r e f o r e , comes down t o showing t h a t V i s "decomposable" i n a g e n e r a l i s e d s e n s e . W e c o u l d have p r e s e n t e d t h i s g e n e r a l i s a t i o n i n c h a p t e r 2 , b u t it would have been cumbersome. W e w i l l b r i e f l y i n d i c a t e how t h e arguments o f c h a p t e r 2 may be extended, l e a v i n g t h e d e t a i l s t o t h e r e a d e r , which s h o u l d n o t , however, c a u s e any d i f f i c u l t y . L e t (xl, x 2 , ) be a fundamental sequence of measurable v e c t o r f i e l d s o v e r Z \ N such t h a t xi E ff; w e -1 can p u t yi = V x i E gl. Making u s e of t h e e q u a l i t y TflV = VTf, w e show, j u s t a s f o r theorem 1 of c h a p t e r 2 , t h a t
...
almost everywhere on Z \ N , f o r any r a t i o n a l complex numbers p l , p 2 , ..., .p, Hence t h e r e e x i s t s a continuous l i n e a r mapping V ( 5 ) of f f ( 5 ) i n t o f f l ( n ( < ) ) , f o r 5 E Z \ N , such t h a t V(C) 5 1 and
11
11
V ( < ) Z i ( 5 )= y i ( l l ( < ) ) f o r e v e r y i, almost everywhere. G e n e r a l i s i n g p r o p o s i t i o n 1 of c h a p t e r 2 , w e conclude from lzhis t h a t t h e V ( 5 ) ' s t r a n s f o r m e v e r y measurable v e c t o r f i e l d o v e r Z \ N i n t o a measurable v e c t o r f i e l d over Z1\N1. Arguing s i m i l a r l y w i t h V - l , t h e r e e x i s t s , f o r 51 E Z 1 \ N 1 , a continuous l i n e a r mapping V ' (51) of ffi(5,) i n t o ff ) , such V' (C1) I 1, and V ' ( < l ) y i ( < l ) = X ~ ( U - ~ ( < ~f )o )r e v e r y i, that almost everywhere; and t h e V ' ( C 1 ) ' s t r a n s f o r m e v e r y measurable v e c t o r f i e l d over z1\ N 1 i n t o a measurable v e c t o r f i e l d o v e r Z\N. Consequently, V ( < ) and V ' ( n ( < ) ) a r e , almost everywhere, i n v e r s e isomorphisms of f f ( 5 ) o n t o f f ' ( q ( < ) ) and of ff' (n(<)) o n t o f f ( 5 ) . A f t e r m o d i f i c a t i o n on a n e g l i g i b l e s e t , ( V ( < ) ) C ~ ~i,s~ an The c o r r e s p o n d i n g i s o q-isomorphism of E 1'2 \N o n t o E l 121\ N 1 . morphism of ff o n t o ff1 a c t s on elements of t h e form Tf"i i n t h e same way a s V , and i s t h e r e f o r e e q u a l t o V. 0
1)
(nd1(cL)
1)
Reference : [ 8 0 ] .
Exercises.
1.
Let
5
+
r
f f ( 5 ) be a v-measurable f i e l d of com-
p l e x H i l b e r t s p a c e s o v e r Z , ff
=
ff ( c ) d v ( < ) , z t h e a l g e b r a of
d i a g o n a l i s a b l e o p e r a t o r s , and A a f a c t o r c o n t a i n e d i n Z ' . Suppose t h a t i s a maximal a b e l i a n von Neumann s u b a l g e b r a i n A ' .
z
a. I f A i s d i s c r e t e , and i f f f ( 5 ) P 0 almost everywhere, t h e r e e x i s t : lo a s e p a r a b l e H i l b e r t s p a c e K O ; 2 O a l m o s t everywhere on Z , an isomorphism U ( < ) o f f f ( < ) o n t o K O ; 3' f o r e v e r y T E A , a unique o p e r a t o r T1 E !-(KO) such t h a t
240
PART 11, CHAPTER 6
(Use c h a p t e r 2 , p r o p o s i t i o n 8 and e x e r c i s e 1, and theorem 3 of t h e p r e s e n t c h a p t e r ) . A s T runs through A , T1 runs through L(K0).
b.
I n t h e g e n e r a l c a s e , f o r every T =
'
re
J
T(<)dV(<)E
A,
T
#
0
i m p l i e s T ( < ) # 0 almost everywhere. [Let Y be t h e measurable s u b s e t of Z c o n s i s t i n g of t h e 5 such t h a t T ( < ) = 0 . Let E be t h e d i a g o n a l i s a b l e p r o j e c t i o n corresponding t o Y . W e have TE = 0 , hence E = 01 [l6], [57], [58]. Problem:
What r e l a t i o n s e x i s t between t h e T ( < ) ' s ?
"51,
[265]. 2. With t h e hypotheses of theorem 2 , show t h a t one can, with t h e n o t a t i o n of t h a t theorem, t a k e f o r Z a compact i n t e r v a l of the r e a l l i n e . (Thanks t o p a r t I , c h a p t e r 7, e x e r c i s e 3 f, take Y t o be generated by a s i n g l e h e r m i t i a n element. Then t h e spectrum of Y i s a compact s u b s e t of t h e r e a l l i n e ) [ 8 O ] .