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1. E. Segal
Decompositions of operator algebras l and II
Memoirs
of the American Mathematical Society Providence Rhode Island 1951
Number 9
USA
ISSN 0065-9266
Memoirs of the American Mathematical Society
Number 9
I. E. Segal
Decompositions of operator algebras I and II
Published by the AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island, USA
1951 Number 9
First printing, 1951 Second printing with corrections and additions, 1967 Third printing. 1971 Fourth printing. 1989
International Standard Book Number 0-8218-1209-2 Library of Congress Catalog Number 52-42839
Printed In the United States of America Copyright Qc 1951 by the American Mathematical Society All Rights Reserved
The paper used in this reprint is acid free and falls within the guidelines established to ensure permanence and durability.
DECOMPOSITIONS OF OPERATOR ALGEBRAS.
I
By I. E. Segal of the
University of Chicago 1.
Introduction. We show that an algebra of operators on a
Hilbert space can be decomposed relative to a Boolean algebra of invariant subspaces as a kind of direct integral, similar to the decomposition as a direct sum of algebras of linear transformations on finite-dimensional spaces.
This decomposition results from an interesting decomposition
formula, for the "states" of operator algebras which we have treated in [8].
If the Boolean algebra is maximal, and with a certain separability
restriction, the constituents in the direct integral are almost everywhere irreducible.
It follows that in the case of a separable Hilbert space, a
weakly closed self-adjoint algebra is a direct integral of factors.
Any
continuous unitary representation of a separable locally compact group is a direct integral of irreducible such representations.
If
G
G
is uni-
modular, then its two-sided regular representation is a direct integral of irreducible two-sided representations.
Any measure on a compact metric
space which is invariant under a group of homeomorphisms of the space is a direct integral of ergodic measures. Our basic results are closely related to results of von Neumann in [161.
The decompositions obtained by von Neumann are from a formal view-
Received by the Editors on March 1, 1950.
I. E. Segal
2
point nearly identical with ours, but there are important technical differences in the approaches as well as in the results which allow us to give considerably simpler proofs of the key theorems, and which yield a theory better adapted to the study of group representations than that of von Neumann.
These differences are notably, first, the use of states, and
second, the use of perfect measure spaces, rather than a measure space over the field of Borel subsets of the reals.
Each of these features simplifies
the serious measurability problems involved in obtaining decompositions. The concept of direct integral of Hilbert spaces is awkward because the Hilbert spaces may vary in dimensionality, and it is unclear to begin with how a measurable function to such Hilbert spaces should be defined.
As a
state is a numerical-valued function, there is no such awkwardness about direct integrals of states, and by virtue of the known correspondence between states and representation Hilbert spaces, a decomposition of a state as such an integral induces a decomposition of the Hilbert apace into "differential" Hilbert spaces, so to speak.
The utilization of perfect
measure spaces (on which every bounded measurable function is equivalent to a continuous function) eliminates the need for various kinds of sets of measure zero which occur in von Neumann's theory, and greatly facilitates the reduction of group representations.
Our theorem concerning maximal
decompositions bears the same formal relation to a theorem of Mautner [5]
that our decomposition theory does to that of von Neumann, but the logical roles of these two theorems are very different, as we use our result to decompose a general algebra of operators into factors, while Mautner's result is derived directly from von Neumann's decomposition theory for general operator algebras.
By virtue of the difference between our basic tech-
niques (some of which apply to inseparable spaces) and those of von Neumann,
our proofs are for the most part necessarily of a different character from those of von Neumann, and in particular no use is made of the theory of
DECOMPOSITIONS OF OPERATOR ALGEBRAS. 1
3
analytic sets. Definitions and notations.
2.
We introduce here a number of
terms and symbols which we shall use without further reference in the remainder of the paper. A W*-algebra (or C*-algebra) is a weakly (or uni-
Definition 2.1.
formly) closed self-adjoint (SA) algebra of (bounded linear) operators on a The term "operator" will always mean "bounded linear oper-
Hilbert space. ator".
Q of operators on a Hilbert space, the set of all
For any algebra
operators which commute with every element of Q- and denoted by
of
A W*-algebra
a!.
Q which contains the identity
I, and is such that
operator, always designated by (only) of scalar multiples of
Q is called the commutor
I
Q " Q' consists
is called a factor.
The term Hilbert
space will be used in the present paper to denote a complex (generalized) Hilbert space of arbitrary dimension
A measure space is the system composed of a set
Definition 2.2. R, a
cr-ring
valued function is finite.
of subsets of
1f.
r
on
(> 0).
1?.
R, and a countably-additive non-negative-
Such a space is finite if
Such a space, denoted as
sets in 7P .
K
complex-valued function on
S ale,
r(S) - L.U.B.K, Sr(S)
varies over the compact and W
We denote such a space as le
R
is the Q'-ring generated by
R, and if for every
G.L.B.W, Sr(W), where
r(R)
(R, 1Q , r), is called regular if
is a locally compact topological space, the compact subsets of
R a *_ and
over the open
(R, r), and a countably-additive
is called regular if the positive and nega-
tive constituents of its real and imaginary parts are such that the corresponding measure spaces are regular (i. e. are regular measures). measure space furthermore,
(R, 7Z, r)
A finite
is called perfect if it is regular, and if,
for every bounded measurable function on the space there is
a unique continuous function on
R
equal almost everywhere to the given
I. E. Segal
4
function.
Definitions 2.3. tional
U
Q
A state of a C*-algebra
Q such that W (U*U) = 0
on
(,)
and
Q is a linear func-
(i(U*) = c3(U)
for
(a bar over a numerical-valued function denotes the complex-con-
jugate function), and with L.U.B. II U I = (U*U) a 1. The associ1, U E Q ated representation s0 of Q , Hilbert space >F , canonical mapping It
Q into 14-, and wave function
of
with the properties. 1)
Q
a mapping on
z, are the essentially unique objects
a to 1+, j(Q) 4)
and z
in Q
V
[8].)
for
; 3)
element
72(U) 1(V) = 7((UV)
z
for any
U
and
V
in Q ; and
of unit norm such that cJ(U) _ (f(U)z, z)
A representation
z
is continuous and linear on
(For further properties and an existence proof, see
UE Q.
exists an element
Y7
is dense in >4-, and (1(U),71 (V)) =rJ(V*U) for any
is an element of
VU) = Uz
14, (i.e. it is
to the operators on N which preserves algebraic opera-
tions, including that of adjunction); 2)
U
Q on
is a representation of
72
Q
f of
in
14-
on $ is called cyclic if there
such that ' (Q )z Is dense in
such an
is called a cyclic vector.
Definitions 2.4.
The spectrum of a commutative (complex) Banach
algebra is the topological space whose set is the collection of all continuous homomorphisms of the algebra into the complex numbers which are not identically zero, and whose topology is the weak topology in the conjugate space of the algebra.
If
is a locally compact Hausdorff apace,
i'
C(r')(or 7Q (r)) denotes the Banach algebra of all continuous complexvalued (or real-valued) functions on (a function the set
f
on r vanishes at
[Yj+f( Y)J
E ]
T' oo
which vanish at infinity on r if for every positive number
£.
is compact), with the norm of a function taken
to be the maximum of its absolute value. Definition 2.5.
,
If
V
is a measure space,
L,,(M)
denotes the
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
5
Banach space of octh-power integrable complex-valued functions on the usual norm, where
1 : a ' co,
LO(H)
M. with
designating the Banach algebra
of all essentially bounded measurable functions on
Two functions on a
M.
measure space agree nearly everywhere (n. e.) if on every measurable set of finite measure, they agree a. e., and a set in a measure space consists of nearly all points of the space if the intersection of its complement with any measurable set of finite measure has measure zero. compact group, 3.
denotes
L a(G)
L a (G, m), where
m
If
G
is a locally
is Haar measure on
G.
Decomposition of a state relative to a commutative algebra.
We show in this section that any state of a C*-algebra
a.
can be repre-
sented as at: integral of more elementary states, over a measure space built on the spectrum of a given commutative W*-algebra in Q'.
By virtue of the
known correspondence between states and representations of
C*-algebras
this shows, roughly speaking, that every cyclic representation (and hence every representation) of
a is a kind of direct integral of more ele-
mentary representations, in such a way that the integrals of the elementary representation spaces are the invariant subspaces of the original representation space.
Thus the present section could be described as an investi-
gation of the decomposition of a representation relative to a Boolean algebra of invariant subspaces. In a later section the present decomposition, which involves no
measurability problems, as states are numerical-valued functions, is used to treat direct integrals of Hilbert spaces (where the dimensionality may vary from point to point) and of operator algebras (which vary similarly),
and thereby we avoid the measurability complications inherent in a direct attack on such integrals. hypotheses regarding
0.
It will also be shown later that under suitable and the commutative algebra in question, the ele-
mentary representations which occur are almost everywhere irreducible. In view of the correspondence between states and representations,
I. E. Segal
6
it suffices to consider states W of the form u)(T) = (Tz, z), where is a cyclic vector for
z
We mention also, as is pertinent to the decom-
CL.
position of an algebra of the form
C', where
C is a commutative W*-
algebra, that it is known that for any such algebra there exists a family
(the operators in) if
C such that the contraction of
of mutually disjoint projections in
{P,,}
1
to
C'
P'N'
has a cyclic vector; and that, moreover,
a it-
is separable, then there always exists a cyclic vector for
self. THEOREM 1.
C be a commutative W*-algebra on a Hilbert
Let
cyclic vector for Q .
z
be a normalized
Then there exists a weekly continuous
on the spectrum r of
1' -,. W,
Ct, and let
a be a Ce-algebra in
space i+ , let
maa
C to the conjugate space of
such that: 1) for any X E Q and Se C,
a perfect measure µ on
(SXz, z) _ /r S( 1() u)Y(X) dt( I ), where the mapping S -> morphism of
0.
is an iso-
C onto the algebra of all complex-valued continuous functions
on r j 2) 0y in case
Q. and
is almost everywhere (relative to
(r,µ))
a state, and
contains the identity operator, everywhere a state.
The proof is based on a series of lemmas, mostly of a measuretheoretic character.
IFMMA 1.1. Let let
(r , r.) be a regular compact measure space, and
be a continuous linear functional on. C(r ).
regular countably-additive set function on r
Then if V
is the
corresponding to W (i.e.
dv('a') for feC(r)), then for any Borel subset B in Yr(f) - I f( r the variation of v over B Is
L.U.B.fEC(T'),
11f 11
1
I
B f( i)dv( 1)
1
We recall the definition of the variation of V , which is a numer-
ical function on the Borel subsets of r
denoted by Var V:
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
(Var v ) (.$) = L.U.B.
where
I
{QS}
collection of mutually disjoint Borel subsets of
7
is an arbitrary finite B Now if f(1)d
with fE C( M, is approximated by sums Ek f( I(k),M( 16 k), then as the absolute value of this sum is bounded, when
11f 11
5 1, by Y:k(6k)
it follows that (Varp)($) t L.U.B. [fsC(j'), IIfII- 1]
I
On the other hand, in a regular locally compact measure space, it is plain can be defined by the equation (Var,,u)(D) = L.U.B.
that
(K1}
where
disjoint compact subsets of number, let
Kit ---, K
n
is an arbitrary finite collection of mutually B .
Now let
E
be an arbitrary positive
be mutually disjoint compact subsets of
l3
such that (Var,A)(B) fF-i t,( K1)I + g , let -ni (i = 1, ---, n)
be
mutually disjoint open subsets of P such that n 1 D Ki , and with
(Varp)(fl1 - K1) < En-1 , and let fi be an element of C(r) which is
1
on
Kit
outside of a it and has values between
0
elsewhere. Setting f(7() =E: 1f1( /)sgnp()(1), then
0
and
IIf 11 _ 1
1
and
Z f( i( )d'p( 7') =Z:, /K1 fai)egnj(K1) dr( 2'+ F- I/81^(n 1
f
Now
K1)
f1(i()sgnj( K1)d)-t( i) .
K1 f1( 1() sgnu( K1) dt ( 7() _ I,)A( K1) I
and
K)i
I/)3^(S2 i
f1(Y()sgnj(K1) dp(Y)I = (Var,M)(fl1 - K1)
by the inequality obtained at the beginning of the proof.
=i
1/u(K) I
(Var/-,)(D)
:
/B
I
Z()
It results that
I + E , and hence that
f( 7l) d/a (') I + 2E, which shows that (Var/`)($)
L.J.B.fEC(r), IIfII
=1
I $ f( /) dy( /)I.
I. E. Segal
8
LEMM 1.2. and c-'
Let
be a compact Hausdorff space, and let
I
be continuous linear functionals on
such that
C (T')
real and non-negative on non-negative functions, and
for all
fa
and some fixed
(x
countably-additive set functions on then
.
If p and
j'
.p'
L.
j,'(ISI)
Ic-'(f)(
are the regular
o
corresponding to p'
is absolutes-2-'T continuous with respect to
a'
,of
and c',
Moreover, if in
JO.
accordance with the Radon-Nikodym theorem we set a(B) = /k( ')dyo( a' ), where B
Is an arbitrary Borel subset of T' , and
function, then Let
Ik(s')I ` CL
8
+ Var or.
k
almost everywhere with respect to p.
be a compact subset of
on which p vanishes and let
F'
Then A is a regular measure on r
exists a sequence (f' n)
of open sets in
nn= nn+l ' and T(Qn) -> X(I). with values between
0
and
is a p-integrable
such that_ n= )6 ,
1
fn be an element of C (TI)
Now let
1, which is
and hence there
on A
1
and
0
outside of
Sln .
If (4
fn(
-> ."xo(i ) a.e. with respect to ?. , and hence a.e. with respect
to JO
and Var a' also.
is the characteristic function of A , it results that
limnA(l') dc(Y) a limsupn
I
and
'a-(A) 1
= "Mn Ic-'(fn)I f c(limsupnIj(fn)I =
ffn( 1) dp( I) I = ap(a) = 0.
Thus let
By the Lebesgue convergence theorem, o'(,6)
G' vanishes on any compact set on which JO vanishes.
b be an arbitrary Borel subset of F for which p(13) = 0.
regularity of G there exists a sequence
such that
f K1}
Now By the
of compact subsets of
KiC B and C( K i ) -> c'( B). But C'( K1) = 0 as of K1)
p(B) = 0, so C(B) = 0.
It remains to show that Ik( f )I Lemma I.I. (Varc')(B) = L.U.B.fEC(F), Fixing 13 , let fk n}
0. a.e. with respect to
j o.
By
5 1 I/f(Y)djo(%()I. and (n n} be sequences of compact and open subIIfnn
sets of T' , respectively, such that Kn _
and x(fn - Kn) --> 0.
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
is a continuous function on F which is
Then if h
n
side of (, and between with respect to
0
and
It and we have
If
1 I
elsewhere,
f( 4)hn(7()
on
1
Kn,
out-
0
hn(1) -> X B( ') a.e. do'( t() j
do (a()I = Ia-'(fhn)I
->a'If(I
Up'(IfhnI) = a 'If( a')I n( I') dp( zl) follows that if fE C (T')
9
d,/O(
It
/Bf( a') dG (7() I f 1, then a jO (B) . Now it is immediate from a well-
and
II f II
I
up(B ) , and hence (Var o') ( B ) known result that (Varc')(B) _ /Ik(I')Id jo (z(), and setting PE _ [ a'I Ik(e) I = OX + E ), where C is a positive number, it results that ( of + E) (Var C) (PE) p (PE ). on the other hand, by the preceding
(VarG-)(PE) O
oc] =
argument, ao
U n-l Pa
1
'
it follows that
LEMMA 1.3.
Let
Ik(z')I
(r',Jo)
the following properties: 1) if
Cr
a.e. with respect to J
be a compact regular measure space with {fn!
is a monotone decreasing sequence of
non-negative continuous functions on r , then there exists a greatest lower bound
ff(1)
f
of the sequence in
and
limn ffn( 1') djD( {) -
d p(-'); 2) the measure of any nonvold open set is positive.
Then
is a perfect measure space. Let
{fn}
K
be a compact subset of
n I
Then there exists a sequence
of non-negative-valued continuous functions on r such that
xK , fn( ',) -> ?(K( 7() a.e. with respect to p , fn+l( /), where x x
and
fn
fn( I( )
is the characteristic function of K (such a sequence
is readily constructed by induction, using the regular character of
f be the G.L.B. of the fn in R (r,) . As fn( 1) -> XK( 7' ) a.e., f(i() 1 )CK( 7() a.e. On the other hand, f( 1() djo( a() = limnfn( 9) dp( i-) = /XK( 1) d jo( 2/), so that /( -k( e ) - f( e)) d p ( 7 ( ) = 0, from which I t follows without difficulty Now let
10
that
I. E. Segal
"l'K( 2() = f( a') a.e. Now let
be any Borel subset of
13
and let
1'
be a mono-
{ K i}
B such that jo ( Ki)
tone increasing sequence of compact subsets of
>
p(B). Let fiE 11(P) be such that fi( i') = xKi ( i() a.e. Now it is easy to see from the fact that a nonvold open subset of P
It follows that as
measure that the complement of a null set is dense.
(fi(
)2 = (It Ki
f1(i'), so that
I < J , then
(7f ) )2 :
a.e. so that
Kj
( I) = fi( 20 a.e., for all ' (fi( Y) )2
Is everywhere either
fi( i')
xKi
xKi
has positive
Similarly, if
0.
or
1
= X K i , and it results that fi( Z' )f'( 2j) = fi(1' )
fif' = fl.
Hence the sequence
is monotone increasing,
{fit
and applying the first condition in the hypothesis of the lemma to the sequence
{xr- fi}
,
it follows that the
{fi}
have a L.U.B.
f
in
J( i)
and that J'1( 7() duo( a') -> dp( ii). Since f( i) f1( 2( ), f( I() : -X CKi (1) a.e., which implies that f( 7T) 1 'X U1 K
() 1
a.e., or f(1)
X ( 1) a.e. On the other hand, A Z() djo( j)
_
i) d10('1 = limn( Kn) _ p(B) - AB( ') duo( 1), and it f( i) = xg(d) a.e.
follows as in the preceding paragraph that Next let
f
prove the existence of an element of
which is equal a.e. to
C (P)
Is clearly sufficient to consider the case in which negative.
If
Is such, say
f
) ,
then putting
If( a') - fn(a() I
2n
) > f (I)
is a measurable set, and if
M i ,n
istic function of M i n
xi n
0 _ f( i ) : Ot
a (1 -
M i, n to be the set 2,
.
(I ,p).
be a bounded measurable function on
and if
f1 n
Pn = oc(1 i=0 a.e. and also
1
in ,
it
, then defining
)]
(i
0, 1,
is the character-
is in 1(P) and equal a.e. to
I )f i,n
2
I Pn+l(
complement of a null set is dense, and as
P
for
)C
f
is-real and non-
f
a (1
To
f nE R ( P ) , and
) - fn( a!)+ .
fn+1 - fn
is in
71
a.e. As the
C(P), the
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I last inequality holds for all
is uniformly convergent, so that element of
C(P)
elements of
equal a.e. to
1(fn+l(Z()-fn(d))
fl, an
converges uniformly, say to
fn
It is plain that
1 (P ).
no
It follows that
a".
11
fl = f
a.e.
If there were two
f, then we would have a continuous
function (their difference) equal a.e. to zero, and by an earlier remark zero everlwhere. LEMMA 1.4.
With the notation of the theorem, let µ be the
Then
T( tr) dpt(7' ) for T C C.
such that (Tz, z)
regular measure on
is e perfect measure space.
( P, 1A)
T
Observe first that the measure of a nonvoid open subset of
is a nonvoid open set of measure zero, let K
For if
positive.
any nonvoid compact subset, and let 1
on K ,
vanishes outside ,L , so that F
is the operator in
jHFzIf2, and hence have
f
be an element of 1(T')
outside fl , and between
0
I.
1
and
Now if
X
[ XzI X6a ]
Now
1
elsewhere.
Jf(6 ) )2 d),L(/ ) = 0.
C which corresponds to
Fz = 0.
XFz = 0 = FXz.
0
be
which is Then
f2
It follows that if
f, then
(F2z, z) - 0 =
is an arbitrary element of Q , we is dense in 1+, and thus
F
vanishes on a dense set; being continuous, it must vanish Identically. Hence
f = 0, a contradiction.
It remains only to show that the hypothesis of Lemma 1.3 regarding monotone sequences of functions in
1 (J')
is valid.
Let
{fn}
tone decreasing sequence of non-negative continuous functions on let
Fn
be the operator in
algebraically isomorphic to its subring
If
and we have
Fn
e which corresponds to
fn.
As
, and
C is
C(r) (in an adjoint-preserving fashion),
of self-adjoint elements is order-isomorphic to Fn+l ,
be a mono-
Fn ! 0.
Obviously tho
F.
1Q(r'),
commute with each
other and by a known result in the theory of Hilbert space, the strong
limit of the seqience {Fn} exists, say limn Fix = Fx for x E 14. If
12
f
I. E. Segal
C(P)
is the element of
Fn
F
we have
fn 1 f.
Fn
H
for all
n, where
This would imply that
corresponding to
Now if
F, from the inequality
and
h e 1'(J')
fn ! h
n, then
is the element of 1 corresponding to
H
(Fnx, x)
for
(Hx, x)
x ei+, and taking
both sides of this inequality yields the inequality This means that
for all
F ! H, so that
f '- h, and hence
h.
limn
on
(Fx, x) = (Hx, x). is the O.L.B. of the
f
fn. Moreover, we have limn An (2() d,p(') = limn (Fnz, z) = (Fz, Z)
X( I) dt (a') . LEMMA 1.5. operator
S, then
If
Ic)(S)I
We explain that S
is a state of a C*-algebra containing a normal
a)
is the absolute-value function, applied to
ISI
by the usual operational calculus.
S
The C*-algebra generated by
is
iscmorphic to the algebra of all continuous complex-valued functions
f
var.ishing at infinity on some locally compact Hausdorff space
Riesz-Markoff theorem, algebra,
cJ (T) =
being the corresponding element of
t(5 )
f
a regular measure on corresponding to
Iw(s)I
t(g ) d or (A ), for all
Putting
.
S, we have w (S)
for
in that o
being
C(,r,)
and hence
We first prove the theorem for the case in
defined by the equation c (X) = (Xz, z ) ,
function of
By the
=w(Isl)
laOI
PROOF OF THEOREM.
f a(E ) d c(g )
Q contains the identity operator
which
T
and
for the element of
a
.
S E C for fixed
I 1 (SX) I = II SX II = II S11
I.
Let
X60 .
cJ
Then
be the state of Q -)(SX), as a
X e Q , is clearly linear, and it is bounded
II XII .
This linear functional induces, via
the correspondence between C and C (I')
a bounded linear functional on
c(r), and by the known form for such a linear functional we have w (SX) =
f
S( e) ds.t (1), for some unique eountably-additive regular
set function on r' , Au X. We observe next that
Iw (SX) I . a u)( ISI )
for
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
a=
X E Q and S e C , where being SA.
(j = 1, 2),
and
X3
S
( W (SX1) I
and
Iu)(SX')I f '(1SXf1).
X2
I W (SX2) 1;
S, and their adjoints mutually commute,
is normal so by Lemma 1.5 generated by
xl
+ 11 X2 a , X = x1 + i12
I1 (SX) I = l a)(SXI) + i ' (SX2) I
For X
since the
11 X1 II
13
X'S
Now the Ce-algebra
is algebraically isomorphic to
for
C (_:E' j)
some locally compact Hausdorff space ._, ', and it is easy to deduce from this that
ISXi I - ISI
Ix iI
II XjII Isl.
It follows that
(uJ(SX')I
'(IIX1 11 ISI), and the stated inequality follows.
Hence Lemma 1.2 applies and asserts that ^X is absolutely con/.A - /"I, and that if we write (by the Radon-Niko-
tinuous with respect to
dym theorem) ,AX(B) - X kX(' ) dy( i ), subset of r , then
ous function
is essentially bounded with respect to. Now
kX
is perfect and hence there exists a unique continu-
(P,p.)
by Lemma 1.4,
B being an arbitrary Borel
on r which coincides a.e. with respect to /
kX(7()
1.( as the functional on Q given by the equation cJ'(X) = kX( i( ), then for each X6 Q., c.)' (X) is a continuous function of 2( . Now as u) (SX) is linear in X for fixed S, ,.UX is a linear function of X, and it follows that k'X+Y ( 1) = kX(7() + kI( ') for any X and Y in a . Hence kX'+Y (?r) a.e. on ( I Defining
with
kX(1() + k4(d)
a.e., but as both sides of this equation represent con-
tinuous functions of
, by a familiar argument we have equality every-
In a similar fashion it follows that for any complex number /3
where.
7r) = /3k4 ( -( ,
7'
cJ r
17
)
for all a'E P . These results mean that for each
is a linear functional on a .
SA nonegative operators is a
muting
a positive linear functional, of /-LX
is then a measure.
nonegative, so u1r (X).
kX(1 )
To show that
,
SA
Since the product of two comnonegative operator,
S, for fixed SA non-negative
It follows that for such
X,
kX(i')
L) (SX)
X, and hence is a.e.
is everywhere nonegative, and the same is true of
Wy
is for each
is
{ a state of Q it remains
14
I. E. Segal
only to show that
wy(I) = 1
dy.(1'), As
S
(Sz, z) =S (I)d,M(I)
ranges over all continuous, and so,as
S(. )
is perfect, all bounded measurable, complex-valued functions on
r, and therefore tion of
o' (I) = 1
a.e., but as
we have equality everywhere.
a'
I F_ Q , for the continuity of o -j
case
Now
a' .
(1 - )y(I)} d,(ao) = 0, for all Se C.
so
ranges over C ,
(r,y,)
for all
in the conjugate apace of
is a continuous func-
a) 1( (I)
This completes the proof for the
1' with values
as a function of
is equivalent to the continuity of
U.
(A)
X)
as a function of e for all X E Q. Now suppose that
a
is not necessarily in Q , and let
I
be 1
the algebra obtained by adjoining
I
that Q 1
Q
Then
is a Ce-algebra.
proved there exists a continuous map of
Q1
to Q ; it is not difficult to see 1 C Ct
and by what has just been
Y-> W4
T on
topologized by the
(i.e.? the collection of all states of Q 1
X e Q1
weak topology on its conjugate space) such that for
.(SXz, z)
%
traction
aY,
of
a( -> o)7(
to
and Y
is continuous on P to the conjugate space of Q.
Y
in Q ,
a (cf. [e]), i.e.y
(fi
r
1
Y -> X for all X e Q. Then for any X
e a, and
(Vy, Xz, Vy. Yz) -> (Xz, Yz).
(V,u x, VN y) -> (x, y)
Thus the equation
holds for a dense set of
z
and
y
in 1*, and
is bounded, it must consequently hold for all a and y in /'r' .
as
In particular
in
3 E C,
a is a.e. a state, for it is clear that the
Let {Yµ} be an "approximate identity" for
for all JA ,
and
S( z() 4 (X) d p ( a') . It remains only to show that the con-
&5j
mapping
to the state space
I
Q
(
(Vµ z, V,
z) --> (z, z), and so there exists a sequence fUn}
a subsequence of the
(Unz, Unz) -> 1.
V },,,)
such that
1
and
It follows from an equation above that
,/J, Oy(I r d j& ( a(Y -> 1, or
(1
- u) j (UnU)) d}+ ( 6' ) --> 0.
1 - u)d(U*Un) : 0, the sequence of functions of verges to zero in
n Un N
L1(r, µ
).
If
`Uni)
As
-)a((UnUn)J
is a subsequence such that
con-
DECOMPOSITIONS OF OPERATOR ALOEBRAS. I
1 - We (Uni Uni ) -> 0 1
a.e. on
a.e., which shows that 4.
(-(
( r,, ), then we have
15 L.U.B.i u)j(Un Un )
i
Is a state a.e.
Direct integrals of Hilbert spaces.
In this section we define
and treat direct integrals of Hilbert spaces, and show that every state decomposition such as that of the preceding section gives rise to this kind of direct integral.
In this way an arbitrary
a- can be de-
C*-algebra
composed with respect to any commutative W*-algebra in
a
(or alter-
natively, with respect to any Boolean algebra of closed invariant subspaces).
Our definition of direct integral is somewhat similar to that given by von Neumann [16], but we find it necessary to consider two types of integrals, a "strong" and a "weak" type, whose relationship is analogous to that of strong and weak integrals of vector-valued functions. Definition 4.1.
Let
suppose that for each point
bolically 1+ R
to
p E R
there is a Hilbert apace
is called a direct integral of the
Hilbert space
on
Up C
(x(p), y(p))
(x(p), y(p)) dr(p), and
A
1.1p.
Y
over
(symx(p)
x(p)c 7 p, and with the following prop-
R 1)p , such that
z = Otx + Py, then
-/+p
M, and
there is a function
p dr(p) ) if for each x e l6c
erties (1) and either 2a) or 2b)): 1) if
x
y
and
is integrable on
z(p) =ocx(p) + p y(p)
are in M,
7'j-
and if
(x, y) _
for almost all
p E R;
for all p, if (x(p), z(p)) is measurable for all x fP (z(p), z(p)) is integrable on M, then there exists an element
z(p)e H
2) if and if of
be a measure space
(R, R , r)
',
z'
)4 such that almost everywhere on
a)
z'(p) = z(p)
b)
(z'p), x(p)) = (z(p), x(p))
M, or
almost everywhere on
M,
x r.*.
The integral is called strong or weak according as 2a) or 2b) holds. The function
x(p)
is called the decomposition of
x, and we use the following
16
I. E. Segal
notation for this:
x = JR x(p) dr (p).
A linear operator
T
on
7+ is said to be decomposable with re-
spect to the yppreceding direct integral if there is a function to
UP
R
on
P
tors on
-W., such that
for all
x
and
y
in
T(p)E p for all
the decomposition of T(p) dr(p).
p
(T(p) x(p), y(p))
1',
(T(p) x(p), y(p)) dr(p) = (Tx, y).
T
T(p)
is the collection of all bounded linear opera-
E R /cep, where
and with the property that is integrable on
The function
M
and
is then called
T(p)
T, and we symbolize this situation by the notation If
T(p)
T
is almost everywhere a scalar operator,
is called diagonalizable.
The basic theorem of this section asserts that a state decomposition such as that of the preceding section induces a decomposition of the
Hilbert space as a direct integral, in which every element of
Q, is de-
composable, and in which the diagonalizable elements are exactly those in
C. Before giving a precise statement of this theorem we make two remarks. First, It is not difficult to show that in case H'
is separable, a weak di-
rect integral becomes an essentially equivalent strong one when the 74'p
replaced by appropriate closed linear subspaces of themselves.
are
Second, the
analog, of condition 2b for direct integrals of spaces, in the case of direct Integration of operators, is valid without further assumption: if for all
T(p)E Z3
p, if
M T(p)II
is essentially bounded on
M, and if
P
the integral
fR
(T(p) x(p), y(p)) dr(p)
exists for all
x
and
y
*, then there exists an (obviously unique) bounded linear operator such that (1'x, y) _ mT(p) x(p), y(p)) dr(p) For setting
for all
x
and
y
x and conjugate-linear in
T
in 1+ .
Q(x, y) =J `T(p) x(p), y(p)) dr(p), it is clear that Q
conjugate-bilinear (linear in
in
is
y), and that,
setting Cl = ess supp e,R u T(p{) JJ , IQ(x, y) J = J'(T(p) x(p), y(p)) I dr(p) 1 1/2 fOcII x(p) l A y(p) JI dr(p) = Qj / x(p)11' dr(p)) t J Y(P) II2 dr(P))f = Ocq xHI
11 yII, so q is bounded.
It follows readily from the Rieaz rep-
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I resentation theorem for linear functionals on
-14
17
that an operator
T
with the stated property exists. THEOREM 2. M = (R, 7f ,
0, C ,
Let
and
be as in Theorem 1, and let
z
C is alge-
be a measure space with the properties: 1)
r)
braically isomorphic (in a fashion taking ad joints into complex conjugates) M, the ele-
with the algebra of all complex valued bounded measurable o n ment
S
of C corresponding to the function
there is a state
3)
of Q , and for
cJ
S ( . ) ;
T E Q,
for each
2)
p E R
is measurable on
o) (T)
M;
for TE0_ andpSe C, (STz, z) =/a C.)p(T) S(p) dr(p).
p,
Then if /
and
,
p
are the representation space,
z
P
p
representation of Q , canonical map of a into respectively, associated with
we have weakly, and in case Q is
u)p
1 _ 1# dr(p)
separable in the uniform topology, strongly,
way that for U E Q,
fR rp(U) dr(p), and
II =
and wave function,
in such
p(U) dr(p).
Uz
Every operator decomposable with respect to this direct integral is in C
C.
and an operator is diagonalizable if and only if it is in
We begin by defining R
the space of functions on
(more precisely, a residue class of
x(p)
Up 1+ , with x(p)E 7=
to
modulo the linear subspace of functions a.e. zero), for by the equation suppose that
U - V, so
To see that
x(p) = 7tp(U).
Uz - Vz, with U
and
g c)p(W*W) dr(p) - 0. Now equation implies c.)p(W*W) = 0 a.e., or
Q
in
.
1p(W) - 0
() (W*W)
a.e.
a.e. on
0
of the form Uz,
Then Wz - 0, where
for all
This means that
R, and hence
x(.)
p,
is single-valued,
(Wz, Wz) = 0, but by 3) in the hypothesis,
=
0
V
x(p)
x
for all
W -
(Wz, Wz) - (WIIWz, z)
p, so that the last ( rip(e),
rlp(W)) -
18 unique (modulo
the subspace mentioned).
Now let
Q
x
be arbitrary in
such that Unz -> x.
Then
1f and let {Un} be a sequence in
a Unz - Umz I -> 0 as m, n -> co , and
I.E. Segal
18
f
p((Un - Um)'*(Un - Um)) dr(p) IJJnz - Umz II2 = ((Un - Um)*(Un - Um)z,a) = = f I'7p(J1n) - Y t p(Um) IP dr(p) ---), 0. '.1e now apply the procedure utilized in the proof of the "iesz-Fischer theorem to the selection of a subsequence of
flip(Un)I whose limit exists a. e. and defines the function which we Let
shall designate as x(p).
fn.)
be a subsequence of the positive intedr(p) < 8-i for n
Rers such that ni+l > ni, and with fIlp(Un) and m greater than ni.
> C is, for
The set of p's for which II Yp(Uni) -
p(Uni+l)II
C> 0. clearly of measure less than (5-2 8-1. and taking
C = 2-i, it results that IIu1p(Uni) - 7(p(Uni+1)p < 2-i except on a set of measure less than 2-1.
<
2-1
Therefore the inequalities II7p(Uni)
- j p(Uni+l)II
hold simultaneously for all i > j except on a set D, of measure less
than Zi>j
2-1
- 2-3+1.
It follows that the series
is uniformly cmvergent for p 4 Dj, and
Z11 f tb(Uni) - 'Zp(Uni+l)J hence that lima Yp(Uni)
exists uniformly for p 4 D3. Putting x(p) for
that the
that limit, it is clear from the fact that r(D') k 0 as limit exists a. o. so that x(p) is defined a. e.
(and nay be defined arbi-
trarily on the null set on which the limit fails to exist).
From the equation /IIifp(Ur) - fp(Unj)II2dr(p) < 8-i for m and ni > nip it results that /R_DkII 11 p(U.) - Y,(U,,J) Ipdr(p) < 8-1 for m and nj greater than ni, and for any k. Now Yp(Un') converges uniformly to _II Ytp(Um) - x(P) II2dr(p) < 8-i for m > nis x(p) as j -- co , on R-Dk, so /'' and for any k. Letting k , oo , it follows that RI( Yp(Um) - x(p) 11 dr(p) < 8-i if m > ni so JAII fp(j) - x(p) II2 dr(p)---3, 0 as m --moo.
f
We show next that the function quence
{U.1
utilized.
Unz -> x, and let
z'(p)
fashion as that in which
x(p)
is independent of the se-
Suppose that Cu} nis a sequence in Q such that be a function obtained from x(p)
{Un1
was obtained from (Un) Then
fR II'?p(U' ) - x' (p) II2 dr(p) --> 0 as m -> oo. Thus both
in the same
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I {II hrp(Um) - x(p)JI2}
Now
f
that
and
19
[n (P(U'm) - x'(P)II2) converge to zero
II p(Um) - P(U'm) A
{JIB P(U )
2
= I IImz - Umz JJ 2 -> 0 as m --> co , 80
- I p(IIm)JI2J also converges to zero in
a common subsequence
{mi}
Ll(M).
II x(p) - x'(p)1 <
II x(p) - 3 p(Umi) II + Hlp(Umi) - rjp(IImi) II + H y (IIi) - x' (P) II 11X(p) - x'(P)II = 0 e.g., i.e.p x(p) = x'(P) R.G. If y is any element in Ilk and if V n z -> be a subsequence such that
{Vni}
we have
Choosing
such that the corresponding subsequences of all
three sequences converge a.e., we have
let
L1(M).
dr(1p)
(x(p), y(p)) = limi,j
so that
y with Vn E Q,
7(p(VmI) -> y(p)
a.e.
Then a.e.
(I P(Un ), j p( m )), and i
which is a measurable function of P.
y{p(Vm )) = W P(Vm Un i
Thus
f
(x(p), y(p))
is a.e. the limit of a sequence of measurable functions,
and is hence itself measurable. and has
(x, y)
to show that
Moreover,
for its integral.
(x(p), y(p))
Is the limit in
functions
(-p(Un1yrp(Vm3)), as
( I p(Un i ),
1
p(Vm
(x(p), y(p))
L1(M)
i, 1 -> on.
Now
(x(p), y(P)) -
i
equation is bounded by )II
of the Integrable
)) _ f(-(p), y(p)) - (j p(Un ), y(p))J +
1p(UnI), y(p)) - ( VUn1), 1p(Vm ))J,
p(Un
is integrable
To show the integrability, it suffices
II x(p) -
P(Un
II
so that the left side of this
II Y(P) II + II Jp(Uni) II II y(P) -
Hence, applying Schwarz' inequality,
3
f(x(P), y(P)) - (jp(UnI), yp(Vmj)I dr(p)
f 4I x(P) - )(p(Uni) 11 2 dr(p)
I Y(p) 11 2 dr(P) 1
1/2 +
20
I. E. Segal
p(U) + jp(Um'),
Y(P) II jp(U0
so
II Y(P) II f
(
II y(P) - rrp(UnJ) II2dr(P) r 1/2
that
II y(p) -
y(p)
Now
Yrp(Umi) It
+
f1/2,
By Minkowski' s inequality, fl, y(p) II2 dr(p) f
) II 1
j
1/2 .
II y(P) - 1p(Un )II2 dr(p)j
TP(Uni) II2 d1(p)
[
is integrable.
II y(p)II2
+
Also,
which shows
II 1p(Un1)II2 dr(p)
Un z, z) = II UnI Z II2 , which is bounded as I -> ni /1) p(U*niUni ) dr(p) = (U* i oD.
It is easy to conclude that
dr(p) -> 0
as
j -> m .
1,
and that its integral is
'(x(p),
y(p)) - ( Yip(Uni),
This shows that
lima,
(V* Un
li ma,)
)I
is integrable
(x(p), y(p))
j
i
1p(Vm
p(Vm )) dr(p) f
dr(p) - limi,' (Uniz, Vm z) = (x, y).
That
x(.)
is
f")p a linear function of
is clear from the fact that OL jp(Un ) +
x
1
on the one hand converges a.e. as i -> oo to q x(p) +Py(p),
VV (Vmf )
and on the other, equals
7i
Q U.
+ /3 Vmf ),
of which a subsequence con-
(for (QUn + 1Vm )z -> Lkx + py as
(ocx + Ay)(p)
verges a.e. to verges
I
1
i,j->CD ). Before concluding the proof that
the /
/- is the direct Integral of
we consider the decomposition of operators.
Q.
arbitrary in
uf( Y/p(TU),
Then
(TUz, Vz) s (V*TUz, z) =
P(T) j p(U),
dr(p) _
be
fp(V*TU) dr(p) _
yrp(V)) dr(p).
This shows
y(p))
'
that the equation
T, U, and V
Let
Fp(T) x(p),
(Tx, y) _
dr(p)
holds for
x
and
`
y
of the forms
xn -> x, where (Txn, y) -
`
x = Uz, y = Vz. (xn}
Now let
is a sequence in
p(T) 7n(p), y(p)) dr(p)
/fix
Qa. and
be arbitrary in /V Then if
and let
y = Vz, we have
DECOMPOSITIONS OF OPE1ATON ALGZ:BRAS. I
f
99 p(T) n(P), y(p)) dr(p)
f(
-
21
(T) x(P), y(p)) dr(P)I =
99
p
I( (pp(T)(xn(P) - x(p)), y(p)) I f Il(pp(T) II
II xn(p) - x(p) JI
IJ 9p(T) JJ
, and
N y(p) IJ
II T JI , as this is true for any representation, so
'_
D
g
if xn(p) - x(p) II
IIr`TJJ
Now
say.
I f T p(T) xn(p) - x(p), y(p)) dr(p)J, = D
JJ T11 (f xn(P) - x(P) N dr(P)}
dr(P)
fI y(P) JI
1/2
l/" y(p) 1I2dr(p) 11/2, which has the limit zero as n -> co. On the (Txn, y) -> (Tx, y), so in this case we like-
other hand, it is plain that (Tx, y) =
wise have trary in
mating
, let
f
fynI
?p(T) x(p), y(p)) dr(p).
Oz
be a sequence in
Next let
be arbi-
y
yn -> y.
with
Then esti-
p(T) x(p), yn(P)) dr(p) - J( 7p(T) x(p), y(p)) dr(P)I
f
as in
the case of a similar expression above, it results that the present expression has the limit zero as (Tx, y)
formula for
T E Q and
x
1S(p) (x(p), y(p)) dr(p), for all
to
x
and
lx.}
and to
{yn}
y, then
Now if
x
in # .
and
For x = Uz
and
follows
are arbitrary in,
y
which converge respectively
are sequences in Qz
(STx, y) = limn (ST"., yn)
limnjS(p) ((pp(T) xn(p), yn(p)) dr(p). tion of
(STx, y)
(STx, y) = S(p) (f p(T) x(p), y(p)) dr(p)
trivially from the hypothesis. and if
in 1/.
y
(Sx, y) = y
and
and
We shall show that
S F_ C.
and that
S(p) (Tp(T) x(p), y(p)) dr(p)
y = Vz, the equation
x
is valid for arbitrary
Now suppose that =
It follows that the preceding
n -> co.
Now
S(p)
is bounded as a func-
p, and this observation together with an argument used above in a
similar situation shows that
J(p)
f(p) (9P(T) x(p), y(p)) dr(p), as
U 6 Q and
y E 114, and putting
Q, we have (SW,
( 9p(T) xn(p), yn(p)) dr(p) -->
n -> co.
fW
Again, if x = Uz
for an approximate identity for (S
x, y) _ ZS (P) (5p(W',L. ) x(P), y(p)) dr(p)
= f(p) (99p(W, U) z(p), y(p)) dr(P).
with
Since
r p(WL
Uz, y)
U) -> 92p(U)
uniformly, relative to ,(,t, i.e., JJ 9p(Wp U) - 9 p(U) II -> 0 uniformly on
R, so that a sequence
01i}
exists such that P p(W
U) -> (JOP(U) 1
22
I. S. Segal
uniformly relative to
1, the last expression converges to
/8(p) ( cpp(U) z(p), y(p)) dr(p) are both arbitrary in
y
and
converges to
x.
Then
/S(p) (x(p), y(p)) dr(p).
11', let
Now, if
be a sequence in Q z
(xn}
x
which
(Sx, y) = limn (Sxn, y) = l1mn
/8(p) (xn(p), y(p)) dr(p), which last expression Is readily seen to equal ,/ S(p) (x(p), y(p)) dr(p).
We observe finally that
(Sx)(p) = S(p) x(p)
J' (Sx) (p) - S(p) x(p),(Sx)(p) - S(p) x(p)) dr(p)
a.ej., for
f{((Sx)(P), (Sx)(p)) - (8(p) x(p), (Sx)(p)) - ((Sx)(p), S(p) X(P)) + (S(p) x(p), S(p) x(p))) dr(p) = (Sx, Sx) - (Sx, Sx) -(S*(Sx), x) + ((S*S)a, x) - 0
(the assumption that the integral exists being justified
by the given expansion of the integrand).
We now conclude the proof that p.
Suppose that
w1(p)
1''
is a function on
is a direct integral of the R
such that
for
w'(p)e.JYP
p e R, (W'(p), w'(p))
able on
M
for all
is integrable on
x E Y. Then
M. and with
(x(p), w'(p)) measur-
Is integrable on
(x(p), w'(p))
M, for
by two applications of Schwarz' inequality we have
f(x(p), w'(p))Idr(P)
x(p)II
II w'(P)II dr(p)
{f I X(p) II2dr(P) f I w' (p) II2dr(P) } 1/2 The same inequality shows that setting
L
L(x) -
(x(p),
w'(p)) dr(p), then
is a continuous linear functional on N. Hence there exists an element
w e IV such that
L(x) _ (x, w).
`x(p), w1(p) - w(p)) dr(p) = 0
S e C and recalling that equation
It is obvious that for all
x e #.
(Sy)(p) = S(p) y(p)
J(p) (y(p), w1(p) - w(p)) = 0.
As
Putting
x = By
a.e., there results the
S
ranges over C ,
ranges over the space of all bounded measurable functions on (y(p), w1(p) - w(p)) - 0
(w(p), y(p))
a.e., i.e..,condition 2b) in the definition of a direct
integral of Hilbert spaces is valid. U1; 1=1,2,...,1 dense, then the
If
rip(U1)
(w'(p), y(p)) _
Q is separable, say with are dense in
S(.)
M. and it
follows that
a.e., or
with
1q- , and as
DECOMPOSITIONS OF OPEhATOR ALGEBRAS. I
(w'(p), jp(Ui)) = (w(p), that
results
w'(p) = w(p)
-)'(p(Ui))
simultaneously for all
gonalizable, then it is, respectively, in is decomposable, so that
x
and
ator on
S
Hp
for
is decomposable or dia-
or
C'
C. Now suppose that
(Tx, y) = JP( T(p) x(p), y(p)) dr(p)
in 1, and some function
y
i, a.e., it
a.e. so that the integral is strong.
It remains to show that if an operator T
T
23
p e R, and with
is arbitrary in C , we have
such that
T(p)
for all
is an oper-
T(p)
11 T(p)fl essentially bounded on
M.
If
(TSx, y) _ JT(p)(Sx)(p), y(p)) dr(p) _
JT(p)S(p)x(p), y(p)) dr(p) _ fT(p)x(p), S(p)y(p)) dr(p) = jT(p)x(p), (S*y)(p)) dr(p) _ (Tx, S*y) _ (STx, Y). Hence
ST = TS
T a C1.
or
It is trivial to show from the fact that C
is isomorphic with the algebra of all bounded measurable functions on that a diagonalizable operator must be in
C
M,
.
The proof of Theorem 2 is thereby concluded, and we have incidentally established the following corollaries.
COROLLARY 2.1.
If S 6 C , T E Q, x E 1, and if x(p) is the de-
composition of
x, then the decomposition of
composition of
Tx
is
COROLLARY 2.2. exists a subsequence
Sx
is
S(p)x(p)
and the de-
Tp(T)x(p). If (ni}
xi a 11-(i = 1, 2, ...) and
xi -3- x, then there
of the positive integers such that
xn (p) -s i
x(p)
a.e.
We close this section by obtaining a result which will be useful in the treatment of separable Hilbert spaces. THEOREM 3.
With the notation of Theorem 2, lat
(in the uniform topology). ators in
a
C2
be separable
Then every strong limit of a sequence of oper-
is decomposable relative to the decomposition of * des-
cribed in Theorem 2.
24
I. E. Segal Let
{Tn}
strongly to an operator [Ui}
Q
be a sequence of operators on T; it must be shown that
is decomposable.
T
be a countable dense subset of Q,, and set
is dense in 14 .
which converges
xi = Uiz; then
(x1}
By Corollary 2.2, there exists a subsequence fn 1,3.1
T ni(p) x1(p)
the integers such that
converges a.e. to
(Tx1Xp).
Let
Of
Next,
,l
there exists a subsequence fni,2} converges a.e. to
(Tx2)(p).
or the
such that
ni,l
Tni 2(p)x2(p)
Proceeding in this fashion by induction, and
employing the Cantor diagonal process, it follows that there exists a subsequence
of the integers such that
(ni}
converges a.e. as
Tni (p)x (p) i
I -> co
to
are, for each ft"}
(Txf)(p).
Now as the
U3
are dense in Q , the
Moreover,
p e R, dense in
is strongly convergent, and hence
is bounded because
II Tn(I
(p)II
II Tn
-Ip(U')
is bounded for
p E R
i
and
I = 1, 2,,...
.
A bounded sequence of operators which converge on a
dense set is strongly convergent, and hence {Tn (p)}
has a.e. a strong
1
limit
T(p).
It is clear that that
(Tx, y) =
is one of the 1'/
x'
and if fx1}
clearly
T(p) x3(p) = (Tx3)(p)
and
is arbitrary in
is a subsequence of the
(Tx;, y) -- (Tx, y)
Now
x3
II T(p) II
(T(p) x(p), y(p))
equal to limi(Tni (p) x(p), y(p))
I
_ff
.
Now if such that
is arbitrary in
x
xJ -> x, then
and on the other hand
fi x;(p) - x(p) II2 dr(p) -> 0. so that (noting that
It follows easily
in the special case in which x
`T(p) x(p), y(p)) dr(p) y
a.e.
limsup1B Tni(p)II
Is measurable on
II TII ,
M. being a.e.
)
f T(p) x3(p), y(p)) dr(p) - J1T(p) x(p), y(p)) dr(p)I =
ji(T(p) x'(p) - x(p), y(p) ) ( dr(p)
II TO
11 x3 (p) - x(p) II II y(p) 11 dr(p)
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
25
if x' - xfl yfi -> 0. It follows that the preceding equation for (Tx, y) holds for arbitrary x and y in 1.
f
II Tii
if
Definition 4.2.
Q
in
such that
If
{Tn(p) _ P
to
p E R
T(p), then
(with respect to
is the strong limit of a sequence {TnJ
T
(T )} n
is called the canonical decomposition of
T(.)
0, and
0 ,
converges strongly for almost all
z).
To justify the preceding definition it should be
Remark 4.1.
shown that the canonical decomposition of
{T'} n
on
T'(p) n
converges strongly to
Q
dense subset of
II Tn(p)
T
T
and that a.e.
Then a.e., for all
T'(p).
Ui
T'(p) -1 P(Ui)1I -> 0.
Now
II Tn(p) lp(Ui) -
II T11 - Tn')U1zii2 =
1p(Ui)II2 dr(p), and so there exists a subsequence {n,}
of the integers such that
7p(UI)II -> 0
If Tnf(p) yrp(US) - Tn,
j -> oo, a.e. simultaneously in
as
By Minkowski's inequality,
I.
H T(p) ip(U1) - T' (p) jp(U1) II = II T(p) jp(Ul) - Tn(p) I p(U1) II II Tn'(p) Ip(U1) - Tn f(p) T' (p) vI (Ui) II
a.e. simultaneously in
I.
1.}-p, It follows that a.e.
Q , if
x
3.1.
h , and if
tions hold a.e.: (a T)(p) - 01-T(p);
II TI (p)
p(U1) II +
and it results that
COROLLA}
in the
which occurs in the proof of the preceding theorem,
7 (Ui) - T(p) -jp(U1) if -> 0, and
(Tn(p) - T'(p))
Suppose now that
is unique.
Q which converges strongly to
is a sequence in
M,
T
If OC
canonical decompositions of
(P(Ui) ..
II T(p) -k(P(Ui) - T' (p) y(p(Uj) II
As for each
p e R, the
71p(Ui)
-0
are dense in
T(p) - T'(p). T
and
U
are strong limits of sequences in
is a complex number, then the following e9ua-
(T + U)(p) = T(p) + U(p); (Tx)(p)
+
T(p)x(p). T
and
U
Here
and
(TU)(p) = T(p)U(p); T(.) x(.)
and and
U(.) (Tx)(.)
are the are the
26
I. E. Segal
decompositions of Lot
U
x
and
Tx, respectively.
{TnJ and LUn }
respectively, and such that
T(p)
and
T + U, and a.e.
and
AI(Tx)(p)
-
T(p) + U(p),
Similarly, it follows
(a.e.).
(ckT)(p) = O'T(p). II(Tx)(p) - T(p)x(p)II
is a.e. equal
On the other hand,
Tn(p)x(p)II2 dr(p) = II Tx - TnxII2 -> 0, and so by Corollary
2.2 there exists a subsequence II
and
converges strongly to
converges strongly to
limn II (Tx)(p) - Tn(p)x(p)II .
to
T
a.e. converge to
{Un(p)}
{Tn + Un }
is arbitrary in 74-,
x
and
(T + U)(p) = T(p) + U(p)
(TU)(p) = T(p)U(p) If
Then
{Tn(p) + Un(p)}
which shows that that
1Tn(p)}
respectively.
U(p)
Q which converge to
be sequences in
(Tx)(p) - Tn (p)x(p)II -> 0
{ni} a.e.
of the integers such that It follows that
II(Tx)(p) -
1
= 0
Tn (p)x(p)II
a.e., i.e., (Tx)(p) = T(p)x(p)
a.e.
1
Remark 4.3.
A slight modification of the proof of Theorem 3 shows
that every strong limit of a sequence of decomposable operators is itself decomposable.
For as
is separable.
If
then if
(x
Q is separable, I = 1, 2, ...}
(x1( 1'); i = 1, 2, ...}
is a subsequence of
{xiJ}
Qz
is separable, so that AP
is a countable dense subset of ]1
is a.e. dense in 74f , for by Corollary 2.2, fxi}
which converges to
U3z, there is a
subsequence of this subsequence whose decomposition function converges a.e. to
3,r(U3), and the
proof is the same.
j,,(U3)
are dense in N'
.
The remainder of the
We note finally that as a weak limit of a sequence of
operators is a strong limit of a sequence of finite linear combinations of the operators (cf. [18]), the set of all decomposable operators is closed in the weak sequential topology, when 5.
Maximal decompositions.
Q is separable. The complete reduction of an algebra
of linear transformations on a finite-dimensional linear space is determined by the selection of a maximal Boolean algebra of invariant subspaces under
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I the algebra. space.
27
Such a selection is likewise possible in the case of Hilbert
Q
In fact, it is not difficult to show that if
is a
Ce-algebra
on a Hilbert space 1/-, and if C is any maximal abelian self-adjoint subalgebra of
Q', then the ranges of the projections in
C constitute (Actual-
a maximal Boolean algebra of closed invariant subspaces under a-.
ly, the Zorn principle shows that such a selection is possible on any linear space, but in the case of Hilbert space, it can be made in the foregoing way, with complementation in the Boolean algebra coinciding with orthogonal complementation.)
The main purpose of this section is to show that If 0
is separable, then the components in the reduction of Q relative to such That
C as in the preceding sections, are a.e. irreducible.
an algebra
are a.e. irreducible under the
is, the
A similar result,
cp p( 0).
P
based on the von Neumann reduction theory, is due to Mautner [6].
In the
next section we apply our result here to obtain a decomposition into factors of an arbitrary We-algebra, similar to that obtained by von Neumann, for "rings" with respect to their centers. THEOREM 4.
With the notations of Theorems 1 and 2, let the
state decomposition hypothesized in Theorem 2 be that obtained in Theorem 1, so
M = (r , }A )
and let
c be maximal abelian in
at.
Then
CPS
is almost everywhere irreducible.
We first prove a lemma on inverses of continuous maps of compact spaces which plays a role somewhat similar to that of a lemma of von Neumann [16, Lemma S] concerning inverses of continuous functions on analytic sets.
LEMMA 4.1.
metric space of
C
Let
f
be a continuous function from a compact
to a compact metric space
C, there exists a Borel function
f-l(y) " E for yEf(E).
g
on
D.
If
f(E)
E to
is an open subset C, such that
g(y)6
I. E. Segal
28
We shall present the proof in stages, first considering the case
E = C, then the case in which E
is closed rather than open, and finally
This arrangement of the proof is not logically necessary,
the general case.
but seems to clarify its structure.
The Lemma is valid in case
SUBLEMMA 4.1.1.
(n = 1, 2,
n # m
be a countable dense set in
{xn}
Let
(we shall assume that
...)
sult is obvious for the finite case).
x
tance from follows:
x', and
to
g1(y) = that
xn
e > 0.
sect.
d(x, x')
denotes the disgl(y)
on
among the
A # B means that the sets xm
if
denote the set of all
Se(x)
We define a function
It is clear that such points
xm
xn
is not finite, as the re-
which has the least index n
S1(xm) # f 1(y), where
that
such that
C C
d(x, x') < e, where
such that
x'e C
points
Let
E = C.
B
and
A
D
as
m such inter-
exist, for otherwise there would
whose distance from the
xm
was
_ 1.
Now
g1
is a
exist points in
C
Borel function.
To show this it suffices, in view of the circumstance that
gl
is (at most) countably-valued with values among the
for any y
g11(xn)
n,
such that
assertion
Up
S1(xn)#f-l(y), and
a)
S1(xs)#f-l(y).
Since the assertion
Clearly b)
f(S (x )). p
Now
is the least m such that
n
e
tinuous image of a compact set,
among m before,
g2(y)
By the compactness of a conis a countable union of compact
f(Se(x))
It results that
as that
xn
gil(xn)
is Borel.
which has the least index
for which S1/2(xm)#f-i(y) ^ S1(g1(y)). g2(y)
f(S1(n))
gll(xn)
hence a countable union of compact subsets.
Next we define
is equivalent to the
is a countable union of closed sunsets, and
S (x)
sets, and so is a Borel set.
is the set of all
gi1(xh)
Si(xm)#f-1(y)
f(S1(xm)), it follows that
y 1
is a Borel set.
xn, to show that
m
By the same argument as
exists, and its values are among the
xn.
Now
g21(xn)
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
29
[yI g2(y) = X'] = Um[yI g2(y) - xn, g1(y) = xm] = Um[yin
is least
[71g1(y) = xm].
p
such that
Since
S1/2(xp) ^ f-1(y) ^ Sl(xm) # 03
is Borel, so is
g1
[ylgl(y) = xm].
[yIn
is least
p
such that
S1/2(xp) ^ f-1(y) - S1(xm)
[yln
is least
p
such that
y E f(S1/2(xp)
f(S
(x ) ^ S (x )) n I 1/2 n
p
We note also that
to be Borel. then
- U
0]
S1(xm))] =
(x ) ^ S (x )), which is easily seen m 1/2 p l
d(g1(y), g2(y)) < 3/2, for if g2(y) =
n
31/2(xn)#S1(g1(y)).
Now we define xn
Moreover,
by induction as follows:
gr
which bs the least index
p
g (y) r
is that
such that
S1/2r_1 (xp) # (f-1(7) ^ Sl/2r_2 (9r-1 (y))). Then clearly y
g- '(X.) _ [YI 8r(Y) = xn] =
mU
[yIn is least p such that
S1/2r-1 (x p)#(f1(y)^ S1/2r-2 (xm)] ^ [ylgr-1(y) = xm].
[ylgr-1(y) ' xn]
Now
is Borel by the induction hypothesis, and the other set
U
involved in the intersection behind
is
[yin
is least
p
such that
m y 6f(S1/2r-1 (x p)^ S1/2r-2 (xm)] = f(S1/2r-1 ( n) ^ S1/2r-2 ( m))
Upm f(Sl/2r-1 (xp)^ S1/2r-2 (xm)), Borel, and hence so is
gr,.
Finally,
(1/2r-2), since a sphere of radius of radius
1/2r-2
around
Now
ga(y)
and hence
{ga(y)}
d(gr(y),
1/2r-1
Thus
g-1(xn)
is
gr-1(y)) <(1/2r-1) +
around
gr(y)
meets a sphere
gr-1(y)
Clearly the series 'K hence the sequence
which is Borel.
ooo_1
d(gr+l (y), gr(y))
is convergent, and
converges uniformly to a Borel function
is of distance less than
1/2n-1
from the closed set
g(y)E f-1(y), concluding the proof of the sublemma.
g.
f 1(y),
30
I. E. Segal
SUbLEMMA 4.1.2. closed subset of such that on
D
With the notation of Lemma 4.1, let
C, and let
go(y)f f-1(y), for
to
C
be a Borel function on
g0
F - f(G)
0
to
Then there exists a Borel function
y EF.
which coincides on
be a
G
with
F
go, and such that
g
g(y) a f`1(y)
for yeD. Let
be dense in
{xn }
We define a function if
y
gI(y)
F, then
which S1(xp)#f 1(y). C,
g1
C - G
D
on
Now
F
K
y e F, and
for
p
is any closed set in
gil(K -G) = g 1(K ^ G), which is 0
and hence a Borel subset of
is borel, it suffices to show that
9i1(K - G)
if
is the least
n
is Borel, for if
gI
g11(K) = g11 (K ^G)" g11(K - G).
a Borel subset of
gl(y) = go(y)
such that
xn
is that Then
(which we may assume to be infinite).
as follows:
To show that
D.
g1 1(x n)
can be done just as in the proof of the first sublemma. be completed by constructing a sequence
is Borel, and this The proof may now
by induction, again just as
fgn}
in the proof of Sublemma 1.
Let E =
PROOF OF LEMMA.
are compact, and let a sequence {hn} f-1(y)
and
y e Fn, and
Borel. that g
of Borel functions
bb+l
agrees with
gn(y) -Z for
Now for any fixed gn(y)
gn
hn
on
hn
f(E)
Fn
on
Fn.
on
y f Fn.
to
Let
z
by setting
and the
Gn
such that
Gn
g(y), with
hi11(y)E
be an arbitrary fixed gn(y) = hn(y)
Then it is easily seen that
y e f(E), y e Fn
converges, say to
Gn+l' Gn
By Sublemma 2 and induction there exists
Fn = f(Gn).
C, and define
point in
UUGn, where
for sufficiently large g(y) e f 1(y).
for
gn
is
n, so
It is plain that
is a function satisf;ing the conclusion of the lemma. The following lemma shows roughly that the
T'/
are irreducible
"on the average". LEMMA 4.2.
With the notation of the present theorem, suppose that
Wr= (k,( e).J, + !3(r)cr,, where
Of ( 70
and
0( 1) are measurable
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I functions on
r' to the interval
(0, 1)
for all 1', and where for each Ua Q, functions of
',
such that
jO,,(U)
and
31
O!(2() + (3(i") = 1
C,/(U)
are measurable
cl, being in the conjugate space of Q and
j j,( and
a.e. states of a. Then a.e. ,Pj( = o)". If a (l') : 1/2 we put p,, - 2((t ( I)jo,, + ((3(1) - 1/2) C,, ) and G'y = Cd,; if a(,( a() > 1/2 we put JOB = jo/ and Gile = 2( c ( 1() 1/2)o', + /3 (V)a,, ). Then Wi _ (1/2)( jOy + cry ), jY(U) and O' U) are measurable functions of ' for U c a , and and cr; are states a.e. Hence it suffices to consider the case in which ca( I) _ 3( a') 1/2.
As proved in [8, page 80] (though not stated formally as a theorem) the equation 2(,),' = fJ,( + Ci implies that j3/U) = (Tt CPj(U)z j , z,() for Uc Q , where T,, E ((Pr(Q) )1 and II T,r 2. Now If U and V 11
are in Q ,
(Ta(-7v, (U), rf,((V)) _ (Tr (Qa, (U)z" , (P,((V)z,( ) C>j (V*U)z,, , z' ) = ,0 j(V*U), and so is a measurable function
of I . It follows readily that for arbitrary x and y in 'M , (Tj x( 1'),y(a) ) is a measurable function of 7 . Hence there exists an operator T on Y' such that (Tx, y) = (T,/ x(' ), 'y(a() d,, (,f) for all x and y in 14. As T is decomposable, T 6 C! On the other hand, T E Q; for if (T,(
U, V, and W are in Q, (TUVz, Wz) _ (T,( 'Y,( (UV), -J., (W)) dJL( 1) _ (P,,(U)TY ) ' (V), yj,r (W)) d ,,m JP ( T , . (P,((U) I./(V), r j a , ( W ) ) dy(I) /T., -2,,(V), `9,1(U*) 1>(W)) dit( 7l) _ (Ty) ,(V), (UaW) I) (TVz, U*Wz) = (UTVz, Wz).
V
As
and
W
range over Q ,
Vz
range over dense subsets of AP, and from this it follows that
_ (UTx, y)
for all
x
and
y
in W', so that
(i)
and Wz (TUx, y)
TU = UT.
T E Q'^ C; but by assumption, Q'n C - C. Now let T( -61) be the continuous function on f corresponding to T and let S be arbitrary in C . Then (STx, y) _ (Tx, Say) - /(T,( x( a/), S( "')y(I) ) dr( -;0 (by Corollary 2.1) - fS(2()(Tax( I), y( a()) dj ( I). On the Thus
32
I. E. Segal
other hand,
(STx, y) =
S( a')T( /)(x( a"), y( ?())
d,,o.&( j).
From the arbi-
trary character of S it results that (T' x(d ), y( a)) = T(s') (x( t(), y(a" )) a.e. In particular (T,( ?Z,((U), )'(d,(V)) = T( d') ( Yft(U), v ((V)) a.e. if
of Q,
and
U
V
(Tr _1a,(U1),
eously for all
I
)7
and
fore
Te
ya,
)
:sow assuming the
?14/(U1)
are dense in
are both states,
PhOOF O6 THEOhEtd.
a"s
Then
a
xy
U1
to constitute a
/,'y , and hence a.e.
and y In N' .
Let N for which
be a Borel set of measure zero which Wj(
is not a state.
be
Let
generated by d and the identity
consists of all operators of the form MI + U, with
and so is separable.
There-
W,( = p,( a.e.
the state space of the Ca-algebra Q 1.
holds simultan-
is proportional to We , and as
a.e. so that a.e. fps.
contains the set of
is any sequence of elements
a.e.
j
= T( I') (x1-, yy ) for all
= T(a')
,Oy and W j
JUi}
(Uj)) = T( 7( ) ( r?.,(UI), Y/,,(Uj))
dense subset of 0 , the
(T j x-f ,
and if
are in
It follows that n Is compact metric (if
U E Q, fu1}
is
a1, d( p, c) =,V-i 2-1 11 Ui r1 11(U1) - o' (U1) I
a dense sequence in
is a metric inducing the weak topology). W' denote the extension to
For any state
QI defined by the equation
cJ
of Q. , let &J'( al + U) =
c + W (U); then W' Is a state of Q1 and is pure if and only if W is pure. That W' is a state is clear with the exception of the requirement that W'( cci + U) '_ 0 if CLI + U 0. Now OLI + U ? 0 means OC I + U = (0 I + V)2 with V 6 Q and t3 l + V self-adjoint. We can suppose that (3 is real and V = Va, for otherwise the equation PI + V = ( OI + V)* implies that I = (A - /3 )(V - V*) so that I G Q and Q1 = Q. Plainly W'( (3I + V) = (32 + 2(3u)(V) + W(V2), which is nonnegative for real
(3
the arithmetic mean.
by the fact that the geometric mean is dominated by It is immediate that
and the converse is proved in [8], p. 87.
W
is pure if W
'
is pure,
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I Now let
C
be the product space a X -a,
33
D
the apace
(f,', c) -> (1/2)( fO'+ Cr'), and E the subset of C consisting of all elements of the form (p', a,') with fJ' Cr'. Now C - E is the diagonal set of all is compact and with O'E L1 ; as continuous, the mapping fJ' -> ( JJ', /o) C - E is compact and hence E f
the mapping
It follows that Lemma 4.1 applies and states that there exists a
is open.
P on the set
Borel mapping
of non-extreme points of CL to
A-1
fix fi such that if )tr (u) ) = (p', o''), then (J' _ (1/2) (p' + C') . 0 an extreme point W' of
we extend }(r by defining
is then defined on .n., and is Borel. actually, let
all elements of
To show that
be any closed subset of - X - ` and
K
J(!-1(K) =
which are diagmal. Then (K - K1) . Now Kl = K'' (C - E) and hence is compact; K
is a homeomorphism on into
C - E, and as
to
,0',
)",-1(K1)
E; as
relative to be Borel.
Finally
'
the set of
-,141-'(K l) V
takes
C - E .
and then
and
)L
wise Borel, being compositions of Borel with continuous functions. u)' -> u)
must
)//-1(K - K1)
I. Borel, being the union of two Borel sets.
We put )v (W') = be the mapping
Now
with a closed set and so is closed
before extension was dorel,
-)(1-1(K)
Kl
as P' -> (p' Jj) on
-Y/-l
is Borel
)G
is compact and hence Borel in
is the intersection of E
K - K1
For
of states of
are likeLet
J
into the continuous linear
a 1
functionals on a ; then it is clear from the definition of the weak topol-
- Jg( u1' and Off= 3X(- ) for ;( f N , so that &, and O', are states of Q when /O/V) and W2( is a state, and (1/2) ( fJ,( + o'1)) = We. For any Usa, for taking the case of are measurable functions on C,((U) - N
ogy that
J is continuous.
Now let fOy
)
,
1
.10Y(U), it is the product of the successive maps 1
-
N to the state space of a ;
Q to Si
;
(c) W' -> s` (tJ')
on
(b)
1-1
(a)
7( -> &),(, on
LJ -> cJ' on the state space of
to n ;
(d)
u1' -> Jw,
34
I. E. Segal to the state space of
on of
Q
A; (e)
to the complex numbers.
o) -> u)(U)
on the state space
Now (c) is a Borel map, (b) is easily seen
to be continuous, and the other maps are obviously continuous. if
is any closed set of complex numbers,
G
[ jl1o (U) e G.
Therefore,
14( f N I
is
ak3orel sunset of r - N , and as N Is a Horel set, a Horel subset of
F W,,
Hence Lemma 4.2 applies and shows that ,/J,. - W,( a.e.
is extreme, i.e. pure, a.e., so that by
?'
[8]
It follows that
is irreducible.
PART II. APPLICATIONS Decomposition of a W*-algebra into factors.
6.
section that any
We show in this
W*-algebra with an identity (= ring in the sense of von
Neumann) on a separable Hilbert space can be decomposed into a kind of direct integral of factors.
A similar decomposition of a W*-algebra into more
elementary P*-algebras is valid also for inseparable spaces, but we are not then able to assert that these elementary algebras are factors.
We
begin by stating just what is meant by such a decomposition. Let the Hilbert space
Definition 6.1. of the Hilbert spaces
111k
,
14 be the direct integral
An algebra Q of oper-
as in Definition 4.1.
p
ators on
7*
ators on
14p, with respect to the given decomposition of
is said to be the direct integral of algebras
integral, if. a) every
T e Q
has a decomposition
a.e.; b) every decomposable operator
is in d .
T
We then write symbolically
THEOREM 5.
Let
Q
on ]*
T(p)
a p
of oper-
H'
as a direct
with
T(p)6 Qp
such that
T(p) E Q p
a.e.
Q = A dr(p).
be a weakly closed self-ad joint algebra of
operators on a Hilbert space ht, which contains the identity and has center C .
Then
0
is a direct Integral of factor-, relative to a decomposition
of J as a strong direct integral whose algebra of diagoralizable operators
is Ci .
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
a contains a
IEMMA
Ce-subalgebra
35
which is separable (in
the uniform topology), whose strong sequential closure is Q , and which contains the identity operator.
This lemma follows at once from von Neumann's theorem that a contains a countable subset dense in the strong sequential topology [17
p. 386.3 PROOF OF' THEOREM.
taining
I
Let
J5
Then
.
',,%
E" =
(211
= .0' n E' =
7''
respectively, in the strong
Q'
and
.9
is separable, for rational linear combinations of monomials
a countable dense subset of 7.
that
C*-algebras con-
be the C*-algebra generated by
in the elements of a dense subset of
a', that
be separable
Ci
Q and
which are dense in
sequential topology, and let
E
and
= Q
and a dense subset of
P1
yield'
Ci
It is not difficult to see that
.6' =
(by a well-known theorem of von Neumann), and
Q' n Q =
C.
C
As
is abelian,
every self-adjoint maximal abelian subalgebra of $
C' contains
C
which contains
Now such a maximal abellan suoalgebra is known to have a cyclic element, z, which we can take to be normalized.
say
Now
j# .
`9r`
clearly, but °"
= C'
the theorem of von ueumanr. just cited.
A fortiori,
is dense in
C'z
is the strong closure of 7, by
Thus 7 is strongly sense in
C'
and it follows that 7z is dense in j,' . We are now in a position to apply Theorems 2 and 3 with
placed by ' . T
Let
(gibe the strong closure or
is the strong limit of a sequence
is decomposable by Theorem 3.
Putting
{T n)
in 1f , but D f--
T(:;) = strong limn Tn( 1')
suitably chosen in
.L0, then for almost all
if
r
, 7so that
T
for its canonical decomposi-
T(%()
tion, so that a.e.
limit of a secuence In
re-
T E Q , then
If
99,,(ff ).
a
a" ,
if the sequence T( a()
T(a")E QV, so that (Tx, y) = ``P( a()x( I), y( a())
is
is the strong
and her.ce is Itself' in Q
is a decomposable operator with decomposition
{Tn)
On the other hand,
T(.)
(i)
such that a.e.
for all
x
36
and
I. E. Segal
in 1+, we shall show that
y
ment just made
U(i)
such that 3.1, and
and
UT
also commutes with each element
commutes with each element of the strong closure of 99,, (& ). U(7 ')
and
T(71)
commute.
It results that
In
UT = TU,
T e IT', i.e., T e a.
that a.e.
,dd
U(.)T(.)
commutes with each
algebras) that each element of the strong closure of
.Si.
Q. is the direct integral of the
Thus
(Py(
9,,(11)
.b
It rcllows easily (using the identity of the strong and weak
particular, a.e.
or
As each element of
U(.)
By Corollary
are both decomposable, with decompositions
respectively.
99$, (6).
g7y(.5)
is decomposable with a canonical decomposition
(6, each element of
closures of
U e Q', then by the argu-
If
Is a.e. in the strong closure of
TU
T(.)U(.)
element of of
U
T 6 Q .
).
Oz(
is a factor.
By Theorem 4,
14y
Q y-.
It remains to show
is a.e. irreducible under
An equivalent way of stating this is as follows: (y" (7)), _
a.e., where
dy
is the algebra of all scalar operators on
is generated as a C*-algebra by Zr
Ake .
Now
and E , and it follows readily
that (P,, (7) is likewise so generated by 9f (D') and 99y(C). It follows that (99e(_4'))1 = ( (19 ) )' (9 ( E ) )' . Putting and for the strong closure of
9,(&), as noted earlier
( Q ,)' = ( 97y(17))', so
(Qy )' n ( N(y)' _ Jy a.e. Now each element of Qy commutes with each element of )f, , i.e., ()Y),()' =) Q., so ( Qy )''Qy a ( Q y )' ^ ( )1(y)' = y. It follows that ( Q y)' ' Qy = Sy a.e., for as I e Pf, both a, and so that Q y is a factor a.e. (Q,)' contain 7.
Decomposition of a
representations.
rou
representatior. into irreducible
We show next that every measurable unitary representation
of a separable locally compact group is a kind of direct integral of irreducible continuous unitary representations.
This generalizes well-known
results of Stcne and Ambrose concerning locally compact abelian groups (but its apolicat_on to the special case yields a result which is considerably
37
O COMPOSITIOiS OF OPF.HATOH ALGEBhAS. I
less sharp than either that of Stone or that of Ambrose), and similarly generalizes a well-known analogous theorem for compact groups.
In view of
the known correspondence between positive definite functions on groups and ccntinuous unitary representations of groups [4], our result generalizes the representation theorem for positive definite functions on locally compact abelian groups by showing that on a separable locally compact group, every measurable positive definite function can be represented as an integral of "elementary" positive definite function, where an "ele-
mentary" function is defined as one which is not a nontrivial convex linear combination of two other such functions (or alternatively, as one for which the associated group representation is Irreducible).
A result closely
resembling that presented in this section has been announced by F. Mautner [5] and is proved by him apparently with the use of his result resembling our theorem on maximal decompositions (see Section 5), which we use in the following.
Definition 7.1.
U
be a unitary representation of the topo-
on a Hilbert space t'.
G
logical group
Let
We say that
13
is decomposed
into irreducible representations by the (strong or weak) decomposition
14 =
f7 dr(p)
if for every
a c G,
U(a)
is decomposable, and if for
P
p e R. there is an irreducible unitary representation
nearly all such that
is the decomposition of
U(a)
(strong or weak) direct integral of the If
G
is locally compact, G
measurable on ally, if
G
U
U(a).
We then say that
Up, or symbolically,
is called measurable if
relative to Haar measure for all
x
y
of
G
is the
U
U = AUpdr(p).
(U(a)x, y) and
Up
is
in * (actu-
is sepaerable such a representation is necessarily strongly
continuous; cf.
[12]).
The regularity conditions which the method of proof of the following theorem could be used to establish are significantly stronger than those implied by the theorem.
In particular it is possible to define in a natural
38
I. E. Segal
fashion, in case the representation has a cyclic vector, for all M, a continuous unitary representation
perfect measure space
is jointly continuous in
(Up(a)x(p), y(p))
p
and
a
ing over a certain dense subset of 7# , as well as with a.e. and
U =
fUp dr(p).
Up, such that x
for
Up
in the
p
rang-
y
and
Irreducible
The method o2' proof also yields a decomposition
strongly continuous representations of inseparable groups, in which
for
the constituents are irreducible in a kind of average sense (as in Lemma
4.2, but with continuous i'u.)ctione of e. Every measurable unitary representation of a separable
THEOREM 6.
locally compact group
is a direct integral of strongly continuous
G
irreducible unitary representations of Let
U
be a given strongly continuous unitary representation of
on the Hilbert space
G
G.
H'.
Vie put
a0
as is readily shown, a SA algebra; cf.
f(a)f(a)da with
form
for the collection (actually,
[91) of all operators on 1* of the
f E L1(G), where as in [9],
tes the operator which takes an arbitrary element
j(a)xf(a)da.
integral
j(a)f(a)da
x 6} '
designa-
into the strong
(For a proof of the existence of this integral
and for other facts concerning the operator thereby defined, of. [9],and [8], esp. p. 83, and 84.)
Q is a C*-algebra. on
lity of
Q0, and
to
L1(G) G
Let
Q be the uniform closure of
Now the mapping L1(G)
f -> J (a)f(a)da
Is continuous
is separable, for the topological reparabi-
implies the separability of
G
as a measure space (relative to
a regular measure) which in turn implies the separability of results that If
Q0 zl
(recalling that If
is separable, and hence
z2
Q
L1(G).
It
is separable.
is ad arbitrary nonzero element of N', the closure of
is a closed linear manifold
Liz I
Qo, so
which is Invariant under the
Q is inva^iant under multiplication by
U(s)) for
is an arbitrary nonzero element in the orthogonal complement
U(a) a e G.
1(1)
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I of
H(1) in, then the closure of
74(2)
In
19z
2
which is invariant under the
7+h
39
is a closed linear manifold and orthogonal to
U(a)
It follows readily by transfinite induction that there exists
14(1).
a collection N (g ) ( A E -) of closed linear subspaces of 14 , mutually orthogonal, with direct sum equal to /f, and each invariant under the and containing an element
U(a)
zg
such that
Qz I
Is dense in
a is cyclic
This shows that it is sufficient to consider the case in which on
1
if U( )
Is thecontraction of
measure space
US) -
Then
For suppose the result has been established in this case.
.
Jr )
7",g
U
to
$(
and de compositions() _ A ()
(').
d
points is U, f= 0. disjoint), in which a
there is a
for each
dud( ')
and
Let T' be the measure space whose set of
(we can and shall require that the
are mutually
c -finite measurable set is one which meets at most T+
countably many
the measure j1
of such a measurable set is the sum of the -Itr, -measures
of its intersections with the
11
in a measurable set, and in which
, and meets each
.
Pg.
is the direct integral of the
It is not difficult to verify that
N.('3 ), over ( F, 1A), the only
condition which is not trivially verifiable being 2a) (note that as separable, so is
Q z
we can take the direct integrals of the
14 ,,(S
follows from the fact that if
z
(z(p), z(p)) = 0, except or. a
G'-finite set of
Q
to be strong).
satisfies the condition in
This
2a), then
p's, say for
00
P F Ui zi(p) = z(p)
and we can set a.e. for
zi, where
z'
and
p 6
zi(p) - 0
zi
is such that
for other values of
p,
i
the sum which defines
Jzi(p),
being convergent because
z'
0
when
i
(zi, z;) _
J. and Z:i II zip2 =
!I
J II zi(p) Il2dpgi (p) _ z(p) II2 dju(p). It is clear that nearly all the Ur( are irreducible and that U = Uyf ) dp(p).
C_i
is
are separable, and
z, so that the
for any
40
I. E. Segal
Suppose now that Q has the normalized cyclic element and let in
Q',
C be a W*-algebra which is maximal abelian and self-adjoint
C exists by Zorn's principle.
3, and 4.
in
z
We can now apply Theorems 1, 2,
Utilizing the notations of these theorems,
irreducible.
is a.e.
(P,,(Q )
Now every (uniformly) continuous self'-adjoint representation
9 of Q induces a unique continuous unitary representation V of such that crl (JU(a)f(a)da) _ I(a)f(a)da for feI,l(G) and with the property that
99
is irreducible if and only if
Ue for the representation of
If we put
V
is (see [8] and [9]).
induced by
G
G
'
, ,
it follows
that Ur is irreducible a.e. and that U = JUdµ( /). 8.
Decomposition of an invariant measure into ergodic
ap
rts.
We show in this section that a regular measure on a compact metric space which is invariant under a group of homeomorphisms can be represented as a kind of direct integral of ergodic measures on the space.
We recall that
an ergodic measure is one relative to which every invariant measurable set is either of measure zero or has complement of measure zero. Definition 8.1.
A finite measure
be a direct integral of measures a measure space
mp, where
M = (n, 1Q, r ) , if for each
measure on 7', ann if also for each E e 7,
m
on a o'-ring
p
is said to
rarges over the set p e R mp
mp(E)
R
of
is a finite
is integrable on
M
//++
and
Jmp(E)dr(p) = m(E). The proof of the following theorem yields a kind of maximal decom-
position of invariant measures into invariant submeasures in the inseparable (compact) as well as separable (compact metric) case, but we are unable at
present to establish ergodicity of the submeasures except in the separable case.
A number of similar decompositions have been obtained by quite
different methods, for the cases of one-parameter and infinite cyclic groups, the first such result being due to von Neumann, and the most general one being that in [2a] which applies to a class of separable measure spaces
DECOMPOSITIONS OF OPERATOR ALGEBFIAS. I
41
including the one considered here.
A regular measure on a compact metric space
TH5O EM 7.
is invariant under a group of
of homeomorrhisms of
G
G-ergonic regular measures on
is a direct Integral
M
M.
There is clearly no loss of generality In assuming that where tion
is the measure in question.
m f
m(M) = 1,
We call a bounded measurable func-
f(a(x)) = f(x)
invariant if
M
on
which
M
a.e. on
M
for all
a E G.
Let 1+ be the Hilbert space ::f all complex-vel c:. functions square-
m, the inner product of two elements
integrable relative to /#-
f(x)g(x)dm(x).
being defined as
Let
g
and
f
of
Q be the algebra of all
id
operators
on 14
Qk
of the form
f(x) -> k(x)f(x),
is a continuous complex-valued function on
k
Qk
algebra of all
C
(N1;
Q is sepsr::ble in the uniform topology
kf
be a basis for the open sets in
i = 1, 2, ...)
be for each
{fin; n = 1, 2, ...}
tions on
i
a sequence of continuous func-
(E.g., if
Ni.
sequence of closed subsets of
LCid is a monotone increasing
such that
N1
N1 = U Cin, then
be taken to be a continuous junction with values in Cin
M, and
which are uniformly bounded and converge (noin.twise) to the
characteristic function of
on
be the
is a We-algebra.
Let let
M, and let C
complex-valued, bounded, measurable, and
k
We show next that
invariant. and that
with
f e L2(M, m), where
and
0
outside of
n [0, 1]
which is
1
N1, such a function existing by Urysohn's Then the rational
lemma and the normality of a compact Hausdorff space). linear combinations of the
can
fin
fin
are dense in
C(+).
For otherwise, by the
Eahr.-Banach theorem there would exist a nonzero continuous linear functional 4 on C (M)
which vanishes on the
functional has the form
4 (f) _
additive set function
on
n
M.
fin.
Now it is known that evbry such
f(x)dn(x), for some regular countablyIt follows easily that
limn 1(fin)
42
I. E. Segal
J Ni dn(x) = n(N1),
so that
n
vanishes on all the
difficult to show that any finite union of the which
Var n
N1
It is not
.
differs by a set on
N1
is arbitrarily small from a finite disjoint union of
which implies that
vanishes on all finite unions of the
n
on all open sets, and so by regularity vanishes identically. C(M).
are fundamental in
k -> Qk
the map
uniform topology (in faot For
a F -G, let
C(M)
in the
to the operators on
Q must be separable.
ll k(lt JJQkJ ), the image
be the operator on
Us
fin
Thus the
is separable, and as
It follows that C(M)
is continuous on
Ni's,
N1, and hence
defined by the equation
1!i
(Uaf)(x) = f(a(x)), f c14.
Then it is easily seen that a bounded measurable
complex-valued function
on
for all
QkUa
a e G.
M
is invariant if and only if
It follows that if
with bounded measurable
[15] that
k
Y1(
)'
= )!f', so that in particular
`I)(
Qk
is the algebra of all
Now it is known
C = ' ^ [ Uara e G ]'.
k, then
UaQk =
is weakly closed, and as it
is easily verified that
Of, = (Qk)*, m is SA and so a Ws-algebra.
Plainly,
is weakly closed, and it is easily seen that
[ U
a
I
a G G ]'
is unitary, so that Hence
(Ua)* - U -1
showing that
R is a We-algebra.
[ Ua+ a e G ]'
Thus
[ Ual a e G ]
Ua
is Sk.
C Is a W*-algebra containing
the identity.
Now let 0.z
be the function which is identically unity on
z
consists of all continuous functions on
regularity of
m,
is dense in 14 .
a z
with
, W ,( , and rt.
I
known; for each W Cji(Qk) _
all
A
,
,(x)dmj m l(
M, and so, by virtue of the
T e d
,S( 7() cJ.(T) dji( al)
as in Theorem 1.
for all
is an invariant
k 4E C(M).
S E C,
and
,
Now the states of
there is a regular measure me (x)
Then
It follows that the conditions of
Theorem 1 are satisfied, and hence for any
(TSz, z) =
M.
on
M
Q are well such that
We show next that for almost
G-ergodic measure.
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I It is easily seen that
UaQkUa-l = QUak
,
43
so that UJU
a-1
E Q
if Te Q, and (UaTU1t, Z) = JS(') Wj.(UaTUa_l)d).&( (). On the other hand, if
fM
T = Qk
and
k(x)p(x)dm(x)
fk(x)p(x)dm(x)
that
S = Qp
and
with
T E Q
and
S E C, then
(UaTUa-13z, z) _ fk(a(x))p(x) dm(x) = m
(for
and
p
are invariant) = (TSz, z).
(2() OJ j(UaTUa_l)djt(2() = Jg(7() WE(T)
the arbitrary character
being an
S(.))
of//
S
It results
It follows from
(every bounded measurable function on T
W,,(UaTUa-1) = WW,(T)
that
(TSz, z) _
for almost all
2(, and since
both sides represent continuous functions, the equality for all But if
f(x)dma,(x)
T = Qk,
so for all continuous functions
It follows that mi
k
and
on
M.
is Invariant for all
We have for any
/
k E C(M), putting
above formula, that fk(x)dm(x)
=
follows.
u)j(UaTUB_1) _
fk(x)dm t
k(a(x))dmj(x).
(x)
2(.
T = Qk
and
3 = I
r [/M k(x)dm,, (x)
if k is the set of all bounded Baire functions equation holds, it is easily seen that
7T
k
on
in the
dAA(). Now M
for which this
)Y is closed under bounded point-
wise convergence, and as k contains all continuous functions, it consists of all bounded Baire functions.
Now if
E
its characteristic function is Baire and so
m
is the uire'ct integral of the
my
It remains only to show that m y preceding decomposition of
m
the sepafability assumption on
is any Borel measurable set,
m(E) over
( I
, J.L).
is a.e. ergodic (in fact the
into invariant sub-measures is valid without
M). Now the ergodic invariant regular proba-
bility measures on a compact space are precisely the extreme points of the set of all invariant regular probability measures on the space (the proof of this in [9] is for the group of reals under addition, but applies to an arbitrary group with trivial modifications). probability measures
n
on
M
The set of invariant regular
is also known to be In one-to-one convex
linear correspondence with the set : of all invariant states
V
on Q
44
I. E. Segal
(V being invariant if V (UTU-1) = v(T) for and
n
correspond if
V
v(Qk) =
T E A and
j(x)dn(x) for all
a 6G), where
k e C(M).
Now
is a convex set which is compact in the weak topology (recalling that the state space of a Ca-algebra with an identiti is compact). is metrizable, for if
set of Q , with no
(i = 1, 2, ...))
(Ti
is a countable dense sub-
d(n, p) = : 2 1
T1 = C, the metric
Moreover,
it T1ff-1 fn(T1) -
i
is easily seen to induce a topology on 37
p(T1)I
identical with the weak
topology.
It follows, by an argument used in the proof of Theorem 4, that if
mj
is not ergodic a.e., then there exist for each
fJy and
such that
o
and
101(T)
T E 12,
G, (T)
an operator
a'
S.
(sr T,(T)z( a'), z( a')) and
invariant states
&)f= (l/2)( fJr + Cr ) for all ate; 2) if
are Borel functions on T ; 3) pr
e's of positive measure.
a measurable set of each
1)
-K
in
(973(A))I
for
C,,
As before, there exists for
such that
jO,(T) =
(Sr r,,(X), 1r (Y)) (Sr 9'y(Y*X)z( 7( ), z( a')) = p, (Y*X), and is a measurable function of a( .
It follows that and
y
in
ff S1
(Sr x(a'), y(a'))
2.
is a measurable function of
, and this function is integrable for
211 x( 7( ) 11 11 y( Y ) fi
is a decomposition.
S
Now if ffnj
f(S. x( 6' ), y( a'))I
By Theorem 2,
S
S E Q'.
on i for which
jSY d/4 ( a')
We show next that
Q1 = 3r,
is a multiplication by a bounded measurable function on is a sequence of bounded measurable functions on
uniformly bounded and which converge a.e. to a function verges weakly to
x
for
a(
, which bound is integrable by SchwarzI Inequality.
Hence there is a decomposable operator
so that
Now
Q.f, by a simple computation.
M
f, then
M.
which are Qf
n
con-
On any finite regular
measure space, every bounded measurable function is a limit a.e. of a bounded sequence of continuous functions.
Q,w Owl
. Hence a contains = al, and Q' c 7i(' = Zi(. of
It follows that the weak closure
but
(2w - a", so
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
Thus
S E 177
arbitrary in
G
and
S 6 C, for if
and so S = Qk. Moreover,
T of this equation shows that
is O, ,
Integration over
(SUaTUa_lz, z) = (STz, z)
(UaTUa 1z)(() = T,(UaTUa_1)z(7f ), by Corollary 2.1 ).
(noting that
S = @k
If
and
fk(x)p(a(x))dm(x) = /k(x)p(x)dm(x), which
T - @p, this means that
fk(a 1(x))p(x)dm(x) = fk(x)p(x)dm(x), from which it follows
implies that
readily that k(a 1(x)) = k(x)
a.e. for each
That is,
a e G.
is a.e. a scalar multiple of the identity, and 0,, = O
a contradiction. 9.
a
arbitrar, in Q , then by the invariance of
T
(Sy 91" (Ua -1)z( ?l ), z( 7( ) ) = (Sr 97,, (T)z( ]j ), z( $)) .
Be
45
Hence my
S 6 C, so
a.e.,
= o),,
is s.e. ergodic.
The Fourier transform for separable unimodular groups.
We
show in the present section how the Fourier transform as defined in [111 can be correlated with the Fourier transform as an integral whose kernel Is an irreducible group representation.
F
F(x*) =
the equation
G
G*
of
G. defined by
The generalized (Weil-Krein) IF(xa)l2dx* G*
f6 L2(G).
Is an integrable function
on the character group
x*(x)f(x)dx.
Plancherel theorem then asserts that f for
f
G, its Fourier transform is usually
on the locally compact abellan group defined as the function
If
= f jf(x)12 dx, G
The Fourier transform can be extended to compact (not
necessarily abeliao groups by replacing irreducible unitary representations of
G.
by the collection of continuous (which is simply the character
G
is abelian); one has then F(p ) = f)O(x)f(x)dx and the G generalized Plancherel theorem (usually called the Peter-Weyl theorem in group when
G
this context) asserts that f jf(x)12 dx G
where
d(jG)
is the degree of p ,
tr
denotes the usual trace, and the
sum is over any collection of representatives of equivalence classes of irreducible representations of
G.
In the case of an arbitrary separable unimodular group, it turns out that the same formal relations are valid, provided "irreducible
I. E. Segal
46
unitary representation" is replaced by "two-sided irreducible unitary representation".
As indicated in [10], this does not materially affect the
situation in groups which are either compact or abelian. in
More specifically,
it is shown that if the Fourier transform is defined through the
[11]
use of the von Neumann reduction theory, then the Plancherel formula for a separaole unimodulae group holds, the trace now being that defined by Murray and von Neumann for factors, and the integration being over a measureG*.
theoretic analog of
As it can be verified that the reduction obtained
in Theorem 2 satisfies von Neumann's conditions (cf. the last theorem in this paper), the Plancherel transform
F(a')
of a function
f e L1(G)"
can also be defined through the use of the present decomposition
L2(G)
f
he shall show in this section that the Plancherel transform of
theory.
t( e r , where
can also be obtained as follows: for each
( r .,At)
is
the perfect measure space on which the decomposition is built, there is a two-sided continuous unitary representation
tL,', Rj , where
and
L,(
are respectively the left and right ordinary representations of which
Ry
the two-sided representation is composed, which is a.e. irreducible, and
F( 7l) = f
such that
(a)f(a)da.
G
We begin by considering the decomposition of conjugations in We recall that a function
suitable situations.
to itself is called a conjugation if it satisfies the conditions:
1-f
1) J2 = I,
2) (Jx, Jy) = (y, x)
has the properties y
on a Hilbert space
J
in
14
for all
x
J(x + y) = Jx + Jy, and
and complex
..
JWJ
by
1.G
J(a x) = 67x
and that it
for all
, and that
With the notation of Theorem 1, let
such that
exists for each
in H ;
x
1( 6
SIT = e for all a conjugation
S e C, and T,(
on
and
is a ring
(JWJ)* = JW*J.
WJ, and designate the automorphism W -> WJ by
THEOREM S. tion of
y
It follows that the map W -> JWJ
automorphism of the set of all operators on 1 We denote
and
J
Jz = z.
J.
be a conjugaThen there
such that for an
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
a
and
in
y
47
Jx, y) _ Jr (Ji x( 7f ) , y( e ))d (Il ) .
,
We first refine Jr
as follows: Jr r?,(T) =
rJ,(A)
on
To see that this definition is single-valued, observe that if
rjr(T) - 1,,(w)) = 0, so that 17r(W), then ('1,,(T) cJ ((T - ')*(T - K)) = 0. Now W is transformed into W by (J)i'' so that u)(J(T - W)*(T - W)J) = 0 = oJ((J(T - W)J)*(J(T - W)J)) _ 1,((T3) - )1/ (WJ) 'I, (TJ) - 1y(WJ) ), so that (TJ) = Y(r(W J and for T e Q, tJ,'(TJ) = Wr(T). We show next that for all
'
Let
be an arbitrary self-adjoint operator in C .
S
Then W (STJ)
J S( )() (Jr(TJ)d}.l( I(). On the other hand, u) (STJ) = W (SJTJ) (for a SA element of C is invariant under J) = a1((ST)J) = ia(ST) = f&r(TJ) S( 8() Wf(T)d1.c(s(). As S = S*, it results that Wd(T)} d1.A( 7( ) = 0. As S(.) can then be an arbitrarJ real-valued
/(2()
bounded measurable function on
T ,
1
Wf(T) = 0
it results that
a.e., and since the left side is a continuous function on
Wf(TJ) =
, we have
1
Wy(T).
Now for any
T
and W
in
Q
(Jt, yr(T), Jr ',,(W)) _
we have
( r (Ti), I{ (WJ)) = wi(WJ"TJ) = L)f(W*JTJ) = Wj((W*T)J) = wr(W*T) y1(T), r(,,(W) ). In particular, II Jr'1r(T) 11 1 II 11((T) 11 , so that J,( is bounded on
the closure
I r (Q ), and therefore has a unique continuous extension to 14,,
From the equation
of
71r(Q ); we denote this extension also by J e .
(J, )r,(T), Jr -J,(W)) = (Yr,(T),
continuity that for arbitrary xr (xr , y,. ).
Jd
Thus
Now
Jr
yr
r ,
it follows by
(Jr x ,
J,' yf ) _
j,(T), so that
?,(Q ), and hence also, by continuity, on ff
is a conjugation.
It remains to show that for arbitrary
(Jx, y)
in
J,(T) = Jr (Jr >7,(T)) = Jr JI(TJ) °
is the identity on J,'
and
r1r(W))
x
and y
in
?Y
,
(Jr x( a'), y( d'))d,u( J(). Now if x = Tz and y = Wz with
48
T
I. E. Segal
in Q , (Jx, y) = (JTz, Wz)
and W
(W*JTz, z) = (W*TJz, z)
(using
the fact that Jz = z) = / ((W*TJ)d1L( ) = ` rj.,(TJ), rj f (W))dj.c( s') _ JJ -1,(T), r(i(W))d)h( f ). Thus the equation is valid for a dense set of
and
x
y's, and it follows as in the first part of the paper that it
is valid for all
x
y
and
in
When
76j.
is not a state,
r
can
J,(
of course be defined arbitrarily. The next theorem asserts that the conditions of the preceding theorem are satisfied (with a suitable choice for certain conjugation on THEOREM 9.
Q
group, and let
be a separable unimodular locally compact
G
LfRg, with
f
g
and
are respectively left convolution be
and right convolution be
f
?(x 1), every self-ad oint element of C morphism induced be z
in 1* such that
Then if
J
(i.e.
-14
all
f
and
as 0. Clearly
and
g
in
and all he 1¢.
h * gn -> h
T e Q.'
lgn}
L
and
L and
R, where La
R
(Raf)(x) = f(xa),
and
if and only if TLfRgh - LfR9Th
TLfRg = LfR9T for such
L1(G)
f
the
(of. [10]), this shows that
TLfh = LfTh.
Hence
TLf - LfT, and as -t
T C P. Similarly
for and
g
such that
(Th) * gn -> Th, it results from the equation
that
are respec-
Ra, these operators be-
and
is a sequence in
LfRgnTh Lf
is the conjuga-
J
and that the weak closure of
(Laf)(x) = f(a lx)
L1(G), i.e..Pif
Now if
and
Let
Jz = z; and Q' = C
C = Q'
Wa-algebras generated by the
ing defined by the equations
fe
g.
S e C ); there exists an element
is dense in 1+ and
Q z
W*-algebra generated by
tively the
Rg
is invariant under the autoif
JSJ = S'w
We show to begin with that is the
and
Lf
the equation Jf = f* for f e Jµ, where f*(x) _
defined
jV
L1(G), where
in
C be the center of the weak closure of Q .
tion on
G.
be the C*-algebra generated bX all operators on >4 =
of the form
L2(G)
for a unimodular group
L2(G)
Let
z) in the case of a
TLfRgnh
is generated by T e 1 ', and so we
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
49
Q ' C L' ( -t. It is shown in [10] that ZI = 9, so
have
which shows that
the other hand,
Lf e .L
and
n -e or
V Rt
Hence
and it follows that
Q"
T -> TJ
operator topology.
S of C ,
As the finite linear combinations of projections in
S
is such a linear combination.
Now let ) be the range of such a projection 7
is then a two-sided ideal In the
every such ideal is invariant under P(JPx) = JPx, or
Applying
PJP = JP.
PJP = P.
shows that
is conjugate
J
It is shown in [11]
P.
0
I,2-system of
in the sense
Now
J.
Hence for
, i.e.,
JPx a
x e 74,
Multiplying the last equation on the left by Is SA, so
P
(PJP)* - P
and
PPJ = P.
to both sides of the last equation shows that
J
in C
P
According to a theorem of Ambrose (loc. cit. Th. 7),
of Ambrose [2].
J
As
3J = 3*
for every projection
PJ = P
linear, it is enough tc show that
that
SJ = 3*.
is easily seen to be continuous in the strong
C are strongly dense in C , it therefore suffices to prove for the case when
is
On
121 .
(loc. cit.),
f E L1(G)
Next we show that for an arbitrary element The mapping
C =
Q", so that
for all
Rf e 7f
which implies that Q c
Q"
It follows that the center of
is abelian.
0_1
Q Is SA, its weak closure is
Q1, and as
Q1C R eNIV,
PJP
PJ, so
P = FJ.
It remains to show that there exists an element that
of
O z 1'#
is dense in
and
Jz = z.
Let
fzi}
of H
z
such
be a family of elements
which is maximal with respect to the properties 1) Jzi - zi is orthogonal to
zi # 0, 2) Q zi
over which
1
Q zj
if
I A J.
and
Then the index set
ranges is at most countable, for the closures of the
Q zi
constitute a family of mutually orthogonal closed linear subspaces of
which by the separability of # must be at most countable. i = 1, 2,
...
and put
show finally that
Oz
z =
n 2 11 zn11 1 zn.
Is dense in 1+.
Plainly
We assume that
Jz = z, and we
50
i. L. Segal
Assume on the contrary that exists a nonzero element oe the closure of
yI{ I.
in
and
Qzi
is not dense in H. Then there
which is orthogonal to
7q-
OZ.
the projection operator on
Pi
(Tz, Sx) = 0
for such
S
and
T.
the last equation is valid for all
S
and
T
hli
Let
with range
7ti-
(Tz, x) = 0 for T C. Q, so (SeTz, x) = 0 for
We have
in Q., or
x
Q z
S
and
T
It is easy to deduce that In the weak closure of
Q- .
Now it is easily verified that )1 is invariant under a , so that
PIE Of = C, T E Q,
to
C
.
PjE a" .
Hence we have in particular, for
or
(Tz1, Pix) = 0.
Fix = 0,
As
x
-k1, so
span
and hence
(TPiz, Pit) = 0
w
Putting
Now
for any nonzero element of
(obviously such a vector exists) and
Qzs = QQzs = QQz' G Q is, so that contradicting the maximality of
but the
P I x E31t1
I - E1 Pi = Q
0,
Tz1
is a nonzero projection
Q 14
such that
ze = w + Jw, clearly
Jw #
and
Jz' = ze
is orthogonal to all the
Qz'
- w
Qzi,
(zi}.
Before proving the result mentioned in the beginning of this section we make appropriate definitions. Definition 9.1. Hilbert space
14'
A two-sided representation of a group
is a pair
(L, R)
on Y such that L(a)R(b) = R(b)L(a)
G
on a G
of one-sided representations of for all
a
and
b
in
G.
If
G
is topological, such a representation is called strongly or weakly contin-
uous if both L If
R
(L, R)
and
is a two-sided representation of
is unitary, and if
fr all
ducible if the 74'
J
G.
L(a)
G
on
is a conjugation on 1#
a e 0, then the system
representation of
of
are (respectively) strongly or weakly continuous.
R
(L, R, J)
R(b)
(a, be G)
(jointly) invariant other than
0
if each of
such that
L
and
JL(a)J - R(a)
is called a two-sided unitary
A two-sided representation and
1'tb,
(L. R)
is called irre-
leave no closed linear subspace and
1'f.
The following theorem shows the connection between the Fourier transform as defined directly thru reduction theory and as defined thru the use
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
51
of an integral whose kernel is a representation. THEOREM 10.
Let
C, J, and
G, 14, Q ,
theorem, and let T, ,M, and
z
be as in the preceding
be as in Theorem 1.
W 1,
L1(0), the left and right convolution operators
Lf
For every
and
f s
are decom-
Rf
posable with respect to the reduction of TM described in Theorem 2, with For almost all
M = (P ,,,u.).
?'F- r there is a two-sided strongly con-
tinuous unitary irreducible representation fLr , Rr , which is almost everywhere on
decomrositions of Lf and
(T, r.)
are
Rf
G
Jy }
of
on ]fir
G
irreducible, and such that the
L,. (a)f(a)da
Ja R,{(a)f(a)da
and
respectively.
We observe to begin with that for any
are in Q
R X a
f
g
and
For let
and X E Q ,
a0 be the algebra generated by the
L1(G); as every
in
a e G
commutes with every
Lf
Rg
LaX
and
LrRg with
this algebra
consists simply of all finite sums of such operators.
Now It is easily veri-
fied by direct computation that
- R a , where
and
fa(x) = f(a Ix)
L L
a f
= Lf
La
RaR
g It follows that
ga(x) - g(xa).
under left multiplication by the
and
g
a
and the
Q0
Ra, and it is not difficult
Q invariant
to deduce by an approximation argument that so also is (cf. [8], p. 80). also that
RaX
is invariant
A trivial modification of a proof in loc. cit. shows
and
LaX
are continuous functions on
G
to Q , in the
uniform topology on Q . The remainder of the procedure for obtaining the
L,,
and the
Rf
is also similar to that used in loc. cit., and we shall merely outline it.
We define Ly (a)
on
-qy(A)
by the equation Ly (a) rtr(T) =
Y( (LgT).
It is easily verified that L, (a) 7(1(T) is single-valued and that for each
a r. G,
L{ (a)
is an isometry on
ly extended to an isometry, denoted by
71 , ,
L ,.(a), of
It can therefore be unique',(
into 1, .
The
n mapping
a -> L, ((a)
is a representation, and hence so is the mapping
52
I. E. Segal
a -> L,,(a).
I,,
Plainly
the identity operator on
tinuous on
to Q , and
G
It follows by continuity that
to 7Wy
x,,E /,r.
for each fixed
properties of
h,..
G
identity representation and
LaRbX = RbLaX
J1
*'
to 1, ,
is con-
so
for any fixed
to
X6Q.
is continuous as a function on
Similarly for the definition and
(These definitions are for the
a state; for the null set of other
Now
Q
is continuous on
L ,(a)xr
a -> LaX
Now the map
is unitary.
yjy
is a continuous function on
L,( (a) ?j1(X)
is the group identity and
e
L,t(a)L((a'1) = Ly(a 1)Ly(a) _
so
Le(a)
I,( , which shows that
0
(where
L ,,(e) = L,,
we take
such that
7(
Ly
and
u){
is
to be the
R,,
to be an arbitrary conjugation on
H 1
for any X a at which implies that
L,' (a)R5 (b) Jf(X) = R y (b)Ly (a) Yjy(X), and as L ((a) and R y(b) are bounded and r(j(Q ) is dense in 1(y, it follows that Ly (a)R y (b)xy = Rt (b)Lt (a)x,, for all x, a 14,., i.e., L,.(a) and Ry(b) commute for all a and b in G. Next, it is easily verified that JLaJ = He, and it follows that for X E Q, JLaXJ = RaJXJ, or (LaX)J = RaXJ. Hence yty((LaX)J) = vt,(RXJ), and by the definition of
Theorem 8,
in Q , we have
16r
and
Jy Ry (a)J, agree on the
and X is arbitrary 99y (LfR9) )1',(X) = y(LfRgX). We note that
f
are arbitrary in
and g
ffLaRbX f(a)g(b)dadb
As
L y(a)
Yy(Q ), and therefore coincide.
Now if
relative
in the proof of
J,, )7f,(LaX) = R3 (a) ?f(XJ), or Jy L1 (a) Yy(X) = R5(a)Jy 7ly(X).
Thus the bounded linear operators dense set
Jy
LI(G)
exists as a strong vector-valued integral (i.e.,
to the uniform topology on a ), and equals
is a continuous linear operation, it results that
J / VLaRbX) f(a)g(b)dadb, but 91y(LfRg)x5
=
(cf. loc. cit.).
LfRgI
rry(LfRe) _
rfy(LaRbX) = L,. (a)R,j (b)'y(X), so we have
/JL, (a)R,. (b)x y f(a)g(b)da db
for xr = y .m. As y(y (a ) is dense in Sy , and as 99r(LpRg) and /1L5 (a)R, (b)f(a)g(b)dadb are bounded linear operators, it follows that
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I the preceding equation is valid for all quence in
such that
Ll(G)
outside of
n nWn
Wn, where
verge strongly to decomposable.
gn(a)
Lf, so that
,
Now if
x1 a 14,,.
`gn
fgn(a)da = 1, and
0,
= {e}
53
LfRgn
then
is a se-
}
vanishes
gn
is easily seen to con-
is decomposable, and similarly
Lf
Rg
is
On the other hand, by the Fubini theorem for vector integra-
tion ffLi, (a)Rr (b)xr f(a)gn(b)dadb = fRy (b) [JL , (a)x,, f(a)da]gn(b)db, which expression is easily seen, by virtue of the strong continuity of
to converge strongly as n --> co, to decomposition of
A . (a)x( f(a)de.
is as stated, and similarly for that of
Lf
It remains only to show the irreducibility a.e. of combined action of the in
L d
G.
(a)
and
g
and the
L r(a)
R
Now if a closed linear manifold in
and the R d (b) in
L1(G)
Thus the
and
a
(b),
1+r
b
Rf.
7YY under the being arbitrary
is invariant under the
it is also invariant under
for all f
99,,(LfRg)
92r(a).
(of* loc, cit.), and hence is invariant under
Now as shown at the end of the proof of the preceding theorem, C
Q
where
w shows that
i9 the weak closure of d
Q, and it follows that
is maximal abelian in
of
f -> If
to Q , implies the separability of
Q
follows.
Q o,
L1(G)
f -> Rf
and
on
from which the separability
fe,(Q)
Hence Theorem 4 implies that
Remark 9.1.
This
The separability of
Q'.
together with the continuity of the maps L1(G)
(Qw)',
is irreducible a.e.
In the special case (for semi-simple Lie groups, con-
jecturally the general case) that L is a direct integral of factors of type I, the situation can be further reduced, in that the corresponding twosided irreducible representations of the group ion
from one-sided representations.
there is a Hilbert space representation
U,,
of
G
arise In an obvious fash-
Specifically, for almost all
( ,
a strongly continuous irreducible (one-sided) 0
on
and a conjugation
that the foregoing two-sided representation equivalent to the representation
{L, , Ry ,
of
Cr
{LV , Re , Jy} J'r}
of
0
1C'
, such
is unitarily
on the Kronecker
I. R. Segal
54
product 74r = 1' # J{y, where Lt. , R'ly , and Ja are determined by the equations L'j (a)(x#y) _ (Uy (a)x)#y, R',r(a)(x#y) = x#(Cr U( (a)C,, y), and J,, (x#y) = (Cr y#Cy x), for all x and y in li(,, and aaG. If y{', is taken as L2(M) for a measure space M = (R,)V, r), then 14' can be taken as L2(M X M) and Jk can then be defined by the equation
(Jf f)(x, y) - f y, x),
fE 7°fY
generated by the
L
f
and the
(a)
W*-algebras
be respectively the
and
To see this, let
a e G. and observe that Z =
R ,,(a),
ed,.t.( 6'), say a - fS, d1u( 6), then St E d?( Y). For if S e for all 2( . Now if W = Rf with f eLl(G), then W - fW y dpL ( 6 ), where
WY
Hence
Sf
fR y
=
(a)f(a)da, and it can be seen that Wr 6 RV, (cf.
Wr
and
commute, and it follows that
which implies S E {Rf}'
or
3
W
and
[10]).
commute,
3 a 'i' _ t . Thus t D Ze
b') .
On the
other hand, it follows from [10] that every element of X is a strong sequential limit of (bounded) operators of the form
Lf, with
f e L2(G)
Every such operator in turn is a weak sequential limit of operators of the
form Lf integral equals
with
f e LI(G), for if
and
fn
f
{K }
L2(G), the
are arbitrary in
exists.
Lf,
It follows from the Lebeague convergence G
is a sequence of compact subsets of
vanishes nearly everywhere outside of
is the product of
{Lfn)
g
(see [11]), and by Fubini'a theorem the integral
fff(y)g(y lx)h(x) dxdy theorem that if
and
exists and by virtue of the boundedness of
/ (y)g(y-lx)dy
(Lg)(x)
h
f
such that
UnSn, and if
with the characteristic function of
converges weakly to
Lf.
I
, then
As weak and strong sequential limits of
decomposable operators are likewise decomposable, it follows that every operator in
Ily d(6').
L
is decomposable, i.e.,
1 cf.Lyd,u( 6'), and hence
Now by the irreducibility a.e. of fL r , R , . , Jy)
.4
7Qr = 8,,, where
/3,,
we have a.e.
is the algebra of all operators on
>/r .
Clearly
DECOMPOSITIONS OF OPSnATOzt ALGLblika. I e7
It follow, readily as in the proof of Theorem 5 that Z
d.
IF, are factors.
-e
Now assuming that
and
e
is of type 1, we shall show that
r
has the special form given above.
[L,. , Ry . Jy }
For this it is
ly sufficient to establish th' following lemma, which includes a rear°_t recently announced by Godement [ 4a J. Let
LEMMA.
{L, R, J1
be a two-sided irreducible strongly con-
tinuous unitary repreaertstion of a topological group
W*-algebra Z generated a the
space ?¢, and let the of type I.
a fO, be
L(a),
Then there exists a one-sided irreducible strongly continuous
recresentation
7?, such that
U
0
of
Re
76', and a conjugation
J' is the conjugation of x
J'(x#y) = Cy#Cx, for all
are the representations of
L'(a)(x#y) = (U(a)x)#y
and
of
C
is unitarily equivalent to the system
He = X# 1(,
where
fined by the equation and
on a Hilbert space
{L, R, J,7,'}
J1, 7'p}
{L' , R' ,
on a Hilbert
0
O
and
in J{ , and
y
the equations
defined
R'(a)(x#y) _ (x#CU(a)Cy)
de-
Jys'
for all
x
and
y
in K. By [7 , pp. 138-9 and 1741,
is mapped into the set operator on
X1, and the
into the set
7Q1
k2.
J
that
Now
J'-t1 J'
1
le
generated by the
S#I
.
J'
of
1¢'
with
R(a)
of all operators of the form IT with T maps into a conjugation
1i
in such a may that oL
of all operators of the form
Wit-algebra
14/' .
is unitarily equivalent to
for suitable Hilbert spaces X1 and
k'1#
on
11'
S
is mapped
an operator
with the property
As the dimension of a Hilbert space is the maximal
number of mutually orthogonal minimal projections in the algebra of all
operators on the space, and as the mapping X -> J'XJ'
morphism preserving adjoints, K
we can set k1 = t(2 = k'.
1
Plainly
equivalence into a unitary operator
is a ring l.so-
and k2 have the same dimension and L(a.)
L'(a)
an
is mapped by the foregoing
on k# 1? of the form
56
I. E. Segal
L"(a)#I, where the W*-algebra generated by the Lt(a), a a G, is .l.
Now
it is easily seen that the map T*I -> T from X1 to the operators on x is strongly continuous.
It follows readily that the map a -> L"(a) is a
strongly continuous unitary representation of G on 11, and that the strong
closure of the algebra generated by the L"(a) is the algebra 6 of all The latter feature implies that L" is an irreducible
operators on k . representation.
Similarly R(a) is mapped by the foregoing equivalence into
I#/R"(a), where R" is an irreducible strongly continuous irreducible unitary representation of G on 1(. For any T 6 16
tion 99 on
.
we have clearly J'(T#I)J' = I# f(T), for some func-
It is readily verified that 99 is an (adjoint-preserving)
ring automorphism of '0 of period 2, and with the property that 9 (a.T) It is not difficult to deduce that
CL 2(T) for complex GY,and T E
there exists a conjugation J" of
such that
(P(T) = J"TJ" for T E 7S
Now let C' be the conjugation of 14' determined by the equation C'(x#y) _
J"y*J"x, for x and y in k.
It Is easily seen that J'C' is a unitary op-
.Ll. Asl is algeb-
erator U' on H' with the property that
raically isomorphic to Z, every automorphism is inner, and so there exists a unitary operator V' in .L`1 such that U'*T'U' = V'*T'V' for all T e t I. It follows from the last equation that U'V'-1 a unitary operator in
'l.
so that U'=V'W' with W'
Evidently V' = V#I and W' = I#W, where V and W
are unitary operators on k, and it results that it = (V#W)C'.
Now J' (T#I) (x#y) = (i## c(TJJ' (x#y) for all x and y in k, and substituting the above expressions for J' and for 9 , it is found that (VJ"y)#f(WJ"Tx) = (VJ"y)#(J"TJ"WJ"x).
It results that WJ"T = J"TJ"WJ", and
multiplying on the left by J", it follows that J"WJ" commutes with T. T is arbitrary in
19, this implies that J"'NJ" = I and hence .d = I.
As
The
equation J'2 = I implies that J12(x4y) = x#y for all x and y in k, and
DECOMPOSITIONS OF OPERATOR ALGEHRAS.
57
I
substituting J' = (V#I)C' there results the equation (Vx)#J"VJ"y) = x#y. Hence V = I and J' = C'.
The proof of the lemma is concluded by the observ-
ation that as JL(a)J = h(a), a e G, we have h"(A) = W(L"(a)) so that R"(a) = J"L"(a)J". 10.
Deflation of decompositions.
In this section we consider the
problem of replacing the regular measure space ( T, )p.) which has figured in the preceding decompositions by spaces which are measure-theoretically equivalent, but which have different topological properties.
Our first
result asserts roughly that under appropriate, but fairly general, circumstances, ( r ,,.L) may be replaced by a regular measure space (
, V ),
which Is a kind of "deflation" of ( F, p ) which arises naturally in For example, in the case of the reduction of the
certain circumstances.
regular representation of a locally compact separable abelian group 0, the measure ring of ( r ,
,
)
is identical with that of the character group G*
of G under Haar measure; but r is topologically much "larger" than 0*, roughly speaking.
The general process described in the next theorem could
be used to replace r in this situation by Ga. THEOREM 11.
Let (2, C and z be as in Theorem 1, and suppose Q con-
tains the identity operator.
Let e be the closure in the uniform topology
of the algebra generated a all functions on r' of the form
T E Q
.
with
Let Q be the (unique) compact Hausdorff space such that
isomorphic to C( A ), and let 99 be the continuous map of r onto
is
such
that if f 64f, and if f corresponds to F e C (A ) in the isomorphism of E
with C( A ), then f( 3') - F(5P ( a()) for all a' E F . Then setting Wt when dS= 92(,() , there is a regular measure v on Q such that
(1) J (T) d).c(') =
'
J tb(T)dv(c5)
(3) the mapping
state space of Q ;
Zg
;
(2) C51 # r52 implies that
is continuous on
to the
(4) if the hypothesis of Theorem 4 is satisfied, then
I. E. Segal
56
TS is pure for almost all b relative to (8 , V );
(5) if Q is dense
in C' in the weak sequential topology, then there is an algebraic isomorphism S --:* S'(.) of C onto the algebra of complex-valued bounded measurable
functions on ( , V ) such that (STz,z) _ p S' ( G )
)) for
dV(
ailSeC andT eQ The existence of the Q and q described in the theorem is assured by known results [ 14 ]
.
We define V on Borel subsets E of Q by the
equation: V(E) _ /.+(T -1(E)); then it is readily seen that V is a regular If f is an arbitrary real-valued element of
measure on
4f()dp() where F( 9 ( (
D'
=
))
CJ )d v ( ) ).
f Nd[
dV [
i
= f( 2( ), F being in C( O ), and so
image under is not pure ]
c*
for all f 6
Ci
( a '2) if and only if
To see (4), observe that the inverse
.
of the Borel set [ 6 1
-Ca,
is not pure] is the set [ ' I &){
, which by Theorem 4 has measure zero.
need the following lemma.
To establish (5) we
Our method of proof here could be used to give
a simple demonstration of a theorem of Dieudonng [ 3 ] LEMYJ 11.1.
IF(S) < NJ, f( 3' )d/p(
Now defining rb as above, (1) and (3) are obvious, and
(2) follows readily from the fact that 'p ( a'1) = f( 1( 1) = f( 7(2
(6, then
.
Let ( T , rl) be a compact perfect measure apac
and e
a uniformly closed SA subalgebra of c( r ) which is dense in C( r') in the
weak topology on C(r) as the con-
ate space of L1( 1
, )w ).
the spectrum of a and let 9D be a continuous map of F onto A .
Let A be Then
there exists a regular measure v on Q and an algebraic isomorphism A of
C( r
on ( Q
onto the algebra of all complex-valued bounded measurable functions
v ) such that :
(1)
(2) if f s C (P), then
if f 6 & , then (A f) (c (7( )) = f ( ') ,
f( i
Let f be arbitrary in C (1
1) = ).
`
A f)( CS )dv( 6 ).
Then there exists a sequence
{fn}
in
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
E which converges weakly to
59
f, so that in particular
A (a')g(Z()du(K ) ->
for gE
Defining V as
.
above it results that
./fn( a')g( z' )dpt( 7l) _ fIAofn)( c5)(Aog)( c5 )d V ( c5), where A0 is defined by the equation (Ac h) ((P (a') ) = h( ?r), a( F_ r' for h e E .
Now
HfniI
is necessarily bounded by a theorem about weakly
convergent sequences of linear functionals on Banach spaces, and as by reg-
ularity C(A) is dense in L1(L , V ) it follows that L(,(,8, V ).
weakly convergent in
{Aofn}
is
As this latter space is weakly sequen-
tially complete, there exists a bounded measurable function F on 8 to which the sequence
fAofn )
is weakly convergent, and defining A by the f c E , one of the values of
equation Af = F, it is clear that for
is
Aof
ff( a')g(f )d,-( a')
and that
g E E .
f E C (1') and clear that
At
single-valued.
AAf)(b)(Ag)( 5)dv( 6), if
f, g, and a selection of
Fixing
is determined as an element of Thus
Af = Aof
Ag, it is
(A, v ), so that A
L
is
f e E , so that (1) in the conclu-
if
It is obvious that for arbitrary
sion of the lemma is satisfied.
f s
_
At
C(r'), ff(Y)du(e) _ fiAf)c)dV(c). A
It is easily seen that
products, let
f
be in
limit of the sequence sequence
is linear.
C( P) and ffn}
in
g
in E , and let
Then At
Chi .
To show that f
A preserves
be the weak
I. the weak limit of the
fAfn} , and as multiplication is continuous in each factor sepa-
rately in the weak topology, we have
fg = weak limn fng, so
A(fg) = weak limn A(fng) and A(fng) = A(fn)A(g) --I- A(f)A(g), so that A(fg) =
A(f)A(g).
By a repetition of the procedure just utilized, it
follows that the last equation is valid for arbitrary f
and
g
in C(r )
Now A is univalent, for if At = 0, then from the fact that for all
'
g s 8, that
ff( i)g(1()d,u(a') = f(Af)( b)(Ag)(c5)dV(6), it results ( a ' )g ( I )d,M ( a() = 0 for all g e &. From this last equa-
tion it is easy to deduce that
f=
0.
I. E. Segal
60
It remains only to show that A is onto.
C(o) which converges weakly to
in
is a bounded
F
(4 , V ), by regularity there exists a sequence
measurable function on fFnJ
If
fn = A-1(Fn
Now if
F.
from the equation IF, (b) (Ag) (b )d V (b ) = ffn ( ()g( )d,,( d') for g e e , it results that the sequence fJff (1( )g(d )d,u (7l ) ] has a limit for all
g a E .
above, so that
C( P)
Now
IIFnIj
is bounded by the theorem mentioned
is bounded (for an algebraic isomorphism of a
11fnIt
into an algebra of essentially bounded measurable functions pre-
serves norms), and it follows readily that the foregoing limit exists for all
g E L1(T', )u).
Making use again of the weak sequential complete-
ness of the conjugate space of an L1-apace over a finite measure space, it results that there is an element
f
C( P)
in
such that
--e.ff( z')g( t')dJt( i) for all g E E. Clearly for all g c 6, fAf)(t5)(Ag)()dv(b) = from which it is easy to conclude that P = Af.
ffn( If )g(
The validity of conclusion (5) of the theorem follows directly from the preceding lemma together with Theorem 1. Example.
As an illustration of the use of the preceding result,
consider the situation described in Section B;
is expressed in the form variant measures.
the invariant measure m
fm,, d u ( J), where the my are ergodic in-
Taking Q , C, and
z
as in that section, it results
that we can also write, for any continuous function f on M,
= tg (Qf)dv (c ), where Qf Is the operation of multiplication by f, which by the same argument as In that section, leads to the equation m(E ) = m's (E)d v ( for any Borel set E in M. Mf(x)dm(x)
Here
ml
a. e. on
[i(I m.
is the measure associated with the state ( 6, v ), for the inverse image of not ergodic J
13
[6 1 m'
, and is ergodic
not ergodic]
, which has measure zero by Theorem 4 .
is
We note
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
also that m # i
if
ml
61
c52
z We conclude by showing that the measure space
( r,
,
which
)
occurs in our decompositions can be replaced not only by any equivalent regular compact measure space, as in the preceding theorem, but, under a separability restriction, by any equivalent measure space (rot necessarily bearing a topology), two measure spaces being regarded as equivalent if there is an algebraic isomorphism between their measure rings.
lar, if separable as a measure space, (r, ,u)
In particu-
could be replaced by a
measure space over the Borel subsets of the reals, and the decomposition of von Neumann thereby obtained. THEOREM 12. that
Let
Q, C, and
z
be as in Theorem 1, and suppose
Q is separable (in the uniform topology) and contains the identity Let
operator.
=
M
(R, 1e
r)
be a measure space such that
C is alge-
braically isomorphic (with preservation of ad joints) to the algebra of all
complex-valued bounded measurable functions on measure
r'
on
R, an r-null set
the state space of
A
such that:
ous with respect to each other; function of (TSz, z)
on
p
M;
(3)
for
Ro, and a map r
(1) (2)
for
T 6 [j
= fR cJ p(T)U(p)dr'(p), where
corresponding to
and T
Then there exists a
N.
p r'
U )p
U(.)
R-R0
to
are absolutely continu-
e Q , L)p(T)
and
on
is a measurable
S 6 C , is the function on
R
U.
Certain parts of the proof of this theorem closely rese?:ible the
proof of Theorem 1,
fixed T
,
- we shall merely sketch these portions.
(STz, z) can in an obvious fashion be regarded as a continu-
ous linear functional on LQ (M) (in its norm topology). Sn(.)
For any
ldoreover, if
is a sequence of elements of LOD(M) such that 1 > Sn(p) > 0 and
Sn(p) > Sn+l(p) for all n, and limn Sn(p) = 0, all these conditions holding for almost all p F
= F.
N, then lim F(S ) = 0, where we set
To sce this, observe that from the given algebraic isomorphism
I. E. Segal
62
of
C with Lc (M)
sponding to
it results that if Sn
I k Sn > S1
Sn(p), then
this situation there exists an operator converges strongly, so that
C corre-
is the element of
It is known that in
0.
to which the sequence
S
FT(Sn) _ (SnTz, z) -->(STz, z).
{Sn }
On the
other hand, as an algebraic isomorphism preserves order among the self-
adjoint elements, Sn(p) >
S(p) > 0 a. e., which shows that S(p) = 0
a. a. and S =
(STz, z)
0, so that
= 0.
We next show that for any continuous linear functional L,4M)
with the foregoing property, there exists an element
that
4(k)
tion on
= f,,k(p)f(p)dr(p)
defined by the equation
R
characteristic function of and the
for k e LOO(M). Let
E (E a R).
f C L1(M).
for some
when k
R k(p)f(p)dr(p)
characteristic functions of sets in
So
s(Ei).
linear combinations such that a. e.
k, and hence for all
k
kn(x)
(STz, z)
We have
is a finite linear combination of le.
Now if
k
is an arbitrary non-
tT
is the element of
= fS(p)fI(p)dr(p), then
on the set E
3
k
of such a. e., and
e Lo'(M).
in I
, and if
sponding to the characteristic function of
but S2 =
{kn }
increases monotonely to
,T
fi(p) = 0
s
It follows that the same formula holds for such a
Next we show that if equation
Thus
zero; and it is
negative-valued element of LOO(M), there exists a sequence
with kn(x) > 0
is a
< nEi
It results from the Radon-Nikodym theorem that
3(E) = d E f(p)dr(p)
c(k) =
be the func-
a
E - 'j Ui
'
4( xE -2:in 1 x'g i ) -* 0. It follows that s(E) is countably-additive; it vanishes on sets of r-measure 1I+11.
such
f e L1(M)
= UiEi, with Ei 6 it
E
Now if
sequence of functions converging monotonely to zero.
bounded by
on
s(E) = 4 (Y -E), where xE is the
mutually disjoint (i = 1,2,...), then
Ei
41
S
L1(M)
f'(p) >
I
0
defined by the a. e.
For if
is the element of C correE, then
as (XE)2 = xE , so (Sz, Z) _
(Sz, z) = 0 clearly,
IISZI/ 2
and
Sz = 0.
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I
TSz = 0
This implies density of
in
Ciz
for
T
r.
or S(Tz) =
CY
H, it follows that
S
0, from which, by the
= 0, so that
Now it is easily seen that if
measure zero.
0
non-negative-valued, then the corresponding function therefore
tive-valued.
fl(p)
This stows that by the measure E
C
1t'.
(STz, z)
Let
=
r'
fT
r
ex.
must be of
whenever h
now be defined for
T
6 C.
R
= fEfI(p)dr(p),
r'(E)
e 0 by the equation If
T > 0, (STz, z)
is a positive
U
are arbitrary
fT(p) > 0 a. e.
If
T
and
is an arbitrary complex number, then it is easily seen
that fT+U(p) = fT(p) + fU(p) and faT(p) = CtfT(p) a. e. partially normalize
is
is a. e. non-nega-
can be replaced in the integration over
JS(p)f1(p)dr'(p), S
0 and if
f
E
a. e.
defined by the equation
linear functional on C , so in
> 0
63
fT(p)
by breaking
L00 (M)
We now
into equivalence classes,
two (residue classes of) 'functions' (modulo the subspace of null functions)
being equivalent if they are proportional (relative to constants), then selecting one 'function' from each equivalence class and then choosing any representative from the corresponding residue class, in an arbitrary fashion except for the following restrictions:
1) the representative of a
'function' which is proportional to a 'function which is non-negative a. e.
shall be everywhere proportional to a non-negative function;
2) the abso-
lute value of the representative at any point shall not exceed the norm of the 'function' in
L ,(M);
3) the 'functions' which are zero and one a. e.
shall have the representatives which are respectively everywhere zero and one;
4) a 'function' which is a. e. proportional to a real-valued function
shall have a representative which is everywhere proportional to a real-valued function.
It is clear that a choice of representative can be made sub-
ject to th-se restrictions, and that if ing
a.g
g
is any representative, assign-
as the representative of the 'function' a. e. equal to
a representative for each element of
LA(M).
We now assume that
CLg yields
fT is a
64
I. E. Segal
T E Q..
representative, for all We have now
T E Q .
and
p E R-Ro, and
To do this, let
T
and all
U
for all p e R
and
T
and
00.
in
and p 6 R-Ro.
Now if
follows:
LT.}
let
U
such that
R0
fT+U(p) = fT(p) +
T
Q is separable. in
fU+T(p)
fU(p) + fT(p)
=
We define W p(T) =
p e R-Ro
for all
T E Qo
for
fT(p)
is arbitrary in Q , we define W p(T)
T
r.1p(T) = limn Wp(Tn).
Then the set of
is countable, and hence there
o
be a sequence in
of our normalization,
in Q ,
Ro
I, and which admits multiplication by rationals and
(U, T), with
is a null set
U
and
i; such a subring exists because
all pairs
fT*T(p) > 0
be a countable SA subring of a , which is
Q o
dense in 0 , gontains by
and
Roughly speaking, it remains only to obtain a null set
such that for all fU(p).
= a fT(p)
f aT(p)
As
Jg(p) J
Tn --- T, and set
for g e Loo (M) by virtue
< 11gil
'w p(Tn - Tm)I
W p(Tn - Tm) = fTn-Tm(p) = fTn(p)
such that
Qo
as
IITn - Tj , and clearly
= W p(Tn) - Wp(Tm), so the
fTm(p
foregoing limit exists; and it is easily seen to be independent of the sequences used to approximate
readily deduced from the additivity of W p
T and
U
for
Tn =
T
is SA and
Qo
Tn - T
p * Ro, for if
(1/2)(Tn + Tn) E
Q.
p f Rot so also is & p(T).
p t Rot for if TRTn
on
Tn --.'*-T
o
and
Tn
It is
that for arbitrary
in Q , LJp(U + T) = Wp(U) « Wp(T) for p 4 Ro.
is real if then
is single-valued.
T, i. e.,L) (T)
with the and as
Moreover, 4)p(T*T) > 0
Wp(T)
Now
Tn
in
0'0'
& (T') is real for
T e Q,
and
with the Tn 6 Qo, then TnTn --- T*T, and as
a Qo, wp(ToTn) > 0.
If O(, is real and rational, then plainly tJp(aT) = Clu)P(T) for T C Qo and p Ro. If c is any real number and La.nI a sequence of rationals converging to Of., then wp( anT)
wp(iT) =
W
p
--> aT so that
and hence Wp(aT) = o[wp(T) for p * R0. for all p and T e O ct it follows that
( aT)
i Wp(T)
oenT
As
DECOMPOSITIONS OF OPERATOR ALGEBRAS. I.
a.Wp(T)
Wp(or.T) =
for all complex
CC
65
T C Q.o
and
(p
It
f Ro).
is not difficult to conclude from this by the method just used that the same equation holds for all
T E a .
Wp
it follows from the fact that W o(T)
for any
T
E CZ.
It is obvious that W p(I) = 1, and
Thus W p
is a state of
conclude the proof it is sufficient to show that
Q
for
T E Q .
such that
Hence if
Tn -> T, then
the limit of
fT (p).
fT(p)
fIn -- fT
a, e.
in p
Ro).
II fT U
_<
is a sequence in
lTn }
Loo(M), and so
fT (p) = W (Tn) n
n
W p(T) =
As
and if
(p
Wp(T) = fT(p)
This is true by definition for T E 0-0. We recall that T C Q
Wp(T)
is real on SA operators that
(p
To
a. e. N T+1
00
fT(p) is a. a.
f Ro), this shows that
66
I. E. Be gal
REFERENCES 1.
Duke
W. Ambrose, Spectral resolution of groups of unitary operators.
Mathematical Journal 11(1944) 589-595. , The I,2-system of a unimoduler group I.
2.
Transactions of
the American Mathematical Society 65(1949) 27-48. 2a.
,
Duke Math. Jour. 9(1942) 43-47.
ures. 3.
P. R. Ralmos, and S. Kakutani, The decomposition of meas-
J. Dieudonne, Sur le theoreme de Lebesgue-Nikodym III.
Annales Univ.
Grenoble, Sect. Sci. Math. et Phys. (N. S.) 23(1947-48) 25-53. 4.
I. Gelfand and D. A. Raikov, Irreducible unitary representations of
locally compact groups.
Mat. Sbornik (Rec. Math.) N. S. 13(1943) 301-316
(in Russian). 4a.
R. Godement, Sur la theorie des caracteres. I. Definition et classifi-
cation des caracteres. S.
C. R. Acad. Soi. Paris 229(1949) 967-69.
F. Mautner, The completeness of the irreducible unitary representa-
tions of a locally compact group.
Proc. Nat. Aoad. Sci. 34(1948) 52-54.
, Unitary representations of locally compact groups.
6.
Ann.
Math. 51(1950) 1-25. 7.
F. J. Murray and J. von Neumann, On rings of operators.
Ann. Math. 37
(1936) 116-229. 8.
I. E. Segal, Irreducible representations of operator algebras.
Bull.
Amer. Math. Soc. 53(2947) 73-88. , A class of operator algebras which are determined by
9.
groups.
Duke Math. Jour. 18(1951) 221-265.
10.
, The two-sided regular representation of a unimodular
locally compact group. 11.
modular groups.
Ann. Math. 51(1950) 293-298.
An extension of Plancherel's formula to separable uniAnn. Math. 52(1950) 272-292.
DECOMPOSITIONS OF OPERATOR ALGEBRAS. 1.
and J. von Neumann, A theorem on unitary representations
12.
of semisimple Lie groups. 13.
Ann. Math. 52(1950) 509-517.
M. H. Stone, Linear transformations in Hilbert space.
New York 1932.
, Application of the theory of Boolean rings to general
14.
topology. 15.
67
Trans. Amer. Math. Soc. 41(1937) 375-481.
J. von Neumann, On rings of operators. IV. , On rings of operators.
16.
Ann. Math. 41(1940) 94-161.
Reduction theory.
Ann. Math.
50(1949) 401-485. 17.
, Zur algebra der funktionaloperationen and Theorie der
normalen operatoren. 18.
Ann. Math.
Math. Ann. 102(1930) 370-427.
, On some algebraical properties of operator rings. 44(1943) 709-715.
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II:
MULTIPLICITY THEORY
by I. E. Segal
of the University of Chicago 1.
algebra ( =
Introduction.
We determine the most general commutative W3:--
weakly closed self-adjoint algebra of bounded linear opera-
tors on a Hilbert space) within unitary equivalence.
Every such algebra is
a direct sum of We-algebras of "uniform multiplicity" and an algebra of the latter type of multiplicity n is unitarily equivalent to an n-fold copy (roughly speaking) of a maximal abelian W*-algebra.
This last algebra is
unitarily equivalent to the algebra of all multiplications by bounded measurable functions, of the elements in L2 over a suitable measure space, and is determined within unitary equivalence by the Boolean ring of measurable subsets modulo the ideal of null sets in the measure space.
Thus to each
commutative algebra, there is for each multiplicity (cardinal number) n, a Boolean algebra B(n), and this function B determines the algebra within unitary equivalence; conversely, if B is any such function (vanishing on sufficiently large cardinals), then there exists a commutative W*-algebra whose multiplicity function is B.
The classification by Maharam of Boolean
measure rings shows that the measure spaces in question here can be taken to be unions of spaces measure-theoretically identical with the product measure spaces Ip, where I is the unit interval under Lebesgue measure and p is a cardinal number (with 10 defined as a one-point space), and allows the replacement of B(n) as a complete unitary invariant by a cardinal-num-
ber-valued function F(p, n) of two arbitrary cardinals giving the number of
I. E. Segal
2
copies of IP whose measure ring is a constituent of B(n); and corresponding to any such function there is a commutative WK-algebra.
Similar but more limited results are obtained for W*-algebras which are not necessarily commutative.
As in the commutative case, every
W-re-algebra is a direct sum of We-algebras of "uniform multiplicity", and an
algebra of the latter type of finite multiplicity n is unitarily equivalent to an n-fold copy of an algebra of uniform multiplicity one.
When n is in-
finite the last conclusion is invalid except in special cases, notably in that of algebras of "type I".
These are algebras which, roughly speaking,
are direct integrals of factors of type I, and for them we give a complete structure theory and set of unitary invariants.
Specifically, such alge-
bras are characterized by the feature that their part of uniform multiplicity n is unitarily equivalent to an n-fold copy of a Was-algebra en of uniform multiplicity one; and are determined within unitary equivalence by the
knowledge for each n of the unitary-equivalence class of the commutative algebra associated with &h by virtue of the fact that the set of operators commuting with an algebra of uniform multiplicity one is commutative.
Most known results in commutative spectral theory either follow readily from the foregoing classification of commutative We-algebras, or are seen thereby to be equivalent to questions in pure measure theory.
Di-
rect consequences of our classification (together with the known structure
of separable measure spaces) include von Neumann's theorem that on a separable Hilbert space, any commutative W*-algebra consists of functions of some operator in the algebra, and the fact that such an algebra is maximal abelian if and only if it has a cyclic vector.
Any commutative '.V*-algebra
is algebraically isomorphic to a maximal abelian Was-algebra via a mapping which is weakly bicontinuous and which preserves the operational calculus.
The theorem that the We-algebra generated by a self-adjoint operator on a separable space consists of all bounded Baire functions of the operator is
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II
3
extended to arbitrary spaces, and a brief derivation is given of the Wecken-
Plessner-Rokhlin unitary invariants of a self-adjoint operator. Our approach has significant contacts with both the Nakano and Wecken-Plessner-Rokhlin treatments of the multiplicity theory of an individual operator, as indicated below in the specific instances.
In particular,
the present definition of algebra of uniform multiplicity for the commutative case is essentially due to Wecken, and the definition which we use in the not necessarily commutative case (while commutative algebras could be treated in terms of this latter definition, which we show to be equivalent to the former definition in the commutative case, it has seemed desirable in view of the central rAle of commutative algebras to treat this case separately) is a variation of that of Nakano for abelian rings of projections. Much of the present material (notably Theorems 1-3 and 5-6) was given in a course on spectral theory at the University of Chicago in the Spring term of 1949.
We are indebted to members of the course and especial-
ly to L. Nachbin for valuable criticisms and suggestions. PART II. COMMUTATIVE ALGEBRAS 2.
Definitions and technical preliminaries.
Definitions 2.1.
A W*-algebra is a weakly closed self-adjoint
algebra of (bounded linear) operators (on a Hilbert space).
Thruout this
paper "operator" will mean "bounded linear operator on a Hilbert space", "Hilbert space" is complex and of arbitrary dimension, and I denotes the identity operator on a Hilbert space which will be clear from the context. An algebra of operators
copy of an algebra 8
CL
on a Hilbert space '#
is called an n-fold
of operators on a Hilbert space k , n being a car-
dinal number greater than 0, if (1) there is a set n such that
series L.
(consists of all functions SI(f(x)J12
f
is convergent, with
on
S S
of cardinal number to
E
for which the
(f, g) defined as
I. E. Segal
4
Ex 6
S(f(x), g(x)), and (2) @ consists of all operators
(Af)(x) =
B
Bf(x), for some
in a .
A
of the form
(We make the usual convention about
infinite sums of complex numbers: they exist only if all but a denumerable number of terms vanish, and if the sum of this denumerable collection exists in the sense of being absolutely convergent).
A masa
algebra of
operators is one which is maximal abelian in th^ algebra of all operators and self-adjoint
(i. e. closed under the operation of adjunction).
A
2 # 0 Is said to have uniform multiplicity n,
commutative Way-algebra
where n is a cardinal number > 0, if it is unitarily equivalent to an nfold copy of a masa algebra; the algebra consisting of the zero operator only is said to be of uniform multiplicity zero. Definitions 2.2.
R, a ring R on
of subsets of
{Ei}
such that if
1.
R, and a real non-negative-valued function
and r(U
E1)
i
of
R
is convergent, then U 1 E1 E R
Zi r(EI), and with the further property that
ishes on the void set.
M =
If
r
is a sequence of mutually disjoint elements
for which the series Zir(Ei)
of
W
A measure space is the system composed of a set
is called measurable if
van-
r
(R, R , r) is a measure space, a subset N n E E
is said to be equivalent to zero if W' E
7e
whenever E
F-
W
, and
Ti
is a null set for all
E
F. R_ .
A measure space is localizable if the lattice of all measurable sets modulo the ideal of sets equivalent to zero is complete.
A function on
R
to a
topological space is called measurable if the inverse image of every open set is a measurable
set, and two functions are called equivalent if they
are equal except on a set equivalent to zero. plex-valued
The Banach space of all com-
X th-power integrable (complex-valued) functions on
M
(mod-
ulo the subspace of functions equivalent to zero), with the usual norm, is
denoted by Lc, (M)
(1 G
Oc
< oD ); L00 (M)
is the space of bounded meas-
urable functions, the norm of a function being defined as its essential least upper bound.
The Banach algebra whose space is
L ,(M)
and in which
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II.
multiplication is defined in the usual way is denoted as bra of all operations on
L2(M)
5
B(M).
The alge-
(which denotes the usual Hilbert space, as
well as its Banach space) which consist of multiplication by an element of B(M)
is denoted by
_W(M)
and called the multiplication algebra of
M.
The central results of the part of this paper which deals with commutative algebras can now be stated. THEOREM 1.
A maximal abelian self-adjoint algebra of operators
on a Hilbert space is unitarily equivalent to the algebra of all multiplications by bounded measurable functions on the Hilbert space of complexvalued square-integrable functions over an appropriate localizable measure
space. We show in a paper CIO] on measure theory to be published separately from the present paper that two masa algebras are unitarily equivalent if and only if they are algebraically isomorphic (in an adjointpreserving fashion) which in turn is true if and only if the measure rings of the corresponding measure spaces are algebraically isomorphic.
By
virtue of the Maharam classification of measure rings, the last is the ease if and only if the measure spaces have the same cardinal number invariants naturally induced by that classification.
Conversely, the multiplication
algebra of a localizable measure space is masa (in fact is masa only if the We mention finally that a direct sum of finite meas-
space is localizable).
ure spaces (see below) is always localizable. THEOREM 2.
number n > 0
For any commutative W*-algebra 2 and each cardinal
there exists a projection
upper bound of the
Pn
n.
in
.
such that the least
(in the lattice of projections) is the identity
operator, and with the contraction of
multiplicity
Pn
(Z
to the range of
Pn
of uniform
There is a unique such function on the cardinals to the
projections in Q., and the
Pn are (necessarily) mutually orthogonal.
I. E. Segal
6
Before turning to the proof of these theorems we make some further definitions and remarks. Definition 2.3.
A measure space
ly) finite (in the present paper) if (locally compact) space if
R
(R, R , r)
R Cn .
is called (strict-
It is said to be a regular
is a locally compact Hausdorff space,
is
'R,
contained in the o'-ring generated by the compact subsets of
R and con-
tains all compact subsets, and if for any E 6:119 , r(E) =
G.L.B.,Er(W)
=
the compact subsets of
r
W
L.U.B.CcEr(C), where
,
and
C
range respectively over the open and
R, which are also in 79 .
For any compact space
denotes the Banach spaces of complex-valued continuous functions
C(T')
on r , with the usual norm.
A finite measure space
M =
(R,-P,, r)
called perfect if it is regular compact and if for every element of there is a unique equivalent element of
0(R).
is
B(M)
The system constituted of a
complete Boolean ring and a non-negative-valued countably-additive function on the ring is called a complete measure ring if every element of the ring is the least upper bound of elements of the ring on which the function is finite.
If
Mx =
(R7,., Vx ,
index X , and if the
r).
) are measure spaces depending on an
are mutually disjoint (as can be assumed with-
R,L
out essential loss of generality), then the direct sum of the the index set) is the space set of all subsets
E
of
(R, 1Z, r), where R
M,
U,, R7, , R
(over
is the
such that (a) E meets only (at most) count-
ably many of the RT , (b) E ^ Rr 6 R,, and for any such set
R =
for all 'X , (e) Z..r(E ^ RX)
E, r(E) = ZT r(E - R,).
For a discussion of many of the foregoing and related concepts,
we refer to [10] Definitions 2.4. algebra
O.
The spectrum of a commutative complex Banach
is the topological space consisting of the set of all contin-
uous homomorphisms of
Q. onto the complex numbers, topologized as a rela-
tive space of the conjugate space of
a, , in its weak topology.
An
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II
7
"algebraic isomorphism" between two W*-algebras is a correspondence which is an algebraic isomorphism in the usual sense, and with the further property that if two operators correspond, then so do their adjoints.
(In
other words, we take the "algebra" of a W*-algebra to be an algebra with a distinguished involutory antiautomorphism, viz. that of adjunction). An element for a set spans
,3
of a Hilbert space 1+
z
of operators on AA
If
variant under d ,
S e 4 , span m .
-fl,
is a closed linear subspace of 74
The contraction of 43
two
is called the commutor of d If
T
Sz,
3 E, is the
Jyt,, denoted as
of the form x - Sx, with
?It
The set of all operators on _H
(x E )1C).
which is in-
is called a relative cyclic vector when the
z
collection of all operators on
S E
which commute with each element
and denoted J -.
Is a SA (self-adjoint) operator on a Hilbert space
with corresponding resolution of the identity
fX dE T
S 6A
Sz, with
$/' (i. e. 14 is the smallest closed linear manifold containing
those elements).
of Z
is called a cyclic vector
if the set of all
), the spectral measure associated with
T
T
(so that
(ET 3
is the function E(.)
on the Borel subsets of the reals to the projections on
1
determined by
the condition that it be countably-additive, regular, and that if then
the closed interval of projections onfl 14
upper bound and
f.Pl,
indexed by
E(B) = E , then
Ur P.
f1-
If
P
B
is
is any family
denotes the least
the greatest lower bound of the
Pr
in the
lattice of all projections on 1 3.
Structure of maximal abelian War-algebras.
We prove Theorem 1
in this section and obtain some incidental results which may be noteworthy. In particular, it is clear from the proof that the measure space in question in that theorem can be taken to be a direct sum of finite perfect spaces.
8
I. E. Segal LEMMA 1.1.
space 19Z, then
Q is a SA algebra of operators on a Hilbert
If
is a discrete direct sum of subspaces each of which is
invariant under Q and has a relative cyclic vector for a . It is clear from Zorn's formulation of transfinite induction that there exists a collection
of mutually orthogonal closed linear sub-
, each of which is invariant under
spaces of $
Q and has a relative
cyclic vector for Q , and which is maximal with respect to these properties.
If
is the discrete direct sum of the elements of,._. , then
lt,
= ft, for if in I-A ,
x
is a nonzero element of the orthogonal complement of
the closure of Q x
is easily seen to be invariant under Q ,
to have the relative cyclic vector
x, and to be orthogonal to all the
elements of LEMMA 1.2.
A commutative W*-algebra with a cyclic vector is uni-
tarily equivalent to the algebra of all multiplications a bounded measurable functions on
over a finite perfect measure space.
L2
Let Q be a W*-algebra on 1 with cyclic vector
z.
It
follows from any of a number of representation theorems (see e. g. [11) ,
Q
Th. 1) that
is isomorphic as a Banach algebra to
is the spectrum of a . function on
1
.
tion W (T) _ functional on
For
Let W
T £ Q , let
T(.)
C(r ), where r
denote the corresponding
be the functional on Q defined by the equa-
(Tz, z), T E Q , and let W ' C(r ), so that
be the naturally induced
W'(T(.)) = W (T).
Then by the Riesz-
Markoff theorem there is a unique regular measure JA on I'
W'(T(.)) =
,n[
such that
T( i')d,-('), T E Q . It follows from (9) , Theorem 1,
that ( r, ,u) is a perfect measure space. Now let follows:
U0(Tz) =
Uo
be the function on
Q z
to L2(r, ,,A)
defined as
T(.), - T(.) is a continuous function on the finite
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II regular compact measure space Uo
preserves norms, for
9
and hence square-integrable.
(V'
=
IITz+I2
(Tz, Tz) _
(T3,Tz, Z) = W (T.*T)
This shows that Sz =
single-valued,//for if
(S-T)(.)
=0, and so or S(?() =
T( 2')
a. e.
S
and
T
Qz
As
is
Uo
in Q , then
(S-T)z
L2( r, µ ), 1. e. S(') - T( I') =
in
0
with
Tz
Now
a. e.,
0
is dense in 1+ , and by regularity
is (subject to an obvious identification) dense in L2(r ,1u ), Uo
on 1*
can be uniquely extended to a unitary transformation U
to
L2(P, JA). :1e show next that U takes Q onto z'K (T', lA ), the multiplication algebra of (F, /-G). This means that the map T -+ UTU-1 on 0 will be shown to be onto )1t(T', /pL). MT
where
by
T(.).
We show actually that
is the operation of multiplication of elements of
(1', p)
is perfect.
Now as
UTU-1
and
Now if
, ll )
ranging over a dense subset of L2(r ,
MT
are both
UTU-1 x = Mlx
bounded operators on L2(T, 1Lk), it suffices to show that x
L2 (I
MT,
The onto character of the map is then an immediate consequence
of the fact that
for
UTU-11 =
), e. g., over C(j'). ,
x = S(.)
x e C (r ), then
so UTU-1x = as the map
(TS)(.).
T - T(.)
for some
Obviously MTx =
S E Q
and U-1x = Sz,
T(.)S(.), which equals
(TS)(.)
is an algebraic isomorphism.
COROLLARY 1.1.
A commutative W*-algebra which has a cyclic
vector is maximal abelian.
For the multiplication algebra of a finite measure space is maximal abelian (see e.g.
F101
PROOF OF THEOREM.
Let
).
Q be a masa algebra on
be the direct sum of the closed linear subspaces which is invariant under
0
e
and has a cyclic vector for Q ,
, and let j ,
each of these
10
I. E. Segal
exist by Lemma 1.1.
Q
The contraction of
to 1+1
is by Lemma 1.2 uni-
tarily equivalent, say via the unitary transformation
Us
,
to the multi-
plication algebra of the finite perfect measure space M _
r
and it is clearly no loss of generality to take Now let
M =
r)
(R,
E M, let
x
x
,
on
with x
L2(M)
to
1u.
= 0
P
be the direct sum of the M ,
:Ye define a unitary transformation U if
(1'
),
for yF E
as follows:
E Ht , and set
Ux
X
- EA E
(this sum exists in the sense of unconditional convergence
-_ U9 x,
,
for the
U
are mutually orthogonal and L,
x
x
112 =
3 5 It is easily verified that U is, actually,
112 = flxII2).
IIx
II U
unitary.
It remains only to show that the map on the operators in Q to L2(M)
those on
be arbitrary in a., let
T let
t
be the function on
t(p) =
T
(where
(p)
corresponding to
by
Q onto Wt. (M), I. e. that
takes
T t(.
(.)
be the contraction of
Tf
defined as follows:
R
Tj (.)
T, I. e., Ug T . U71
L.U.B.
T
to
p e r
if
, and
, then
is the bounded measurable function on
It is easily seen that
).
Let
y?t(M) = U a U-1.
is the operation of multiplication t(.)
M, and
Now the bound of the operation of
(.)11.
II T
is measurable on
OD
multiplication by a bounded measurable function on a measure space, of elements of
L2
over the space, is readily seen to be identical with the
bound of the function in so that
t
of multiplication by
Hence
LCd
(.)I l
To show that
is bounded. t
IIT
UTU-1
T
Qt
it is enough, in view of the boundedness of these
cular it suffices to show that
function.
= IITII
is the operation
operators, to show that they agree on a dense subset of
a finite union of the
U-ill
UTU-If = Qtf
if
f
L2(M).
In parti-
vanishes outside of
and coincides on each with a continuous
The foregoing equation is linear in
even to show that UTU-lf = Qtf
for
f
f, and so it is sufficient
vanishing outside of
and
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II continuous on
But for such an
(x)f(x), where
on U }1%
= (U
)
invariant, and
t
T
(x)
TU11 f) (x)
agrees with U
U
and vanishes for
equal to
=
for
x
x
E T
(Qtf)(x)
f..
E
(UTU-1f)(x) _ (UTU-1 f)(x)
otherwise; while
U .1
ti (x)
11
and
U-1
(as
(for Ui l f e 14-
(x) =
t
0
coincides with
leaves 791
T
T (x)f(x) for x Er,
on
(by the definition of
U
), and so finally is
(x)f(x) also.
At this point we use, for the first time in the proof, the assumption that U O U-1
Q- is mass.
It follows that
has been shown to be contained in
U Q U-1
is likewise mass.
As
((M), and as the latter algebra
is obviously abelian and $A, it results that
V (M), and
U 1 U-1
hence that U QU-1 = M(M). COROLLARY 1.2.
A maximal abelian self-adJoint algebra of opera-
tors on a separable Hilbert space- has a cyclic vector.
By the known classification of separable measure spaces, every such space has its measure ring isomorphic to that of a finite measure space.
Now it is clear that if
L2
over a measure space is separable,
then so is the measure space, and hence a mass algebra Hilbert space
14,
L
on a separable
is unitarily equivalent to the multiplication algebra of
a finite measure space.
It is clear that the function identically unity is
a cyclic vector for the multiplication algebra of a finite measure space, and hence the corresponding vector in
1/
is cyclic for )K.
We remark that the present corollary follows also without the use of the classification theorem for separable measure spaces, from the
observation that any collection of mutually orthogonal projections on a separable Hilbert space is at most countable, together with Lemma 2.5 and the remark in the proof of Lemma 2.9 to the effect that a separating vector
for a masa algebra is cyclic (the proofs both of the lemma and the remark
I. E. Segal
12
being independent of the corollary). Remark 3.1.
The result established in this section essentially
includes the spectral theorem, and can be used as a basis for the operation. al calculus.
The fact that a commutative War-algebra is algebraically iso-
morphic to a
a( F)
is nearly equivalent to the spectral theorem, and the
theorem follows readily in full from Theorem 1. tion by considering the case of a SA operator
We illustrate the situaon a Hilbert space
T
the same procedure being valid for any finite number of commuting SA operators.
By transfinite induction there exists a masa algebra
T, and it follows from Theorem 1 that ht
ing
is unitarily equivalent to
the multiplication algebra of a localizable measure space Let
correspond to the operation
T
valued bounded measurable function let
of
SA _
let
Cp E RI k(p) < X]
, and let EX
be the operator in M
corresponding to
Ej .
,
It is
a straightforward application of known measure theory to verify that
/AdET and
, r).
be the operation of multiplication by the characteristic function
x SA
M = (R, 7
of multiplication by the real-
T'
k,
contain.
74'(.
in the usual sense (that
g
(T'f, g) = Zoo T d(E1,,f, g)
E, . It
in L2(M)), and that strong lim,
follows that
T
=
fk dE
, where
is a resolution of the iden-
[E,,3
tity in the usual sense, and it is readily seen that all operators which commute with
T'
for any
EA
T, and that the family
commutes with
[E"}
is
unique. Now let
show that
)'(T)
)G
be any bounded Baire function on the reals.
can be defined simply as the operator in
ing to the operation of multiplication by
J&(k(p)).
of a standard sort implies that this operation is follows that the corresponding operator in 77L
Z'YL
We shall
correspond-
Again measure theory
f)[i(1k)dEn , and it
is /Y,( ;l}dE,
particular is independent of the mass algebra in which
T
and so I,
is imbedded.
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II
Structure of commutative W*-algebras.
4.
13
We base the proof of
Theorem 2 on a series of lemmas. LEMMA 2.1.
Let
CL be a commutative W*-algebra of uniform
n, on a Hilbert space 1°1'.
multiplicity,
0, then the contraction of multiplicity
If
is a nonzero projection in
P
to the range of
£j,
P
likewise has uniform
n.
In proving this, there is evidently no loss of generality in tak-
Q to be an n-fold copy of the mass algebra m on
ing
unitarily equivalent to such a copy), so that f
on a set
vxeS and
S, of cardinal number
(f, g) _ Z xeS(f(x), g(x)) for f
Q consists of all operators of the form
"; and
in
Bf(x), with
For Be > , let
B E ' .
3p(T) =
case
be the operator in -
F
lit
Putting Ll for the range of
P.
to
7i 1, and Q
1
is an algebraic
P, '#1
for the contraction of
range of P, it is not difficult to verify that
ml on
(Af)(x)
onto Q , and hence there is a projection P in ZZ(r
isomorphism of
contraction of
90(B)
It is clear that
defined by the preceding equation.
such that
74 consists of all functions
n, to 7t , for which the sum
"f(x)11 2 is convergent, and
g
(rather than
7Z
Q1
for the t2-
to the
is an n-fold copy of
It follows that it is sufficient to prove the lemma for the
n = 1. Now assuming
commutor of
Q1, then N E a 1
be the operator on
Then W
n = 1, It must be shown that if W
e Q), for if
(it is clear that
1
is SA).
defined by the equation Wx = WPx, T
and
tively, .'1Tx = WPTx = WPTPx
x
is in the Let
W
x e 1+'.
are arbitrary in Q and 7# respec-
(for
0-
is abelian) =
WTPx
(where
is the contraction of T to P1*) = TWPx (since W 6 (Q1) r) ; TPl'YPx (as h x C P1#) = TPWPx (by the definition of T) = T4YPx (since 7Px E P7/-) = 'NN. Hence W 6 Q , and it follows that the
T
14
I. E. Segal
contraction of
W
P74 is in
to
LEMMA 2.2.
xµ ,
Let
Q 1, and this contraction is
be a mass algebra on the Hilbert space
where w ranges over an index set, and let
of the 'X .
Let
, be the direct sum
'At be the set of all operators
by such an equation as
Tx
T
P,M x, where
T
k
on
x e
onto , Tµ a and with
pro iection of
W.
,
determined
Pf+
11 TI
L3 the
bounded.
Then m is a masa algebra. In the statement of this lemma, and elsewhere when convenient, we make the obvious identification of a summand in a direct sum of Hilbert spaces with a closed linear manifold in the sum.
that the algebra 24
defined in the lemma is SA.
need therefore only show that if
W E 7l', then
It is readily verified To prove the lemma we W E 71r.
it is clear that P 6 1( , so that Pµ
nition of
which implies that
W
leaves invariant the subspace
P,,,,
From the defi-
and W commute, 7-) = 1C
.
Now
the operation of contracting an operator to an invariant closed linear manifold is a homomorphism of the algebra of all operators leaving the manifold invariant, into the algebra of operators on the manifold.
of W
contraction
to y{
Hence the
commutes with the contraction to the
same manifold of each element of )' , and so commutes with every operator
in II R,.I1
As
39(x,
is maximal abelian, it results that
Wf,
E
Now
so that IIWis bounded, and it
=
follows that
W E N .
LEMMA 2.3.
If
",,j (I
is a family of mutually orthogonal
pro-
jections in the commutative W*-algebra a , and if the contraction of 0
to P,,, 7q- has uniform multiplicity n for all 114 , then the contraction Qo of a to (U,. P', ) likewise has uniform multiplicity n. Let the contraction
of Q to
P,, 1c' be unitarily
DECOi;POSITIONS OF OPERATOR ALGEBRAS. II
equivalent, via the unitary transformation U,
copy, say
is the set which here plays the role of the set the
have cardinal number
S/1
generality to take all the Let
1*, to an n-fold
of the masa algebra on
on
Cl-
on
S
is the projection of is bounded.
to be identical and equal, say, to
S
onto
copy of
T
By the preceding lemma
proof consists in showing that
7( on k
Sf,.
n, and it is clearly no essential loss of
of the form Tx =
/
on
T
If
in Definition 2.1, all
W- be the direct sum of the 7µ , and let
operators
15
QC
7(
S.
be the set of all
R,, x, x E A , where R,,, C YI( , and such that [IT,M.II masa, and the remainder of the
is unitarily equivalent to an n-fold
.
We first define a unitary transformation U
( Ur Pf. )7' to the n-fold direct sum x E 7 , Uf P x is some element of
on
of 7 with itself. say f (. ), and
'A,
For any IIP,,,, x 112
IIUr P x112 = 57,a E S I1fja (a)II 2. Now 11x11 2 = Z IIPI,,x(12, so that the series a 6 S Ilfµ (a)11 2) is convergent, and it follows that the series Laa E Sr Iif, (a)11 2 is also convergent. Now f",, (a) E µ and the are mutually orthogonal subspaces of so that the convergence of 2 implies the convergence of Z (a) to an element f(a) of k , where IIf(a)11 2 As ,Ifs,,, (a)II 2. Z. e S I1f(aJJ 2 = ZaZi Ilf,,,, (a)II 2, which last expression is a conf C Z .
vergent series, we have and observe that
U
We define
is linear and isometric.
U by the equation Ux = f To show that
it remains only to show that it is onto .. Now let and let
be defined by the equation
gam,
'a 11g, (a)11 2 <
for all
x
in
4
Z. 11g(a)II 2 so that , we have
as the and the sum
IIy II
2
g,,
g
U
is unitary
be arbitrary in
g,, (a) = R,,,g(a). c;
Then
and as LORx = x
g = LEI, , . Let yµ
lgf, and set are in P they are mutually orthogonal, is absolutely convergent as 11y/<11 = IIg
11
I. E. Segal
16
so that the sum defining
r (U
U, (Uy)(a) =
tion of
is unconditionally convergent.
y
r )(a) = Z g
Py)(a) _ 2: (U y
R" g(a) = g(a), so Uy = g. Finally we show that
UAOU-1
is an n-fold copy of
We first observe that by the preceding paragraph,
the 4, .
of
T
Then
Now let
h, and let
to II T j
be the operator I I TII , and as the
<
II T,JI
gonal, there exists a unique operator T on
all the T for the map on
W'fk, (a), 5P' E put
operators,
(a) E
, f,
As
.
which is an extension of
/_
and putting
X`.
, we set
,,
(
and
T),,
is easily seen to preserve the bounds of
?µ
is bounded and so
IIT,jj
T'
exists and is in _'V(-
UTU-1 = D (T'), where P is the map on
We show now that
to the operators on t given by the equation
f E t .
As the oCµ
span oZ , and as both
Now
(IITU-if
)(a) _ (Up. T,
and
9(T')fr
T'fµ (a) = T £,,,(a),
(T')r, )(a)
(
(' (W')f)(a) = W'f(a), UTU-1
µ = IITII-If
bounded, it suffices to show that .
are mutually ortho-
for the operator on 1 determined by the equation T'x
T'
F, Tµ Rf x, x E )7
µ
T-1
T
II,,
defined by the equation
to
Uf,le,,
50(T')
for all
for some
T E a0.
Let
T'
be an arbitrary element of
tor on ' which agrees with orthor,onal complement of
ment
,J
IIT
A
Let in Tj,.
P^
fA, E
f )(a) = (T f,,. )(a). It
)(a)
be the contraction of T' to is an operator on
are
On the other hand,
remains only to show that every element of KK has the form
T
on
be the contraction
T,,
U. , TX E
By the definition of
(a)
is the direct sum of
Q0, let
be arbitrary in
T
By the defini-
,
A on
Now
=1
T,
and let PA,
Ty.
T
.
92 -1(UTU-l)
and let T,4-
be
(T). )UU , so that the (unique) opera-
, and which annihilates the is the contraction of some ele-
a- to PP 14 , and clearly PAP, = T- , so Tµ E 2 .
of
µ II
IIT
-
Il 9. (Tj',, )II
II'f;.ll
s IIT'') , the directed set
As
DECO14POSITIONS OF OPERATOR ALGEBRAS. 71 ,,E F Tµ , where
ranges over the finite subsets of the
F
dered by inclusion) converges strongly to an operator ly
T E Q , and it is straightforward to verify that LERMA 2.4.
traction of Q to
( Q. P`,,
(or-
i-l- 's
Obvious-
on
UTU-1 = 90(T'). -
and if
Q such that the contraction of
is of uniform multiplicity n
Pµ 7q1
T
Q is a commutative W*-algebra on
If
is a family of projections in
Q to
17
II
for each ,p , then the conn.
) 1+ is likewise of uniform multiplicity
It is no loss of generality to assume that the index set over which
varies is well-ordered.
u.
Q
then
Setting
Qta = Pt,
(I - U ' Py
),
(as the projections in a N*-algebra constitute a complete
lattice, which is a Boolean algebra when the W:-algebra is commutative),
Q to
and as the contraction of
is of uniform multiplicity
P,,, Y
it follows from Lemma 2.1 that the contraction of Qpi°1'
is also of uniform multiplicity
n
Q to
Qµ
n.
It is readily
are mutually orthogonal, and hence the preceding
lemma implies that the contraction of a multiplicity
Qt P, !
(assigning all multiplicities
to an algebra of operators on a zero-dimensional space). verified that the
n,
to
( U" Q",) H I. of uniform
Finally, it is not difficult to verify that
U"0- Qt'
U/A P. . The concepts described in the following definition are among those used by Nakano in [7]
, and the next lemma is due to Nakano; the
proof which we give of its non-trivial half is somewhat simpler that that which Nakano indicates. Definitions 4.1.
Q on h'
A W*-algebra
is called countably-
decomposable if every family of mutually orthogonal non-zero projections in An element
is at most countable.
vector for
Q if the only projection
the zero projection; if
R
in 1# is called a separating
x P
in Q for which Px = 0
is a projection in
is
Q', x is called a relative
18
I. E. Segal
Q on R*) if x E R1' and x
separating vector (for
to R.
vector for the contraction of Q Remark 4.1.
0, then
If an element
x
is a separating vector for 2
(TcT)x = 0, and it follows that if
C generated by
W*-algebra
T = 0.
Tx = 0, T E Q , implies that
then the equation
is a separating
W
C
Tx =
is any operator in the
T*T, then Wx = 0.
vector for Q , which contains C ,
For if
As
is a separating
x
can contain no nonzero projection.
Now any W::-algebra is generated by the projections it contains, so
T.>T = 0
(0), which means
LEMMA 2.5.
and so finally
T = 0.
A commutative W*-algebra is countably decomposable if
and only if it has a separating vector. If
vector
Q is a commutative W*-algebra on -H
x, then
with a separating
is countably decomposable, for if
QP1}
family of mutually orthogonal projections in Q , then
llx 112
Q.
7,r1) P x J)2, so that all but at most countably many of the
is any
> P)PI x
zero, which implies that all but at most countably many of the
are
P)_
are
zero,
Now let Q be a countably-decomposable commutative W*-algebra on
, and let Y be a family of projections in
with respect to the properties: P7./, for
P E °7+
;
2)
Q which is maximal
1) Q has a relative separating vector on
the elements of
dently
T is at most countable:
and let
xi
.7 are mutually orthogonal.
let its elements be
Evi-
{Pi; i = 1,2,...}
be a relative separating vector of unit norm for
Q on
P1}{'.
y = ZI 2-ixi is then a relative separating Q on (UiPi)75b. Now UiPi = I, for otherwise there
It is easily verified that
vector of
exists a nonzero element
z
in
upper bound of the projections and for which
(I - U ipi) Q
and if
Qp
is the least
in Q which are bounded by I -
Qz = 0, then Q has the relative separating vector
UiPi z
an
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II (I
and
U, 1Pi - Qo)H
which by the fact that
y
I
UI
QO
0
P i - Qo
19
is orthogonal to all the
contradicts the maximality of W. Hence
Q on 1'.
is a separating vector for
If Q is a commutative nonzero countably-decompos-
LE1,21A 2.6.
able W*-algebra on 'J- , then there exists a nonzero projection
Q
such that the contraction of
Let J
be a set of separating vectors for a which is maximal
and
for the projection of
Px
x
and
is orthogonal to Qy. Put
t ?x
bound of the projections
Q
7'-1-
onto
Qo # 0, for if
C
y
are any two elements of
x
for the closure of ax Let
1Cx.
in Q for which
be the closed linear subspace of
let
in Q
P
P1+ has uniform multiplicity.
to
with respect to the property that if
J , then
Pi,
Q,p = 0, then
1''f'
QO
be the least upper
Q(I - U .P.) = 0, and spanned by the X.. Then
I - U xPx # 0
and the contraction of
Q to (I - UXPx)7Ef = _H Q 4 is an isomorphism, by the proof of Remark 4.1, and putting
.$I O j (which exists by Lemma 2.5), then such that
Q z
0- on
for any relative separating vector for
z
is orthogonal to all the
z
is a separating vector for Q
Qx with x e J , - which con-
tradicts the maximality of d .
for the range of
the contraction Q x
P
tive separating vector traction of
a result in
Q to [10]
:
Qom.
By Corollary 1.1, Ox
is masa.
Now we utilize
It follows that there exists a masa algebra
such that for each x e
unitarily equivalent to
n
Q to k a has the rela-
if two masa algebras are algebraically isomorphic, then
Wt on a Hilbert space Q,oH, and it follows
of
1,6x'
Qox, and is algebraically isomorphic to the con-
they are unitarily equivalent.
that if
Px = QOP. and
Qo = Ux QoPx, and putting
Clearly
A?f
on -k 1.
on
Now the direct sum of the
PX 14, is Px .F- Is
without difficulty by a technique previously employed
is the cardinal number of .d , then the contraction of Q to
20
I. E. Segal
is unitarily equivalent to an n-fold copy of )k
Q,oy¢
LEMMA 2.7.
to
PJr
and such that for any
I
Q whose least Q,
is countably decomposable.
Q
Let 7 be a family of projections in
2) if
P E
able.
We show that
, then the contraction of
zero element
U PE
Rz =
and such that
O', is nonzero as (I-Q-R0)/
0, I-Q-Ro
which is maximal with
Sr are mutually orthogonal, to
P1k
is countably decompos-
For otherwise, there exists a non-
Q = U PE 7
(I-Q)7#, where
in
z
P = I.
Q
the least upper bound of the projections
on
then the
,
the contraction of
P
respect to the properties: 1) the elements of
I-Q
1 j .
is a commutative W*-algebra on
7' of mutually orthogonal projections in
exists a family upper bound is
0
If
on
R
P, and putting
in Q which are bounded by
is orthogonal to all the elements of
0, and Q has the relative separating vector
z
so that by Lemma 2.5 the contraction of
,
for
Ro
a
to
z
(I-Q-Ro)74
is countably decomposable. Let
LE6131A 2.8.
Q be a commutative W*-algebra containing
For each cardinal number n > 0 projections
in
S
form multiplicity
Q
I.
let
Pi
of
Q 1
and
n.
Un Rn =
Then
P1(I-Q)1°{{
P
in
0.
Q to
nonzero projection
such that the contraction 02
is countably-decomposable; by Lemma 2.7, P1 If
PI
is the contraction to
P1(I-Q)1N = P(I-Q)11- . N2
has uni-
S1+
be denoted as Q 1, and
Q (that such a projection
then we can write
to
I.
be a nonzero projection in Q 1 to
A
and as the basis of an indirect proof, assume
Let the contraction of
P1(I-Q)1-
jection
be the least upper bound of the
Rn
Q such that the contraction of
= Un Rn
Q
Set
let
in
I.
Q2
P
(I-Q)1*
exists,
of the pro-
exists is easily verified),
By Lemma 2.6, there exists a
such that the contraction of
Q2
to
DECOIIPOSITIONS OF OPERATOR ALGEBRAS. N2P(I-Q)14
has uniform multiplicity, say
Now if
N1
is a projection in
and if
N
a
21
n, and obviously
whose contraction to
e 2l
is a projection in
II
N2P(I-Q)N 76 0. P(I-Q)11
whose contraction to
Is
N2,
N1
is
(the existence of these projections being clear), then the contraction of Q 2
to
N2P(I-Q)*
It follows that
NP(I-Q) < Rn and
Rn
nitions of
is the same as the contraction of Q
nlicity n
on
Q be a commutative W:-algebra of uniform multi-
Let
P
mt"o
Then nN,
.
is a nonzero projection in the algebra of operators q
14- such that the contraction
posable, then by Lemma 2.1, Q 1 m.
NP(I-Q) = 0, a contradiction.
and also of uniform multiplicity m. If
NP(I-Q)11-.
NP(I-Q)Rn = NP(I-Q), which by the defi-
implies that
Q
LEMMA 2.9.
or
to
a1
of
Q to P9 is countably decomand
is of uniform multiplicities both n
Hence by Lemma 2.7 it suffices to prove the present lemma under the as-
a is countably decomposable.
sumption (which we now make) that
Let Q
be unitarily equivalent on the one hand to an n-fold copy of the masa algebra
on 7t'
, and on the other to an m-fold copy of the masa algebra 7Z
on
By Lemma 2.6 these
hence both of these algebras are countably decomposable. exist separating vectors spectively.
projection
u
and
P
for
Obviously
and )f
Y(
on Z reso that the
(I-P)u = '0, and since u
As
is a
I-P = 0, which shows that the closure of 11(u
I. e., u is a cyclic vector for
vector for
79( on k
on this closure commutes with each element of 71(.
separating vector for It ,
v
Now the closure ofu is invariant under
is maximal abelian, P E 70(.
is
and x , and
a is algebraically isomorohie to both _M
Now
'A(.
Similarly v
is a cyclic
on 4.
It follows from the definition of uniform multiplicity that is the direct sum of
n mutually orthogonal subspaces
kµ
M
( ,u e S,
I. S. Segal
under a
on 7C ; and 1
,4
orthogonal subspaces
Xv
Q to 7k^
and such that the contraction of
equivalent to
where
(
V E T, T being an index set of cardinal m),
leaves each of the Z y
Q-
is unitarily equivalent to
onto
x1,
invariant and whose contraction to any
on ° .
7'L
V
set of all indices
for which
for all ,u , then clearly , Aye )
In particular, (x
(A*xf, , y, as
)
=
such that 0 x,,.
is
Now the projection xyv
0
0
for A E a-
T
for all
and
A
E O,
u E S.
such that
xt,,,,
x,.
for all
is orthogonal to
xf,,
has cardinal
,tt
there is a p such that
V
to be an index in
?--
=
for some
0
S.
E S, or
and
Now the
E.
A*x,,,
span
A ranges over 2 , and so it follows from the last equation that
(z,,, , y,,
) = 0 for all
results that
(z, y,
contradiction.
z1,
= 0
)
in 'K,- . Since the for all
Thus the cardinal number m of
Let 7
LEMMA 2.10.
in
T is at most
=
r1'.
P
b,
0, a
n.
By
m X. .
Let
x be a cyclic vector for
be an arbitrary sequence of vectors in 1-e,.
a nonzero projection
yT =
be a eountably decomposable mesa algebra of
operators on the Hilbert space X . {yi}
span 1+ , it
It
z e.7¢ , which implies
symmetry, n < Zto m, and it follows that
let
E 'y
xf ,,
n. But for every
Pf,
0, for otherwise, taking 0
By the preceding paragraph,
vanishes, except for countably many V , so that the
/,v
number at most
is unitarily
is also the direct sum of m mutually
and yv there exist vectors xt, E 1 is dense in and ayv dense in of
n) each of which is invariant
is an index set of cardinal number
S
where
in m such that Pyi 6 m' x
7f(, IM
Then there exists (i = 1,2,...).
We note that as shown in the proof of Lemma 2.9, there does ac-
tually exist a cyclic vector for rn on 74; is such).
By Lemma 1.2, we may assume that
(in fact, any separating vector 7)'t
is the algebra of all
multiplications by bounded measurable functions on
L2
over a finite
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II
M = (R, 7 ,
measure space dense in
L2(M)
<
lyi(a)I
[a I
that
r).
x(a)
It is clear from the fact that a. e. on
0
<
Putting E =
I
Ei r(R - Eiji) < (1/2)r(R).
ji
it results that
Now taking
P
is the product of
Pyi
equals
yi(a)(x(a))-1
x
r(R - E
r(R - E) <
Pyi a
E, we have
and the function, bounded by
a e E
for
R
to be the operation of
multiplication by the characteristic function of for
Eij such that
iii
Eiji ,
Ii
is
Eij differs from
U3
by a set of measure zero, and hence there is a r(R)2-1-1.
x
Now setting
M.
, it follows that
j jx(a))]
23
x,
j1, which
and equals zero elsewhere.
Our proof of the next lemma utilizes a simplification of a device employed by Nakano
in connection with a similar result for the case
[6]
of separable Hilbert spaces. LEMMA 2.11.
plicities both
n
Suppose a commutative W->-algebra has uniform multi-
m, where
and
n
is finite and m < r1o
Then n = m.
As in the proof of Lemma 2.9, we can confine our attention to the case in which
Q is countably decomposable and obtain subspaces
of the space * on which Q acts, and vectors
and
.
,
such that Q x .
and Qyv
x,,,
are dense in
and
and
in
y,,
and .
-k
respectively, and with 7 the direct sum of theand also the direct sum of the
finite and
(here
.2,
kv v = 1,2,... otherwise).
formation from fT
onto
the contraction of can clearly take
putting y
fi = 1,2,...,n
(2
xJ' =
v = 1,2,.... m
and
Putting
and
to
U. x'
x'
is
for a unitary trans-
IIµ
which implements the equivalence of
lk and
for a cyclic vector for 7tt , we
without essential loss of generality.
for the projection of
y.
Now setting yv
onto
V)lyv
P
Now
we evidently have
,
it results from the
preceding lemma that there exists a nonzero projection that
if m
P'
in )X
such
for the projection in Q which is
I. B. Segal
24
unitarily equivalent via the given transformation to the n-fold copy of P' U. e. the contraction of P to is IIf P'71), then P 0. It is easily seen from the relation P'y; 6 71( x' that PT,3,, 6 Qx and v
for all
by
, say
=
Pyv,A
shows that we can suppose
P
Ty,,xµ ; multiplying this equation
TV.14
=
PTvf,
It follows that Py = Z v = 1 T assume
m > n
and derive a contradiction.
x,
, for all V
Now we
We use the fact that in an r-
dimensional module over a commutative ring with unit, any r t 1 elements are linearly dependent over the ring (see the module over
[4] , Th.bl). We apply this to
P Q of all ordered n-tuples of elements of
P Q, and in
particular to the n } 1 n-tuples (T,,1, TV 2, ..., Tv n) ( V = 14,...,n+l1 It results that there exist elements not all zero and such that
vi-l Ty
1
Si. S2, .,., Sn+1
of
P Q.
which are
Sy = 0. Hence 'n+1 Sy Py v=1
0, or z vi2 Sv yv = 0. As S yy e ty , and since the S. yv =
are mutually orthogonal, we have
is a separating vector for
0.
Q now implies that
The circumstance that
Si, = 0
y,
( v = 1,...,
n+l), a contradiction.
LEMMA 2.12. With the notation of Lemma 2.8, the
R.
are mutually
orthogonal.
For if
0, the contraction of Q to % If is of uni-
RnF=
form multiplicity m by Lemma 2.4, so that by Lemons 2.1, the contraction of
Q to
likewise has uniform multiplicity m.
Rn(Rm}¢)
the same contraction also has uniform multiplicity n.
Lemma 2.10, that either m = n the other is not greater than
By syumtetry,
It follows from
or else one of m and n
is finite and
By Lemma 2.11, m = n
in the latter
o.
case also.
PROOF OF THEOREM. Ro
for
I-E, where
E
With
Rn
as in Lemma 2.8, we have, putting
is the maximal projection in a, Unnn = I
by
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II Lemma 2.8, and the contraction of
n by Lemma 2.4.
Now if
Pn
Q
to
25
Rn tf has uniform multiplicity
is for each cardinal
n
a projection in
Q.
with the properties stated in the theorem, then from the definition of it is clear that
Pn < Rn.
Now U n
P.
I, but the
disjoint by the preceding lemma, so that
Pm ^ Rn = 0
Rn for
Rn,
are mutually in
,f
n, and
Rn = P.
it results that
5. Unitary invariants of commutative W*-algebras and of SA operaIn this section we first prove a theorem which gives a simple com-
tors.
plete set of unitary invariants for a commutative Wo-algebra.
Assuming, in
order to avoid a trivial complication, that the identity is in the algebra, these invariants consist of Boolean rings
B(n), one such ring being attach-
ed to each cardinal number (or multiplicity) ciently large
n.
n, and vanishing for suffi-
These rings are (lattice-theoretically) complete measure
rings, and all such rings may occur.
The classification theorem of Maharani
[ 3 ] for measure rings is stated for the
0'-finite case, but there is no
difficulty in extending it to an arbitrary complete measure ring.
The use
of this extended classification provides a still simpler set of invariants, consisting essentially of a function on pairs of cardinals to the cardinals, - if
f
is this function, f(m, n)
is the number of direct summands of the
measure ring of the infinite product measure space
Im, where
I
is the
unit interval under Lebesgue measure, which occur in (the direct decomposition into homogeneous parts of) the case
1 < m <
x'o
B(n), but the discrete part of
must be treated separately.
B(n)
and
The validity of these
invariants, whose range is clear, follows at once from the following theorem together with Maharam's theorem, and we refer to C.3 ], from which the mode of derivation of these bardinals is clear.
Thus the most general
commutative Wo-algebra can be regarded as completely and rather explicitly
known.
I. E. Segal
26 Definition 5.1.
be as in Theorem 2.
LPnJ
the contraction of
Q.
Let a be a commutative W*-algebra, and let The Boolean ring
to the range of
Fn
B(n)
of all projections in
(which ring is shown below to
Q for
be a complete measure-bearing ring) is called the measure ring of the multiplicity
n.
THEOREM 3.
Two commutative W*-algebras are unitarily equivalent
if and only if their measure rings for the same multiplicities are algebraically isomorphic, and also the maximal (necessarily closed) linear manifolds which they annihilate have the same dimensions.
For any connotative W*-algebra a , the contraction of Pn}#, where
1#
is the space on which
2, will be called the part of and
.b
Q. acts and Pn
Q, of uniform multiplicity
Cl.
to
is as in Theorem n.
Now if C
are unitarily equivalent W*-algebras it is clear from Theorem 2
that their parts
C. and 4n of uniform multiplicity n are unitarily
equivalent, and hence their measure rings for the same multiplicity are algebraically isomorphic.
It is obvious that the dimensions of the maximal
closed linear manifolds which
C and .U annihilate are equal.
Now suppose that C and
.0 are commutative W*-algebras whose
measure rings for the same multiplicities are algebraically isomorphic, and such that the maximal linear manifolds which they annihilate have the same dimension.
We shall show that
C and
tY
are unitarily equivalent, and
for this purpose we may evidently assume that both
C and V contain the
identity operators on the respective spaces on which they act.
and 0n n
and
be the parts of
C and
.LY
of uniform multiplicity
Let en n, and let
n be masa algebras, to n-fold copies of which C. and P1 n
are respectively unitarily equivalent.
tion of n-fold copy that C n
and
Then it is clear from the defini-
W(n on the one hand and 'ffn
on the other, are algebraically isomorphic.
Now
and
?'In
(n and 1Z n are unique
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II within unitary equivalence, for taking the case of
27
n, if
is also
(gin
Cn
unitarily equivalent to an n-fold copy of the mass algebra c.n, then and
n
are algebraically isomorphic and so Nn and
cally isomorphic.
.Z
n
are algebrai-
and
are both multiplication algebras of 7Zn localizable spaces, within unitary equivalence, by Theorem 1, and it is
shown in [lo]
il(,
that if two such algebras are algebraically isomorphic,
then they are unitarily equivalent.
Let W n
and h n be respectively (unitarily equivalent to) the
multiplication algebras of the localizable measure spaces
Mn
and N.
The Boolean ring of projections in Cn is plainly algebraically isomorphic with the ring of projections in 71(n isomorphic with the measure ring of
jections in b n Hence
('n
Mn.
Similarly the Boolean ring of pro-
is algebraically isomorphic to the measure ring of N.
and Xn have algebraically isomorphic measure rings.
7f1n
result in lent.
which in turn is readily seen to be
their multiplication algebras are then unitarily equiva-
[10]
Thus
7Jln
and Pin
By a
and
7Cn
are unitarily equivalent, and it follows that
are unitarily equivalent.
It is straightforward to show
C and b are unitarily equivalent.
from this that
Next we obtain a complete set of unitary invariants for a SA oper-
ator, this set being due to weaken some of whose techniques we use.
[14]
and to Pleesner and Rokhlin 181
Before stating the basic theorem we use
the foregoing theory to reduce the problem to the situation treated in the theorem.
plicity n
If
T ( =
is a SA operator, and if contraction of
T
Tn
is its part of uniform multi-
to the range of
in Theorem 2, a being the W*-algebra generated by
Pn
Pn, where
T), then
Tn
is as
is uni-
tarily equivalent to an n-fold copy of an operator Sn with simple spectrum (i. e. the W*-algebra generated by Sn
is maser).
By Theorem 1,
Sn
can be taken to be the operation of multiplication by. some function, on L2(Mn), for some localizable measure space
Mn.
It is easily seen that a
I. E. Segal
28
complete set of unitary invariants for the
Sn
is also a complete set for
T, and so the problem is reduced to the essentially measure-theoretic one of determining when two multiplication operators, each of which has simple spectrum, are unitarily equivalent.
The classification of Maharani could be
used to reduce the problem further to the case when the measure spaces in question are homogeneous.
Naturally it is a restriction on a measure space
for it to admit a multiplication operator with simple spectrum, but we shall not discuss the nature of this restriction, which at present is unclear (except for the fact, which follows from Corollary 5.3 without difficulty, that the separability character of the space must not exceed the cardinality of the continuum). Thus in order to obtain a complete set of unitary invariants for a SA operator, it is sufficient, in view of the foregoing, to obtain such a set for SA operators with simple spectrum, and in the remainder of this We note that if attention is
section we consider only such operators.
restricted to SA operators with simple spectra which are unitarily equivalent to multiplication operators on finite measure spaces (and for operators on separable Hilbert spaces this is always the case, as it means that the W*-algebra generated by the operator is countably decomposable), the operator is determined within unitary equivalence by its spectrum together with its spectral null sets, - for separable Hilbert spaces this was proved
by Nakano [6]
.
The invariants given by the following theorem for the
general case are a kind of generalization of these invariants.
Another set
of invariants for the general case, more closely related to those for the case of finite measure spaces, but in some respects more complicated than the present ones is due to Nakano Definition 5.2. space 1
[7]
.
For an arbitrary SA operator
, the weighted spectrum
E(T)
T
on a Hilbert
is the family of all (finite regu-
lar) measures m on the reals of the form m(B) =
(E(B)x, x), where
B
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II is an arbitrary Borel subset of the reals, E(.) associated with
m
and
T
is the spectral measure
is arbitrary in 1* .
x
is concentrated on the spectrum of
29
(It is easily seen that
T, as this term is usually defined).
THEOREM 4. (Wecken-Plessner-Rokhlin).
Two SA operators on Hilbert
spaces with simple spectra are unitarily equivalent if and only if their weighted spectra are the same. It is clear that if two SA operators are unitarily equivalent, then their weighted spectra are the same.
Now let
ators with simple spectra on Hilbert spaces whose weighted spectra T'
and
f
e'
T
and
7f and
are the same.
-t-t'
T'
be SA oper-
respectively
To show that
and
T
are unitarily equivalent we require two lemmas, which are essentially
contained in the work of the authors mentioned.
In connection with these
lemmas we recall that two measures (on the same ring of sets) are said to be orthogonal if the only measure absolutely continuous with respect to both of them is the zero measure. Let
LEMMA 4.1.
x
and
ated with
T.
orthogonal to
Then Q x
be elements of
y
Wo-algebra generated bg T, and let
is orthogonal to
my, where for any
Q y
, mz
z E 19t
19', let
if and only if mx
We observe to begin with that if
(Vu}
(E(B)z, z).
is in the closure 'M,,
is absolutely continuous with respect to
mx.
of
For if
is a sequence in Q such that Vnx ---p z, then mz(B) =
IIE(B)xJ12 = we have
z
is
is the measure on the Borel
subsets of the reals given bs the equation mz(B) =
Q x, then mz
Q be the
be the spectral measure associ-
E(.)
li%JJE(B)VnxUU2 = limnIIVnE(B)xII2, and if mx(B) =
E(B)x = 0 and it results that
the projection
under a , Px
P. E
of
14
Q', but
onto le x
a, = Q .
mz(B) = 0.
0
We note also that
is in Q , for as k x
is invariant
Similarly the projection
Py
of
I. E. Segal
30
741/1
onto the closure
of
1T-' y
Now suppose that
tion, to show that It
x
mm
E
z
Next we assume that
Py
and
commute.
By the last observa-
^ k7, then as.ahown in the premy, so
mz =
0,
z = 0. is orthogonal to
Q x
are orthogonal.
and
my.
is orthogonal to both mL and
from which it follows trivially that
then mx
P.
are orthogonal it suffices to show that
Y
Now if
`sac
ceding paragraph mz
Thus
is orthogonal to
and 1t
their intersection is 0.
is in Q .
Q y
a y
and show that
As the basis of an indirect proof, let
MY n be a nonzero finite regular measure on the reals which is absolutely
and my.
continuous with respect to both mx
there exist ak
and m7
integrable non-negative functions
respectively such that n(B)
= fB hx( X)dmx( A.)
is an arbitrary Borel set.
B
where
By the Radon-Nikodym theorem,
Putting
_
functions defined by the equations
gx( ) = min (1, h.( X))
for the set
nx(B) = fB gx( 7,-)dm,(A)
= fB gy( A.)dmy( A.), then it is clear that n
and by
hy( ?)dm-( A.),
gy( Ar) = min {1, h7( x)} , and setting nx and ny
and
hx
and
nx
and
n,(B)
are absolutely con-
tinuous with respect to each other, that the same is true of n and y, and that for any Borel set In particular, nx
my(B).
B and
we have both nx(B) < mx(B) ny
JB f( x)dnx(X), for some nx-integrable function f. B
min 0.
ny(B)
Defining m on Borel
by the equation m(B) = Jg f' ( j)dnx( X), where f' (T) =
{1, f( X)) , it is evident that
Thus m
mx(B)
and
m(B) < ny(B), m(B) < mx(B), and m m(B) <
is a nonzero regular measure on the teals such that m(B) < my(B)
for all
B.
Applying the Radon-Nikodym theorem once more, we have
fBfx( A.)dmx( A.) _ g fy( A_)dmy( A.), where fx my
ny(B) <
are absolutely continuous with respect
to each other and so by the Radon-Nikodym theorem we have
sets
and
m(B) =
and fy are mx and
integrable functions respectively, which are bounded by unity.
and my
are regular measures, fx
and
fy
can be taken to be Baire
An mx
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II
Now fx(T)
functions.
the form V2
31
is a positive semidefinite SA operator and so has
for some SA operator
in Q .
V
Clearly m(B) =
B fx( X)d(E, x, x) _ (fx(T)E(B)x, x) = (V2E(B)x, x) = (E(B)Vx, Vx) where x' =
mx,(B) Thus
(E(B)x', x') =
(E(B)y', y')
y' E ay.
for some
Similarly m = M.,
Vx.
for all Borel sets
As the
B.
generate Q in the strong operator topology, it follows that
=
(Sy', y') for all operators
projection on fe x
$
in
Q.
Taking
S
E(B)
(Sx', x')
to be first the shows that
and next to be the projection on
x'
y' = 0, so m = 0, a contradiction. LEMMA 4.2.
contraction of
T
to the closure of
ax is unitarily equivalent to the
X on L2(Mx),
operation of multiplication by the coordinate function where measure
M.
The
x be an arbitrary nonzero element of 141.
Let
is the regular measure space on the reals, -co < 2
mx.
A proof of this can be given which is a straightforward adaptation and simplification of the proof of Lemma 1.2 and we omit further details.
Completion of proof of theorem.
weighted spectrum
E(T)
of
T
Let
3r
be a subset of the
which is maximal with respect to its eleis clear
ments being mutually orthogonal and nonzero; the existence of
For each p c 7 , let
from transfinite induction. x'
(= x$(1o))
be elements of
]t and
7yb'
,/J (B) = (E(B)x, x) = (E'(B)x', x'), where
E'(.)
be its orthogonal complement in 1'.
under Q , and if
z
is any nonzero element of
is orthogonal to all the
mx,.
is the spectral meas-
Oxp
Let 7t be the closed linear manifold spanned by the .4
and
respectively such that
ure associated with T. We show now that the
let
(= x(,p))
x
span 1 , Jp c Qx.p
Clearly
)t
is invariant
S, mz ¢ C(T)
with p 6 X, by Lemma 4.1.
.
, p E 7, and
Hence
andQ mz
X! = 0.
I. E. Segal
32
Thus
is the direct sum of the
fY'
is the closure of
Qxr,
, and by symmetry
where is the direct sum of the
If'
jo E 'X), where N is the closure of
, Ql being the
Qlxr,
V. By Lemma 4.2, the contraction of
W.:-algebra generated by
T
to nib
is unitarily equivalent to the operation of multiplication by the coordinate function on to
hf'
L2 (Mx
and hence equivalent to the contraction of
)
It follows without difficulty that
.
T
and
TO
TO
are unitarily
equivalent. 6.
In this section we make a number of applica-
Applications.
tions of the preceding structure theory, mainly to spectral theory.
Our
basic result is as follows. THEOREM 5.
If
mal abelian W*-algebra
(,L
7x
is a commutative 'W-1-algebra there is a maxi-
to which
(with preservation of ad joints).
Q is algebraically isomorphic
Any such isomorphism
f
of Q onto
is bicontinuous in the weak topology and has the property that if f is any bounded Baire function on the complex numbers, then for any operator T
in a ,
99(f(T)) = f(f (T)).
The algebra
-
is unique within unitary
equivalence, and the dimension of the space on which it acts is not greater than the corresponding dimension for If
'l
and
1'2
CL.
are algebraic isomorphisms of the W*-algebra
Q onto a mass algebra 7fj , then }k phism of
onto itself.
1 21
is an algebraic isomor-
As an algebraic isomorphism between mass alge-
bras is induced by some unitary transformation between the corresponding
Hilbert spaces [101 , there is a unitary operator U on the Hilbert space
k on which )( acts, such that that any isomorphism of
Q onto
}k(T) =
U*TU, T E J1r.
Hence to show
is bieontinuous in the weak topology,
and preserves the operational calculus, it suffices to show that any one
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II Moreover, if
isomorphism has these properties.
33
is algebraically iso-
Q.
morphic to a mass algebra 7Z , by the result just quoted N and unitarily equivalent, i. e.,
is essentially unique.
7Z
are
Hence to conclude
the proof of the theorem it suffices to show that for the given W*-algebra
Q, there exists an isomorphism of Q onto a mass algebra
N'(
which is
weakly bieontinuous and preserves the operational calculus.
be unitarily equivalent to an n-fold copy of the mass alge-
Pn ht (n > 0) bra
on
n
be as in Theorem 2, and let the contraction of Q to
Pn
Let
consists of
By Theorem 1 we may assume that -k
lT
n.
Mn =
over a localizable measure space the multiplication algebra of
Mn.
and that )n is
)
n Let the measure space L2(M)
n, so that
Mn, for all
be the direct sum of the
(Rn, )? n, r
L2
M = (R, 1Q , r) can be identified be
in a clear fashion with the direct sum k of the 1ti,n, and let the algebra on
is readily seen to consist of all operators
Then
Tn
to Kn is in X n
and such that
T 6
mass, for if
and if
Qn
then as plainly Qn E 1(, TQn = Tn
Putting
and V
leaving
any operators
U
TnSn =
for S e )1(.
IITh
, we have
T E
from a result in
[103
.
Q%T T )tin
T
operator U
=
so that
T
'3V( is
Now
onto kn,
leaves An invariant.
to 1tn, then
(UV)
n =
for
UnVn
invariant, and it follows that
IITnII <
).
in Q , let
Tn
9D
of Q onto. For any
denote its contraction to
in the contraction of
a to
Pn1j, let
Pn7w, and for any
,n(U)
be the oper-
is taken into by the unitary equivalence
Q with the n-fold copy of )Yn IIUJ'
is bounded.
(The mesa character of 7% also follows directly
ator in mn whose n-fold copy U of
whose contraction
Tn E ),fn', and as obviously
Hence
Next we define an isomorphism operator
JITn11
T
is the projection of 7
for the contraction of
SnTn
M.
corresponding to the multiplication algebra of
Ile
It is easily seen that
, so that there exists a (clearly unique) operator
(ln(U)II 7) (T)
on 7L
34
I. E. Segal
whose contraction to -kn
is
)rn(Tn), and evidently 9 (T) E `7k
Now
.
is an isomorphism, and it follows without difficulty that so also is
If To see that
is weakly continuous it is sufficient, by virtue
of the linearity of 9 , to show that it is weakly continuous at 0.
IT E Q I and
the
xi
in
a in the weak topology.
y1
are elements of AP, is an arbitrary neighborhood of
weak topology is
CT' C )k
A general neighborhood of
we define
xi
?,'H- be the direct sum of the
Sn
where
and
yf
to be the element of
vlan(x)
n
on
IInj
jn
be any element of
the projection of x
y' in
17,/n
and
y'
on
x'
and
y'
and
1o (y+)).
e
,ie
Pn 1f has uniform multiplicity
Unj x', x' e n
+ L n
in k and xn
n).
fin'
vlJn( n), where n is
and
TO
x'
and
in Nj, (Tnx , -y)
yn and the projections of
is the contraction of
Tn n both sides of the last equation over n
[T`E )G( I
on xn (these subspaces
It is easily seen that for any
,Pn(xn)' pn(7')), where and
in-
194,n3
TO e -Nn, (T'x', y') = ($V 1(T+),,on(x'), 1n(y')), and
it follows that for any
x'
'3jln
we put /o(x) =
onto -Kn'
0 E Sn,
nj
a leaves
-p n(x') =
Sn, and set
Now for any element x e le
1A
unitarily equivalent, via the uni-
exist because the contraction of Q to Let
For any x in 'kn
Fn 7* obtained as follows: let
to 7fnj, to
1r n
F] , where
are in 1t .
n), where
variant and has a contraction on fy nj
0
in 'W_ in the
0
closed linear subspaces
is an index set of cardinal
tary operator
,Ie
(T'xi, 7,1)1 < E
I
are as before and the
F
and
Now
£ >0, F is finite, and
, i E F] , where
I(Txi, 7 )I < £. 1
shows that
n'
Hence the inverse image under
-1
Summing
(T'x', y')
I (T-x', y' )I< E , i 6 F] is CT 6 Q F] , and so is a neighborhood of 0 in a
Next we show that
to
TO
is weakly continuous.
of
(T
. It Is not diffi-
cult to see from the definition of uniform multiplicity that the contractim
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II of
Q to
tion Vn
Pn
on
is unitarily equivalent, say via the unitary transformaPn7°f', to the algebra of all multiplications by bounded meas-
Mn, on
urable functions on whose set is
Sn
L2 (M1 X W ), where
W
is the measure space
and in which each finite subset is measurable and has
Now if x and y are arbitrary in
measure equal to its cardinal. say
35
Vnx = x(p, I)
arbitrary in Q ,
Vny = y(p, 1)
and
then
VnTnVnl
bounded measurable function
(p a Rn, i
6 Sn)
and
Pni*, T
is
is the operation of multiplication by a
tn(p), where
Tn
is the contraction of
T
to
(Tx, y) =
tn(p)x(p, i)y(p, i)dr(p)di, and integrating Mn X W n first with respect to i, this equals tn(p)w(p)dr(p), where w(p) = Mn x(p, i)y(p, 1), so w 6 L1(Mn). Writing w in the form w(p)= E Sn Pn
and
zi
x' (p)y' (p), with x'
y' in
and
L2(Mn)
and with
IIx'II 2
(Tx, y) _ (T'x', y'), where
IIwjIl, it results that
= I'ytII 2 =
To = T -I(T).
x = Zn xn and y = L'n yn, with xn and yn in P.-M, and summing both sides of the equa-
Now for arbitrary x and y
tion
in 1+ we write
(Txn, yn) _ (T'xn', y')
x' =
where
n
and
x''
over
Z n yn', these sums existing in the sense
_
y'
(Tx, y) = (T'x', y'),
n, then
of unconditional convergence because Zn ll xn' II 2 = En M1w Illl i) I dr(p) En An (Z1lxn(p, i)yn(p, i)1 2n JHn I Zi n(P, dr(p) Ixn(p, i)yn(P, 1) I dr(P)di < ZnII nII 117n11 Z-n /Mnx I
Wn
n1l 2) ( vn IIynII 2) = Ilxll 41yIl which I. finite; and similarly when x is replaced by y. Hence the image under of the neighborhood IT E 01 I(Txi, yi)I E , I E P] of 0 in Q, is of the form
(Zn
II
"
[T' E *I
I(T'xi, yi)j < r]
so is a neighborhood of
0
, with the
in
.
Thus
It remains only to show that
bounded Baire function
f
9D-l
and the
yf
in k, and
is weakly continuous.
7'(f(T)) = f( (P(T))
for any
f, and this we show is valid for any weakly contin-
uous algebraic homomorphism when
xi
9
.
The foregoing equation is obviously valid
is a polynomial, and it follows from the Weieretrass approximation
I. E. Segal
36
theorem that it is then valid for any continuous function the boundedness of the spectra of
T
and
(P(T)).
collection of all bounded Baire functions for (for all
E 0-): we show that
T
convergence of sequences. fn( a) --b f( a )
Let
f
(in view of
Now let X be the
which that equation is valid
is closed under bounded pointwise
be a sequence in 7 such that
{fn
for all complex
o!
, and with
`fn(.)`
bounded (n =
It follows from the spectral theorem for normal operators to-
1,2,...).
gether with the Lebesgue convergence theorem that the sequence converges weakly to to
As
f((P(T)).
f((P(T)), and hence
{fn( q(T))}
f(T), and similarly (P
{fn(T))
converges weakly
is weakly continuous it results that P (f(T)) =
-Jr
contains all bounded Baire functions, for it is
clear that all such functions are in the smallest collection of functions containing all bounded continuous functions and closed under bounded pointwise convergence.
The following result is due originally to von Neumann
1131
COROLLARY 5.1. For any commutative W*-algebra 0 on a separable Hilbert space there is a self-adloint element of Q such that every element of
a is a Baire function of Let
isomorphic.
T.
-4 be a mass algebra on 9
to which Q is algebraically
By Theorem 1, )( is unitarily equivalent to the multiplication M, and by the preceding theorem
algebra of some localizable measure space
L2(M), which can be identified with k , is separable. fication of separable measure spaces, M
By the known classi-
can be taken to be (i. e. has its
measure ring isomorphic to that of) the direct sum of a (possibly vacuous) real bounded interval under Lebesgue measure and a (possibly vacuous) dis-
crete measure space containing at most a countable number of points, which can be assumed to lie in some real bounded interval disjoint from the previous one, and to have finite total measure.
The resulting measure space is
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II
37
finite and regular, and for every measurable function on such a space there is a Baire function equal a. e. to it. tion on
It follows that every multiplica-
by a bounded measurable function is a bounded Baire func-
L2(M)
tion of the operation of multiplying by the coordinate function
Thus
x.
V consists of the bounded Baire functions of a single SA operator, and by Theorem 4, Q also is such.
The next result was pointed out to us by I. M. Singer. COROLLARY 5.2. An algebraic isomorphism between two commutative W*-algebras (not necessarily on the same space) is necessarily bicontinuous in the weak topology and preserves the operational calculus for bounded Baire functions. If
-y is an algebraic isomorphism of the W*-algebra Q1 onto
the W<:-algebra Q isomorphic to
2,
0-1
h(1
and if and
respectively, then clearly
4'2
are algebraically isomorphic. are unitarily equivalent.
are masa algebras algebraically
and m 2
rnl
This implies (loc. cit.) that
and 1
X1(2
and 71(2
The corollary now follows from Theorem 4.
In the case of a separable Hilbert space, the W*-algebra generated by a SA operator consists of all bounded Baire functions of the operator.
This is no longer the case for Hilbert spaces of higher dimension.
The
next corollary shows however that with an appropriate generalization of the operational calculus, the result remains valid.
We first make the
following Definitions 6.1.
A projection
P
on a Hilbert apace is called
countably-decomposable relative to a W*-algebra
Cl each of which is bounded by P
mutually orthogonal projections in at most countable.
An operator U
on a Hilbert space
extended Baire function of a normal operator countably-decomposable projection
(.Z if every family of
P
T
on
14'
is
74' is called an if for every
in the We-algebra generated by
T
38
1. E. Segal I, U leaves
and
P1* invariant, and the contraction of
U
P7w is
to
(in the usual sense) a bounded Baire function of the contraction of
to
T
P1*. We mention that a very explicit development of an apparently closely related notion of extended function is due to Pleasner and Rokhlin C 8] , who deal with an operational calculus for functions on the reals
which depend also on a variable ranging over a certain Boolean algebra. COROLLARY 5.3. The W*-algebra Venerated by a SA operator the identity consists of all (bounded) extended Baire functions of
Let Q be the W*-algebra generated by the SA operator and S
I, and let E Q,
that
S
be any extended Baire function of
S
e Q", or that
SU = US
for all U
ing contractions to PW by subscribing Baire function of
SP E (Q.)". (UT)
T., and that
TU = UT
Now As
U
which by virtue of Now let
and
Q.
leaves
"P", it is clear that
is generated by
TP
and
SP
is a
I., so
P14 yields the equation
invariant, it follows that
It results that
rLemma2.7 shows S
P19-
be a
P
(2 which is in 0- Denot-
SPUP = UPS., or
TPUP =
(SU) P = (US)P,
that SU = US.
be arbitrary in the algebra Q defined above.
is a countably-decomposable projection in Q , then clearly (AI,P
T on 1*
To show that
E Q. Now let
and contracting to
P*
UPTP, or UP E, (Q.)1.
P
and
T.
is equivalent, by a well-known theorem of von Neumann, to showing
countably-decomposable projection relative to
(TU)p =
T.
T
is the W*-algebra generated by TP
and
If
SP E QP
IP, so it is sufficient
for the remainder of the proof to restrict attention to the case in which I
is countably decomposable relative to Q.
assumed that Q is mass.
By Theorem 4, it may also be
Hence by Theorem 1, Remark 3.1, and Lemmas 1.2
and 2.5, the proof of the corollary will be concluded by establishing the following result.
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II LEMMA 5.3.1.
k
space, and let
M = (R, 1Q, r)
Let
39
be a finite
fect measure
er
be a real-valued continuous function on
M
such that
the W*-algebra generated be the identity and the operation on
multiplication
k, is the multiplication algebra of
real-valued continuous function h on M .0
h(p) = d(k(p))
such that
the/p
Let
A
k-1(B), where
B
element of
,J
almost everywhere on
is a Borel subset of the reels.
14 = L2(M)
By the Riesz-Fischer theorem,
.
there is a bounded Baire function M.
or-ring of all elements of R of the form
be
be the set of all elements of to
We show first that every
ft
k.
K
Hence
It can be shown
of )91.
which are measurable relative
is a closed linear manifold in
(cf.
[12]
then in 7r , and as the function
K, or that
K
of multi-
) that any such manifold consists of
noting the characteristic function of any set
e. on
T
is invariant under the multiplication algebra
all elements of W which vanish a. e. on some set K
1
in
12
.
But, de-
as ',S, we have x
E
which is identically one on
is also in X. This implies that
k , 1 - XR-K or x"K
Let k
.
1+, and it is easily seen to be invariant under the operation
plication by
of
Then for every
M.
differs by a null set from some element of 2
-R
L2(M)
Y(.K
R
R-K is in
vanishes a.
is a null set.
It is clear that adding a constant to
k
affect the situation, and hence we may assume that
does not materially k(p) > 1
for p E R.
Now let h be an arbitrary real-valued bounded measurable function on and let
't
and W be the functions on the Borel subsets
defined as follows: where
E = k-l(B).
1'(B) = f$ h(p)dr(p)
We observe now that
ous with respect to W , and in fact
m..
k(p)dr(p) =
lihil oc,W(B).
B of the real
t.)(B) = / k(p)dr(p),
It is easily seen that r and
lar measures on the reals.
11h)!
and
M,
i.)
are finite regu-
is absolutely continu-
'Y(B)I < ,Ihjl
oo E
dr(p)
Hence there exists a bounded
40
I. E. Segal
W -measurable real-valued function B
'L'(x)d W(x).
on the reals such that
''
t(B) =
In a finite regular measure space, any bounded measurable
function is equal a. e. to some bounded Baire function, so that
can be
taken to be the latter type of function.
We conclude the proof by showing that a. e.
h(p) /_ //(k(p)).
It is evidently sufficient to show that
g h(p)dr(p) = ,/E )1'(k(p))dr(p)
for all E C l? p, but as every element of
IF
differs by a null set from
an element of .p , it I. sufficient to establish the last equation for
of the form
k-l(B), B
being Borel.
For such a set
E
E, the equation is
valid by the definition of )Ir . PART II. 7.
NON-COMMUTATIVE ALGEBRAS.
Decomposition theory.
In this section we obtain a decomposi-
tion of an arbitrary W*-algebra into parts of uniform multiplicity, and in the following section we determine the structure of the general W*-algebra of uniform finite multiplicity.
Their structure for the case of infinite
multiplicity remains, however, obscure, except In special cases. The present decomposition coincides in the case of a commutative algebra with that obtained in Part I, but the method used in Part I is inappropriate in the non-commutative case, and the present method is not as well adapted to the abelian case as that of Part I.
The basic difficulty
in extending the method of Part I is that an algebra with a cyclic element need not be of uniform multiplicity one, in the non-commutative ease, (with any reasonable definition of uniform multiplicity).
The technique we em-
ploy here is in part an extension to the non-commutative case of a reformulation of the technique employed In Definitions 7.1.
A Was-algebra Q on a Hilbert space /- is
said to be of minimal multiplicity n the cardinal numbers
[6] .
if
n
is the least upper bound of
m such that there exist m mutually disjoint
DECO)POSITIONS OF OPERATOR ALGEBRAS. II in the commutor
projections
Q to
contracting
Q', the contraction of
in the center of
multiplicity
Q and each cardinal
For any W*-algebra
there exists a projection
Pn
tion of a to the range of
> n, and if
plicity
I, and with the contrac-
a to the range of
is likewise of minimal multi-
P
be a family of mutually orthogonal projections in
{Pµ }
Then the
k k.
QP
where
Q', then
n.
Let
each
of uniform multiplicity n.
is a nonzero projection in the center of
P
such that the contracting of
(Z'
Pn
If the minimal multiplicity of a W*-algebra a 2n
LEMMA 6.1.
the contraction of
Pn
n > 0,
such that the
in the center of
are mutually orthogonal and have union equal to
is
has minimal
P
to
Cl
n.
THEOREM 6.
1
It
if for every nonzero projection
is said to be of uniform multiplicity n P
It is said to be
if it consists only of the zero operator.
0
of uniform multiplicity
a such that the operation of
of
is an algebraic isomorphism.
1+
P
Q'
41
PP,
Q,
to
P,1°)'
is an isomorphism, for
are mutually orthogonal projections in
is the contraction of
a to
PPµ * is an isomorphism, for if
( QP)',
P1#, and the contracting of Q
QP, say
_T.
is the contrac-
tion of T E Q, and PPS,. TP = 0, then it follows that
T = 0, so
to
PT = 0
and
TP = 0.
TP E.
The lemma follows now from the definition of
minimal multiplicity.
LEMMA 6.2.
Let
be a W*-algebra on
Q
family of mutually orthogonal projections in tion of alt.
Q to the range of
Then the contraction of
minimal multiplicity > n.
P
a!
4. and
has minimal multiplicity CL to the range of
be a
f Ply' } such that the contrac-
U7. P
?:n, for all likewise has
I. E. Segal
42
If m
is any cardinal
< n, then by the definition of the minimal
multiplicity there exist projections Q
where V ranges over a set
of cardinality m, in
is the contraction of
( Q,)', where
Qf,
Q
to
P., 1¢, which are mutually orthogonal and such that the contracting of Q 14
to
is an isomorphism.
tion orthogonal to
Now let
not difficult to verify that P, Q',,, = Setting
(
(A.J
and
1# such that Aa Br = 0
U,- A, } ( U,t B.t) Q'.
E Q+.
= 0 if V 74 -V', for y, = (Q"',.V Pr)(P-vIQµ, y,) = 0 if V A V', and it is not
Thus
in
and that Qty
Rv = U, Q, , then R. RV
difficult to verify that if tions on
be the unique projec-
Q,,,v
whose contraction to P 1 is Q v ; it is
I -
{Br,}
for any
are families of projecCr
and T , then
= 0.
{R ) is a family of m mutually orthogonal projections
To conclude the proof of the lemma it suffices to show that the
contracting of traction of
Q 1
to
R1,1*
Q to the range of
with T E Q1.
is an isomorphism, where
Uf. P
Q", v TPf. = 0.
is an isomorphism, it results that
to deduce that
T = 0. Let
Q_ be a W*-algebra on
117 of projections in the center of to the range of
P,
contraction of
Q to the range of
C2'
Q'A,,,,,
shows that
TP
7yd and
0.
It is easy a La
{P ,,}
such that the contraction of Q
has minimal multiplicity . n, for all (J
Qua V T
As the contracting of
to Q P/, 1f
LEMMA 6.3.
R y T = 0
Now suppose that
Multiplying the last equation by
0, which implies that
Q1 is the con-
Then the
likewise has minimal multi-
plicity > n. There is no essential loss of generality in assuming that the in-
dex set over which Ja
fining
P, -
varies is well-ordered with first element
Uv9LA
1.
De-
PP , for ,u > 1, and Q1 = Pl, it is not
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II
43
Q <
difficult to verify that the
(which is the same as the contraction to to
Q to
It follows from Lemma 6.1 that the contraction of
Pp .
of the contraction of Q
QA4N,
P,,14-) has minimal multiplicity > n, and Lemma 6.2 now implies that
( U_ Q )14 has minimal multiplicity
the contraction of Q to
The lemma now follows from the observation that LEMMA 6.4.
Q, Ls a W*-algebra of minimal multiplicity n
If
a to
such that the contraction of
at
4.P.
Uf,QI, =
on 14, then there exist mutually orthogonal projections center of
2!n.
multiplicity
n, the contraction of
greater than
n, and with sum
P
P14-
Q
and
in the
is of uniform
Q7* is of minimal multiplicity
to
P + Q equal to the maximal projection in
a If
Q is of uniform multiplicity n, then it is obvious that the
conclusion is valid.
Q is not of uniform multiplicity n,
Assuming that
let
Q be the least upper bound of all projections
Of
such that the contraction of
> n.
a to
tiplicity
> n, and putting
in the center of
is of minimal multiplicity
R1'
Then by Lemma 6.3, the contraction of
R
Q to Q1- has minimal mul-
P = S - Q, where
S
is the maximal pro-
jection in Q , it easily is seen that the contraction of of uniform multiplicity
Q .
Po
P1w-
We define the
Pn by transfinite induction.
for the orthocomplement of the maximal projection S
By Lemma 6.4, E = P + Q where
P
and
of uniform multiplicity > ,1; we set
P1 = P.
in
Q are mutually orthogonal
projections in the center of Q , and with the contraction of Q to
been defined for m < n
is
n.
PROOF OF THEOREM.
We first put
Q to
Now suppose that
Q1*A'
Pm has
in such a way that these are mutually orthogonal
projections in the center of
Q'
with the properties that the
contractim
44 of
I. E. Segal
Q to
Pm1¢ is of uniform multiplicity m, and that the contraction
of Q to
(I - Rm)J-A
U r t m Pr.
is of (minimal multiplicity > m, where
N =
Putting
U m
I -N
Um
Um-,n (I - Rm), and by Lemma 6.1 the contraction of Q to has minimal multiplicity > n.
If
N = I
P. = P, where
a n; otherwise we take
projections in the center of
at
P
such that
P -
Q
for all
Q = N, and with the Q14- are respec-
tively of uniform multiplicity n and of minimal multiplicity > n existence of
P
Q being assured by Lemma 6.4).
and
n'
are mutually orthogonal
Q to P1* and to
properties that the contractions of
(I - N)7*
Pn' = 0
we set and
R, =
(the
It is clear that in
either case the hypothesis of the induction is valid at the next stage, so that the
P.
are well-defined, and it is easily seen that they have the
properties given in Theorem 6. the contraction of
(We note that the minimal multiplicity of
Q to any invariant closed linear manifold is bounded
from above by the cardinality of a set of mutually orthogonal projections in
Q', and hence by the dimension of it3.
We conclude this section by showing that the present notion of
algebra of uniform multiplicity n
agrees in the case of commutative alge-
bras with the notion introduced in the first part of this paper. THEOREM 7.
n
A commutative W*-algebra is of uniform multiplicity
in the sense of Definition 2!1 if and only if it is of uniform multiplio-
ity n according to Definition 7.1. LEMMA 7.1.
multiplicity n in
If a commutative W*-algebra a on
is of uniform
in the sense of Definition 2.1 and of uniform multiplicity
in the sense of Definition 7_1, then
in = n.
It is clear from Definition 2.1 that there exist n mutually orthogonal projections
Pi
in
Q'
Pi n is an isomorphism, so m > n.
such that the contracting of
Now suppose that n > .(o.
A Let
to
N
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II
45
be a nonzero projection in Q such that the contraction Q N
Then N
N1+ is countably-decomposable. is of uniform multiplicity n
is in the center of
there exist n separating vectors for
be a family of
{x, }
of
x,,,,
x.,,,
to
a'N
Q v (N h)
M has cardinality at most Z(', n.
is {QvJ
Q N where m > m' is an isomorphism.
for which x,
v # 0
for
However, for each v there is a
x µ v # 0, as otherwise, taking ? to be an index such
p such that
x,,,,x = 0 for all 1L(, then clearly x,,
that
It is clear that
on Qy N14- vanishes except for countably
many V , so that the set of all indices v some
so Q N
Now assuming m > n, let
-y4t du.'.
A4
> n, such that the contracting of
The projection
Q'
for QN such that C1Nx,
mutually orthogonal projections in
m'
Q to
in the sense of Definition 2.1 and of uni-
form multiplicity m in the sense of Definition 7.1.
orthogonal to
of
for all 1u . In particular
(x,,,,
is orthogonal to Q. N},F.
, Ay,) = 0 for all Au , all A E ON
and all yx E Q T N 7 f. It follows as in the proof of Lemma 2.9 that 0, a contradiction.
yA
Hence m < j n = n, so m = n.
To conclude the proof of the lemma it suffices to show that if n
is finite, then m < n.
graph.
Let y,
Let
N and
be as in the preceding para-
{Q Y}
be a separating vector for
QN in
Qi, N l+.
The remain-
der of the proof is essentially identical with the proof of Lemma 2.11,
with yy
playing the same role in both proofs.
PROOF OF THEOREM
uniform multiplicity n n > 1, as the case
Suppose to begin with that Q on
in the sense of Definition 2.1, where we may take
n = 0
is trivial.
Let
{Pij
of projections satisfying the conclusion of Theorem 2. tion of
Q to
Pi'* is of uniform multiplicity
nition 7.1 and of uniform multiplicity n so
Pi = 0
for
)Y is of
i
be an indexed family Then the contracin the sense of Defi-
in the sense of Definition 2.1,
1 # n by the preceding lenena (the algebra of all
I. E. Segal
46
transformations on a zero-dimensional space being taken to have uniform
multiplicity
n, for all
Q is of uniform multi-
and it follows that
n)
in the sense of Definition 7.1.
plicity
a on 7# is of uniform multiplicity n
Now assume that
the sense of Definition 7.1, where again we may take
n > 1.
Q
to
inition 7.1, so
Pi7+ is of uniform multiplicity
Pi = 0
i
LPiS
in the sense of Def-
Q is of
i # n, and it follows that
for
in
Then the contrac-
be projections satisfying the conclusions of Theorem 2. tion of
Let
uniform multiplicity n in the sense of either definition. 8.
a
algebra
We show next that a W*-
Algebras of uniform multiplicity.
is of uniform multiplicity
1
if and only if
at
is com-
The classification in Part I of commutative Wit-algebras within
mutative.
unitary equivalence thereby induces a similar classification of algebras of uniform multiplicity
n
1.
A W*-algebra of uniform finite multiplicity
is shown to be an n-fold copy of an algebra of multiplicity
finite
n
1.
For in-
the corresponding conclusion is not valid (as is clear from con-
sideration of the case of a factor of type II)
and our results as regards
this case are highly incomplete, except that a basic special case is given a full treatment in the next section. theorem is due to Nakano THEOREM 8.
[7]
The commutative case of the next
.
A W*-algebra is of uniform multiplicity one if and
only if its commuter is commutative. Let Clearly
Q'
0 be of uniform multiplicity 1 is commutative if and only if
is the case provided Q
Q in
projections
Q such that
Q in
Cl', let
at.
R be the L.V.B.
QP = 0, and let
d for which QP = Q.
I
E Q.
as C at and this in turn
contains all projections in
be an arbitrary projection in tions
on '.t , so
S
Now let
P
of all projec-
be the L.U.B. of the
It is easily seen that
DECOMPOSITIONS OP OPERATOR ALGEBRAS. II
RP = 0
SP = S, so
and
R(PS) _ (RP)S = 0.
RS
an arbitrary unitary operator in Q , the equation
U*QPU = 0 = (U*QU)P.
Now if U
QP = 0
is
implies that
R = U*RU, i. a.,, R is in the
It follows that
Similarly S
center of d .
47
is in the center of Q .
Now I-R>P(I-R) = P>PS = 3, so I-R>P>S. By the definition of uniform multiplicity, with case the preceding inequality shows that of
to the range of
Cl
P
I - R - S = 0, in which F_ Q, or the contraction
is of uniform multiplicity
I - R - S
1.
0-1
We show
that the latter alternative is an impossibility.
We show first that the contracting of Q 1
to
P(I - R - 3)h,
is an isomorphism, i. e.1 that if U E G2.1 and if P(I - R - S)U = 0, (I - R - S)U = 0.
then
when U 0
Now it is sufficient to show this for the case To see this, let
is a projection.
and NUU* = 0.
Clearly Nf(UU*) = 0 for any polynomial
f(O) = 0, and hence if K
that
NU =
N = P(I - R - 3), so f
such
is any projection in the W*-algebra gen-
erated by UU*, NK = 0.
As this algebra is generated by the projections
it contains, the equation
NK = 0 for all K implies that NUU* = 0
(NU)(NU)*, so
pose that of
NU = 0.
Now taking U
P(I - R - 3)U = 0.
Then
I - R - S
R, and multiplying this inequality by
both sides) yields the inequality
to be a projection in O L, sup-
(I - R - S)U < R, by the definition
(I - R - S)U < 0, so (I - R - S)U = 0.
Next we show that the contracting of is also an isomorphism.
show that if U 0, then
(which commutes with
Q 1
to
(I - P)(I - R - 3
As in the preceding paragraph, it is sufficient to
is a projection in
(I - R - S)U = 0.
Cl
such that
The equation
(I - P)(I - R - S)U =
(I - P)(I - R - 3)U = 0
can
be put in the form P(I - R - S)U = (I - R - S)U, which by the definition of
S
implies that
equality by P(I - R - 3)
(I - R - S)U < S.
I - R - S
and
shows that
Multiplying both sides of this in-
(I - R - S)U = 0.
(I - P)(I - R - S)
Now the projections
are obviously orthogonal, so what
I. E. Segal
48
has just been proved shows that Q 1
is of minimal multiplicity at least
Hence the second alternative above was impossible.
2, a contradiction.
Now suppose conversely that If P
t?
in the center
Q'
(Z
Q1 of
, the contraction
minimal multiplicity at least
2.
Ql
Let
projections on * which coincide on P1+ with annihilate the orthogonal complement of
Q', for if
T
(for
P
and
Q
Qj
and
Then
P1+.
Q1
T(PQ1) = TQ1.
and which
Q2
and
Q2
Ql
are in we have
Q,T = (Q1P)T = Q1(PT) = Q1(PTP)
is in the center of Q ) = (PTP)Q,
(TP)Q3.
are mu-
be the unique
Q2
and
is arbitrary in 4, then taking the case of
Q1P = PQ1 = Q1)
(noting that
Q2
and
is of
P
such that the contracting of Q 1
= 1,2 ).
QjP1. is an isomorphism (i
a to
Qi
Suppose then that
(Q1)'
tually orthogonal projections in to
1, then for some nonzero projection
is not of uniform multiplicity of
Q t C (2.
is commutative, so
Q t
Therefore
Ql
(as
Q2
and
Qj 6 (Q1)') = are in
a-, and as
Qi
are mutually orthogonal, (Q1P)(Q2P) = 0, which shows that the
contracting of
Q1P1+
to
Q 1
annihilates
Q2P, and so is not an isomor-
phism.
Algebras of uniform multiplicity
1
play a conspicuous role in
the following, and so it is convenient to make the following definition,
which is justified by the fact that a Wa-algebra containing form multiplicity
1
I
is of uni-
if and only if the lattice of all closed linear sub-
spaces invariant under the algebra is a Boolean algebra. Definition 8.1.
An algebra of operators is called hyper-reducible
if it is a W*-algebra of uniform multiplicity COROLLARY 8.1.
1.
If two hyper-reducible algebras are algebraically
isomorphic, then they are unitarily equivalent. Let the hyper-reducible algebras spaces
1t
and
14'2
Q 1
and
Q2 on Hilbert
be algebraically isomorphic via the map
f on Q l
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II to
Then
Q 2.
center
C2
C1
) maps the center
a 2.
of
Let
hrl
of Q 1
49
isomorphically onto the
be any mass algebra on
7¢1
which con-
tains C1 . As Q1 is hyper-reducible, Cl = (Q1)+, so (C1)' (al) 1, = al, and hence 2'1 C al. Putting2 - 9( 7!(1), then 2
is masa on
7¢2, for as
is an algebraic isomorphism, 71f2
9'
Q 2,
imal abelian relative to
1. e.,
(
2) n Q2 = )1(2.
is max-
Taking commut-
ors, it follows that 7Jr2 v a2 = ( 2)', and as
Q2)' = C2 C NO
it results that N2 = ( >2)+, i. e., 1l'2
We can now apply the
result in
is masa.
that if two mass algebras on Hilbert spaces are algebrai-
[lo]
cally isomorphic, then there is a unitary transformation U between the spaces which implements the isomorphism.
C1 is carried onto 2 by
As
the operator-isomorphism induced by U, (C1)' that isomorphism, i. e.) a1 COROLLARY 8.2.
is carried onto
( 2)'
by
Q2.
is mapped onto
A W*-algebra of finite uniform multiplicity n > O
is unitarily equivalent to an n-fold copy 2f a hyper-reducible algebra, the latter algebra being unique within unitary equivalence. Let
n, and let tions in
Q be a W*-algebra on
be an indexed set of
Pl,...,P,
Qt
such that for each
an isomorphism.
of finite uniform multiplicity
7d-
n mutually orthogonal projec-
We show first that the contraction
is hyper-reducible.
Let
Qt
a to
i, the contracting of
a,
of Q to
Q of which
Q'P1 N has minimal multiplicity Rt
and
on
St
onal, and such that the contractings of are isomorphisms.
agree on
Putting
Q'Pi74 with
RI
R and and
S'
(for some fixed
> 2
S
Q'PI 23
Q.i,
is the contraction.
Qt
the basis of an indirect proof suppose that the contraction
there exist projections
Pi7Y'
be a nonzero projection in the center of
and let Q be the projection in
to
is
Pi 1'
in
13
As
of 01
1), so that
1i', mutually orthog-
to R+Q'Pi* and to
S+QtPi*
for the projections on Q1* which respectively, and which annihilate
50
I. E. Segal
Q(I-Pi) 1f, then it is not difficult to verify that
where
Q to
C is the contraction of
both isomorphisms.
to
S, together constitute a set of n + 1
C
Pj
S
and
S
P3Q N'
to
Q 1#
C',
are in
are mutual-
and to
RQ7'
C to
Now the contractings of
likewise isomorphisms and the contractions of and
R
Q 1f, that
ly orthogonal, and that the contractings of
and
R
SQ14-
(.1 $ i)
are
are
(f * i), R
mutually orthogonal projec-
tions in C .
It follows that
n + 1, but Q
is in the center of Q , so this is in contradiction with
is of minimal multiplicity at least
the definition of uniform multiplicity, which requires that
C be of uni-
form multiplicity n. By Corollary 8.1, there is a hyper-reducible algebra that each
Q i
show that
Q
is unitarily equivalent to
70.
I&
such
It is straightforward to
is unitarily equivalent to an n-fold copy of
t3.
The final theorem in this section concerns the uniqueness of multiplicities in the non-commutative case. THEOREM 9.
An m-fold copy of a hyper-reducible algebra .6' is
unitarily equivalent to an n-fold copy of n hyper-reducible algebra
and only if m = n
and
e7
if
and C are unitarily equivalent.
ff
As the "if" part is clear, we assume that the W*-algebra a on 14 is unitarily equivalent to an m-fold copy of . and also to an n-fold
copy of e. Then Q is algebraically isomorphic to both so that
We can now assume that $ = 6 ; for each
8.1) unitarily equivalent. T 6 .
, let
fl(T)
bra of 19
'(T)
and
m- and n-fold copies of
let
V and & ,
and C are algebraically isomorphic, and hence (by Corollary
be the unitary transforms in
T, respectively.
Now let
a of the
)t( be a mass aubalge-
(the existence of which is shown in the proof of Corollary 8.1), C , and let
and SA in .19', as
7)
and
)(/-1( C) = )t .
Then ?I
is maximal abelian
)b are algebraic isomorphisms, and hence mass
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II (of. the proof of Corollary 8.1). tarily equivalent
fold copy of
74(
[10J
As
.
It follows that
ZY(
51
and
are uni-
7T
C is unitarily equivalent both to an en-
and to an n-fold copy of
, it results from Lemmas 2.9
t
and 2.11 that m = n. COROLLARY 9.1. Let
that for each n multiplicity
be a family of projections in the
{Pn }
0 on
center of the W*-algebra
Q to the range of
the contraction of
n, and
{Pn}
Then if
Un?I = I.
is separable, this equation is valid for all If
and
Rhat
5PI
n
514
A
have uniform multiplicities
and
to
infinite, and suppose that RS14'
R
and
S
at
and n respectively, where
is finite, then
RS = 0.
at, and if these are both finite, it follows directly
from Theorem 9 and Corollary 8.2 that
m
, and if
a to RS* has uniform multiplici-
Clearly the contraction of at
(so that if W'
such that the contractions of a to
and at least one of m and n
ties both
is a family with the
n).
Q Ls a W*-algebra on
are projections in the center of
Pn has uniform
Pn = Pn
n
property given in Theorem 6, for all finite
LEMMA 9.1.1.
n, such
indexed by cardinal numbers
/4.
RS 96
RS = 0. 0.
Now let n be finite and
Then the contraction
Q o of Q
is unitarily equivalent to an n-fold copy of a hyper-reducible
algebra 5 .
be a mass subalgebra of .B', let
Let
braic isomorphism of
.
'
onto
(20
(P
be the alge-
induced by the map of .& onto its
n-fold copy and by the unitary equivalence of this copy with Q o, and set is unitarily equivalent to an n-fold copy of
(P())() _
, so that
W( Now as
a is of infinite uniform multiplicity, there exist mutually
orthogonal projections
map
P1, P21...
in
T --- PST is an isomorphism on Q .
that the
on A .
Pi
at
such that for each
I
the
As a = 3Z , Q' - >t' , so
are in n ', and clearly the map
Thus ( is of minimal multiplicity
T -* PIT
is an isomorphism
On the other hand, it
52
I. E. Segal
is of uniform multiplicity n
according to Definition 2.1, hence of uni-
form multiplicity n according to Definition 7.1, so that it cannot be of
minimal multiplicity
j,1', .
PROOF OF COROLLARY.
Q
in the center of
Pn
Let
by Lemma 6.3, Q has uniform multiplicity n on Pn1' .
Un
and
P.
k.
PZ < P ,.
P'
= Un
Evidently
Pn.
By the preceding lemma, the
we have
.
P,
Pn = Fn
As
(
and
separable, it is plain that necessarily
0
and
in, and hence
Finally, if
Pn) V Pco 7¢ is
for n > Ro.
We give a structure theorem for We-alge-
Algebras of tore I.
9.
Pn
n
are mutually
vnck,
I
= Poo .
P,
Off;
Poo
for all
_< Pn
P''
PM with infinite
Pn) V Poo =
for n < 21,
Let
with finite n
Pn
orthogonal, and are also orthogonal to the
orthogonal to
Q
be the L.U.B. of all projections
such that a has uniform multiplicity n on
bras of "type I", where these are, roughly speaking, algebras which are direct integrals of factors of type I.
At the same time we obtain a com-
plete set of unitary invariants for such algebras, which can be regarded as fully known by virtue of this classification.
The following definition of
algebra of type I is equivalent to a definition in
[i]
for certain ab-
stract algebras.
A W*-algebra is said to be of type I if every
Definition 9.1.
projection in
at
is the least upper bound of projections of type
Q', where a projection jection in in
Q'
P
in
Of
is of tune M if whenever Q
such that QS P, then Q = RP where
R
M
in
is a pro-
is a projection
a n of. Roughly speaking, a projection
P
is of type M if when the alge-
bra is decomposed as a direct integral of factors, P
decomposes into an
integral of projections each of which has a range which is of (ordinary linear) dimension at most one, so that it is akin to a minimal projection,
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II as our terminology is intended to suggest.
53
The following theorem asserts
essentially that a W*-algebra is of type I if and only if it is a direct sum of n-fold copies of hyper-reducible algebras. THEOREM 10.
if there exists a family
Pn}
indexed by cardinal numbers
union
a on
A Wa-algebra
is of type I if and only
14/-
Q',
of Projections in the center of
n, which are mutually orthogonal and have
Q
I, and are such that the contraction of
to
(n - 0)
Pn 1+
is
unitarily equivalent to an n-fold copy of a hyper-reducible algebra ffn
of uniform multiplicity n, while The
Fn
is the maximal projection in 0_.
I-P0
are unique as are, within unitary equivalence, the
J"f n.
Conversely, a is determined within unitary equivalence kZ the knowledge of the
.1 'n
as abstract algebras (which knowledge determines the L9'n
within unitary equivalence), or alternatively by the unitary invariants of the commutative algebrea J9 a, together with the dimension of the range of Po.
Remark 9.1.
Naturally a second alternative to the .n as uni-
tary invariants are the invariants of the
&n'
This
as given in Part I.
implies that a cardinal-valued function F(n,m,p)
of triples of cardinal
numbers could also be used in place of the unitary-equivalence classes of F(n,m,p)
the Cr Jn. Here
is the cardinality of the number of summands
isomorphic to the measure ring of the product measure space
Ip
(where
I
is the unit interval under Lebesgue measure) which occur in a decomposition into finite homogeneous parts of the measure ring for multiplicity ,PJn, and it is necessary as earlier to normalize it equal to
1
for the ease when
p
is determined only within the interval ably interpreted.
I
0
and the number
1 - P < &, and
n, m, and
of
P(n,m,p), say by setting
Clearly any cardinal-valued function
vanishes for sufficiently large
in
I0
F
of summands
must be suit-
F(n,m,p)
which
p, then corresponds to a unique
54
I. E. Segal
unitary equivalence class of W*-algebras of type I containing the identity operator.
The first six of the following lemmas are needed in the proof of the "only if" part of the theorem, while the remaining lemmas are for the "if" part.
LEMMA 10.1.
Let
bra O, and suppose that
PT = TP = T, where
and
Then there exists an operator
T = SP
Q such that
be a non-negative SA operator in the W*-alge-
T < P
jection of tyLe M in Q . of
T
and
S
P
is a pro-
in the center
0 < S < I.
The W*-algebra C generated by P
and
is obviously commu-
T
tative and has a unit, and so is isomorphic to the algebra C (r) continuous functions on a compact Hausdorff space P . difficulty that there exist projections
(i,n = 1,2,...) such that for each fixed orthogonal, the
o:
in
in
Qiu
n, the
(.'
of all
It follows without and real numbers tXin
Qin
are mtually
satisfy the inequalities 0< Ocin < 1, and with
Tn -- T uniformly where Tn = rr Lei & in4in Qin. < P
T.
-
so that
Qln = RinP
with
and Tn < Ta«1. Evidently Bin a projection in Q ^ at, and
(Zi ainRin)P or T. = RnP with Bn 6 Q " 0-t. Now let
be the G. L. B. of all such non-negative operators SnP.
Now Sn < Sn+l
elements in C (Sn ^ Sn+1)P
Sn
Tn =
for denoting lattice intersection in the set of SA
by "A", we have
Tn = Tn A Tn«1 = SnP A Sn+1P
SnP = Sn A P)
which shows that
Sn, or Sn < Sn+l
It follows that
(noting that
Sn, by the minimality of
verges strongly to an operator ily seen that
Bn; than clearly
S
0 < Sn < I, we have
in A ^ At
and
Sn A Sn*l = {sn}
con-
T = SP; as it is eas-
0 < S < I.
LEMMA 10.2. &Z projection of type N in a W*-algebra is contained in a maximal projection of type M.
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II
55
This follows at once from Zorn's principle once it is shown that
Now
the L. U. B. of any chain of projections of type K is again of type K.
.1 is such a chain in the Wa-algebra
if
A
the elements of W , suppose that A < Q. that
A = RQ for some
in C " Et.
R
in 6 ^ et
Rp
for the G. L. B. of all such
P20 then
-< SP1.
Now for any
such that
E 'y we have
P
PAP = RPP
P, for if
Pi
PAP = SPP, and C
'
(i
P2P1AP1P2 = P23P1AP2 or W. = SP1P2
Now let
S
be the L. U. B. of the
{SP}
the strong limit of the
Putting SP
0 < Rp < I.
RP, then clearly
monotone increasing function of
2
is a projection in 6 such that
It follows from the preceding lemma that
ror some operator
SP
Is the L. U. B. of
To conclude the proof of the lemma it is only necessary to show
PAP < PQP = P.
P1
Q
, and
'
SP, P e
is a
By
= 1,2)
and
which implies
Then
.
is
S
where this indexed set is directed by the
usual ordering on the indices, and passing to the limit in the equation
PAP = SPP yields the equation QAQ = SQ, or A = SQ.
If
LEMMA 10.3.
algebra C, and If
e ^ Et, then
R
and
P
and
3
Q
are projections of In* K in the W*-
are mutually orthogonal projections in
RP t SQ is also of tyye K in '.
Suppose that E
is a projection in a such that E < RP 4 SQ.
E = ER + ES, and ER < RP
Then E < R 4 S
so
R' E C ^ CO.
Similarly ES = S'SQ with
R'RP + S'SQ =
31
so
ER = R'RP
6 E ^ CO.
with
Thus
E
(RR' r SS')(RP t SQ).
LEMMA 10.4. If
Q Ls a Wa-algebra of tie I on
is any closed linear subspace invariant under
1-01, and if .6
(),, then Al contraction of
a to j is again of toe I. Let
Q1 be the contraction of
nonzero projection in
( 01)0.
Let
P
Q to
.t
and let
be the projection on
Pi be a 1* which
I. E. Segal
56
P e
Then
and annihilates 1f E )f-.
P1
extends
Q'
have for x e t, UPx = Ux = PUx, and for x E 1
UP = PU.
FUx, so
such that
Q < P.
iant; let
Q1
Q annihilates 11 O t , so it leaves £ invar/-.
is of type M for if
Q1
is the contraction to
S1
Sl = R1Q1 LEMMA 10.5.
is a projection in
implies that
with
R1 E
for if
x e Fi-
Q. < P1.
S1
(Q 1)', then
where
Q '
S ' Q; and as
S(1' Q ..)
Q
is of
Q', from which it re-
Ql n (Q1)'.
0 be a W#-algebra on * of type I. and let
Let
be a maximal projection of type M in Q'.
to
in
S
S = RQ with R 6 Q '
type M, it results that
P
Q1 6 (Q1)'
Also it is clear that
o- of the projection
0, and the equation S1 < Q1
suits that
Then
is the contraction of U e Q , then we have for
f, , Q,U1x = QUx = UQx = U1Qlx. nally
E >.t,, UPx - 0 =
Q be any nonzero projection of type M in a '
Let Then
be its contraction to
U1 E Q1, say U1
U C Q , we
for if
Then the contracting of Q
is an isomorphism.
P/4-
It is sufficient, by a remark in Part If to show that if S
projection in a such that
PS = 0, then
L. U. B. of all the projections 0
and U
Thus
is unitary in
S
in
S =
SP = 0.
Q such that
a, then U*(SP)U = 0
Now let
0.
is a
T be the SP =
If
(U*SU)P = 0.
so that
is the L. U. B. of a set of projections which is invariant under
T
the inner automorphisms induced by the unitary operators in Q , and hence
If Q is a projection in
is in the center of Q . such that type M.
Q'
of type M and
T > Q, then by Lemma 10.3, P(I-T) + QT = P + Q
By the maximality of P, Q = 0, and since
If
Q is a nonzero W*-algebra on
then there exists a nonzero projection contraction of
to
Is the L. U. B. of
T = 0.
such Q, it follows that LEMMA 10.6.
T
is again of
P }/-
P
in 12 n
a'
14- of t
e I,
such that the
is unitarily equivalent to an n-fold copy of a
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II hyper-reducible algebra, for some cardinal Let
n > 0.
be an indexed family of projections
°
P
is maximal with respect to the properties: 1) for each of
a to Pis an isomorphism; 2) the
3) each
57
in
which
Of
the contracting
,AA
are mutually orthogonal;
Pf
is of type M (the existence of °.K is clear from Lemmas 10.2
P,
and 10.5).
Let
0. Then
which Q(I
0, and the contracting of Q to
(I -
continuation of this argument, let Lemma 10.4, Q 1
is of type I.
Q1
Q1
and annihilates 1
In
on H. be this contraction; by be a maximal projection in
Q be the projection on
of type M, and let which extends
Let
is an isomorphism.
U)..P,, )1'
Q1
I - Ut.P,
0, as otherwise
,
Qo
in 0- for
Q
be the L. U. B. of the projections
Qo
O 141.
7601' (necessarily in
( Q1)'
at)
Then the contracting of Q
to Q74 (= Ql f,/1) is an isomorphism, Q is orthogonal to all the ?/,-* and it follows as in the proof of Lemma 10.4 that
Q
contradicts the maximality of 7 and so shows that
Qo
0.
is in the
By an argument used in the proof of Lemma 10.5, Qo
center of Q . Putting
P;,,
= QoPp
and K,,,,. = P 1, 14, , then:
invariant and its contraction
leaves
(b) the identity in (Qp )' of type N in
(0.'*)'
(c) the contracting to isomorphism.
a,,
(a) Q
^ is of type I;
to
PA
(i.e. the contraction of
to
/lit
) is
(by an argument used in the proof of Lemma 10.4); of the contraction of
Q to Qo1' is an
It is clear that the identity operator is of type M in a We-
algebra only if the algebra is commutative, and hence the reducible.
This
is of type M.
If n
are hyper-
2,M
is the cardinality of the index set, it follows from
Corollary 8.1 that the contraction of
Q to
tarily equivalent to an n-fold copy of any one of the LEMMA 10.7.
L
Q00'
a^
A commutative We-algebra is of t
Kf,
) is uni-
-
e I.
Let Q be a commutative nonzero W*-algebra on 7# .
To show
58
I. E. Segal
Q is of type I it suffices to show that for any projection
that
Q', there is a projection of type
M
in
at
P
in
which is contained in
P.
a such that
Let W be a countably-decomposable projection in
0, and put Q1 for the contraction of
isomorphism, there exists a separating vector x1 (x,
Ql to Al
for
is maximal with respect to the properties that it contains
ax
is orthogonal to
,
is an
Ax.,
for
V
,AA
Put-
in
for an indexed family of separating vectors for
}
VWP =
Q to *f1 = (I-V)wMAs 121
is countably-decomposable and as the contracting of
ting
0
be the maximal projection in Q such that
V
(cf. Lemma 2.7), let
WP
which
a_1
x1, and that
, a repetition of the con-
struction at the beginning of the proof of Lemma 2.6 shows that there exists
a nonzero projection
in
Q1
a 1 to Q1, and if of Q2xf,. , then Ql =
0-1
such that if
Q2
is the projection of
R,.
U R.
is the contraction of
Q1
and the contraction of a 2 to
n
algebra, where
onto
R1Q1
1
be a projection in
U
R114- invariant, for if R11°1'
U(I-Q)
annihilates
Q'
such that
is in
and
T' < R1.
invariant, T1 = R1U1 projection in
Q14 = Q11 1.
S s at.
As
Q2
Finally If
with U1 6
is mesa on 8111
Q1,
d- leaves
T1
T of
a2 while is any proT
to
Q1+
and leaves T1Q *
Q2 n (Q2)', and putting Uo for a
0 whose contraction to
a is commutative.
Now
Q1# is in
T < 3, then the contraction
Q?
is
not difficult to verify that T = UR, and U and
is of type M in
S
is arbitrary in Q , U = UQ + U(I-Q), and UQ
R11+; therefore
jection in 0.1
and
invariant because its contraction to
leaves
S < P,
Q whose contraction tot is
q 6
and set Q = (I-V)WQo, so
for
3
then it is easily seen that
and we conclude the proof of the lemma by showing that Qo
Putting
is the cardinality of the index set.
1-
the projection of
Let
R
being unitarily equivalent to an n-fold copy of this masa
is masa, Q 2
Q'.
onto the closure
1
U1 E
and U = QUo, it is
Q n at. for U C Q
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II LEMMA 10.8.
Let the We-algebra
Then
reducible algebra.
53
Q be an n-fold copy of A hyper
a is of type I.
It is easily seen that if
is an indexed family of mutual-
{Pn J
ly orthogonal projections in the center of a W*-algebra a such that the
Q
contraction of
to the range of
Q is itself of type I.
I, then
Pn
is of type I and with
Unrn =
It follows without difficulty from Theo-
rem 2 that it suffices to consider the ease in which the hyper-reducible
algebra £ of which a is an n-fold copy has its commuter equal to an m-fold copy of a mass algebra
Let
(I act on
796,
)y(
.O
on 1-6.
act on
.
, and let
be the
copies of X which are (mutually orthogonal) subspaoes of 7'. any operator on into
and
.
-Zv
r
If
then
is the projection on
T
is
P, TP,
and so induces in a natural fashion an operator
maps
Z,,.
T v
on Z ; by the matrix of
we mean this matrix
T
)), which
((T1,,y
is a function on the direct product of the index set with itself to the operators on .1 .
It is not difficult to verify that a matrix
is the matrix of an operator in Q '
corresponding element of . and
only if
Sr, v x'-,
s,,,,,, x,
112
x'"
the element of L
x,,
is the
corresponding
is convergent (this is the ease if and
1f3r,vis bounded for each fixed v
and
(1) it is the matrix
is arbitrary in 7' , if
of a bounded operator, I. e.lif x
to x'4 , then the sum
if and only if:
as a function of
If
is a convergent sum; (2) S,,
}
for
!t, v . We omit the details of the verification of this, as they in/standard methods. volve
each
Now let
(P
be the algebraic isomorphism of
,0'
onto
takes each operator into the operator of which is the m-fold copy.
on b
be the n-fold copy of
is the matrix of
3, then
S,
on k . ,v'
and
If
S
6
and
(('P (S A.v )))
which
Let
((8," )) is a matrix
60
I. E. Segal
of operators in W ; we shall show that is the matrix of an operator in
y('
(relative to the decomposition of the space
,$
as an n-fold direct sum of copies of 7,C ), and that To show that
isomorphism onto.
on which
i(S)
YL
is an algebraic
is the matrix of an opera-
tor in ( , it is only necessary to check condition (1) above.
Now each
is in a natural fashion an m-fold direct sum of copies
For each r', let
.
and for each
y
the projection of
h let y,
in
y
of
be any index in the range of values of
Cr 1
on the
acts
a- ,
be the element of 1 corresponding to
,..th
copy of
in 4 , and further let
It
be that element x in 1'f such that x,, or = yt. and x1,,,d = 1 is the element in 7-6 corresponding to c rte` C, where x 0 if the projection of x on the c'th copy of 71'* in the `.th copy of because 0 is an Then as (1) holds and as II = I)S,.y I J I T-). Sk. v ( 1(y)) (I 2 = algebraic isomorphism, and '? (y)
11
.
11 Z". 0 (S },, v )yy II
2, it follows that the condition corresponding to (1)
holds in the case of the matrix the matrix of an operator in
(( (Sf.y ))), so that
is
((
7Q'.
to is the matrix of an operator S' in show that t is onto we need only show that Ev II Z, 0 -l(S'r1 )xµ 112 is as in the preceding paragraph, we obis a convergent sum. If x1)x,#2 II x".0- , so serve that ( fd-1(s'A )x,,, Now if
( (Sf' V ))
n' = 1 Z L' Fl6I
xN °' 1) 2 x1,
y a II
I1s'II To see that
It follows that
I1 2.
x, 2Z"I1x,,
11
= IIZ r ZI.
2
II 2
=
S' p v X', c y
II Z,
y 1I
11
2 y
= (s y y )x,u 112
xf.r II 2
<
= 113112 Z, . IIxf.P II 2 = IIs112 11x112.
is a homomorphism, observe that it Is a homomorphism on
the subalgebra of operators whose matrices have only a finite number of nonzero coordinates, because
0 is a homomorphism, so that by the valid-
ity of the standard rules for matrix operations and the strong density of
DECOMPOSITIONS OF OPERATOR ALG'bBRAS. II that subalgebra (of.
61
, p. 137) it is a homomorphism on the entire
[5]
algebra.
Thus 41
at
and
are algebraically isomorphic.
type I, as it is commutative, so every projection in of projections of type M.
7(1
Now
'Y(
is of
is the L. U. B.
As L. U. B.s of projections and the concept of
projection of type M are preserved under algebraic isomorphisms, it follows that every projection in
01
is the L. U. B. of projections of type M, -
Q is of type I.
I. e.,
PROOF OF THEOREM.
Let
Q be a W*-algebra on
of type I.
1tk
From Lemma 10.6 it follows readily that there exists a family
{Pp I
of
mutually orthogonal projections in the center of a such that the contrao-
0
tion of
to
Pf,1d'
is unitarily equivalent to an n(u)-fold copy of a
hyper-reducible algebra, and
is the maximal projection in Q .
U "-F,.
The proof of Lemma 2.2 shows that if
{Qy }
jections in the center of a W::-algebra
C to
Q,,74-
are mutually orthogonal pro-
(f such that the contraction of
C to
is hyper-reducible, then the contraction of
U,,Qy )1' is likewise hyper-reducible. Putting Qm = Un(,L& ) -m P/" it follows that the contraction of 0
to
Qm 74- is unitarily equivalent to
an m-fold copy of a hyper-reducible algebra, and clearly the ly orthogonal and
Um Qm
is the maximal projection in
Conversely, let the W*-algebra exists a family I-Po
{P }
Q on
are mutual-
6.
]¢ be such that there
of projections in the center of
is the maximal projection in
Qm
Of
and such that
the contraction of Q to
Pn 1¢
is unitarily equivalent to an n-fold copy of a hyper-reducible algebra. show that
a is of type I it suffices to show that any such n-fold copy
is of type I, and this is the statement of Lemma 10.8.
Finally, the unique-
ness part of the theorem follows without difficulty from Theorem 9. COROLLARY 10.1. commutor.
To
If a W*-algebra is of type I, then so is its
I. E. Segal
62
It is easily seen that it suffices to prove the corollary for the case in which the algebra
Q.
in question is an n-fold copy of a hyper-re-
ducible algebra .. To show that
is of type I is to show that every
Q contains a nonzero projection of type M.
nonzero projection in 1°1T
at
is commutative so, by Lemma 10.7, every nonzero projection in
contains a nonzero projection of type M, and as
d9'
and
Now
&
Q are algebrai-
cally isomorphic, it follows that the same is true of Q . COROLLARY 10.2.
A W*-algebra of type I Is algebraically isomor-
phic to a hyper-reducible algebra via a mapping onto the hyper-reducible algebra that is weakly continuous and preserves the operational calculus for normal operators.
The proof of this is a slight modification of part of the proof of Theorem 5.
The next result is essentially equivalent to the non-trivial part
of Theorem IV of
[5] .
COROLLARY 10.3.
(Murray and von Neumann).
A factor whose com-
mutor contains a minimal projection is unitarily equivalent to an m-fold copy of the algebra of all operators on an n-dimensional Hilbert space, for
unique cardinals m and Let
projection
Pgr
n.
be a factor on 14
E, and let
P
Then the L. U. B. of U*EU as -Xv
whose commuter contains a minimal
be an arbitrary nonzero projection in 7 1.
U ranges over the unitary operators in
is a projection in the center of
that the operator easily deduced that
T = PU*RUP
P = 0, but
'XI, and so equals I.
is nonzero for some T
is SA
It follows
U, as otherwise it is
and non-negative so that it
can be uniformly approximated by linear combinations of projections Q that
o Q < T
for some
oc > 0.
Now
T
<
such
U*EU, from which it follows
DECOMPOSITIONS OP OPERATOR ALGEBRAS. II
Q vanishes except on the range of
that
minimal, so also is that
7
Hence
U*EU.
U*EU, so Qt U*EU; and as
U*EU, and it follows that
U*EU
is a scalar multiple of
T
63
Q = U*EU.
E
is
It results
T =
and it is easy to see that
is of type I in the sense of Definition 9.1.
As the center of
is trivial, it follows from the theorem
that it is unitarily equivalent to an m-fold copy of a hyper-reducible algebra and
.O L'P
on ]ti
Now
.
c
contains the corresponding copy of
n "1
so
follows that
operators on
j
That m and n
.
$
contains this latter m-fold Dopy.
consists only of scalars, so
l
.IY'
on K , It
is the algebra of all
.O
are unique is clear from the uniqueness
part of the theorem. In view of the special role in the foregoing of com-
Remark 9.2.
mutative W*-algebras of uniform multiplicity we should mention a relatively concrete form for such.
bra k on 1', , then measure space
If
Q on
is the n-fold copy of a masa alge-
1
can be taken as the multiplication algebra of a
7x
M = (R, IF, r)
which is the direct sum of the finite per-
(R,, , lfa
, r, );
the collection of all indexed families
fP (p)
fect measure spaces
M, _
µ ranging over an index set 1). E.
L2(M)
for all /u
(bounded) operators on .t
A
on
14,
of the form
A function
T(.)
M), then the function n
JjT(p)Il
(Af)pt (p)
Now let to the
to Z (i. a.
is bounded as
1Im(p)II 2
is
defined by the equation n(p) =
is again a measurable function on M
bounded in case
M.
on M
m is n. e. the limit of a sequence of simple functions, and
T(p)m(p)
f,,,c(4
is called strongly measurable if, whenever m(.)
is a strongly measurable square-integrable function on M
integrable on
with
n, and such that
is a bounded measurable function on
be a Hilbert space of dimension' n.
R
ff,, it 2 convergent; and
and with the sum k
of functions on
of cardinality
Q is then the algebra of all operators k(p)f,u (p), where
14 can plainly be taken as
p
to -; and
ranges over
T(.) R.
is called
Now regarding
64
I. E. Segal
as the space of all complex-valued functions
=C,,
sum
2
ie convergent and with
f*
on fl
(f, g)
such that the
for any
ff, g.,
two such functions, there corresponds to any bounded strongly measurable function and
on M
T(.)
a (unique) operator
in 1- , (Tf, g) =
g
J
on 1+ such that for any f
T
(T(p)f*(p), g.(p)) L dr(p), where
R indicates an inner product in J_ ; this follows readily from the
(.., ..)L
observation that the integral exists and defines a continuous function of f
and
which is linear in
g
f
and conjugate linear in
g.
We can now
state:
The operator T
for every
in
a,, and every element of
Q '
has this form, i, e.
there is a bounded measurable function
a'
T(.)
such that for any f in 7qL, (Tf).(p) = T(p)(f.(p)). If
to T
is in
T
are elements of
S(.)
oc
tions are valid n. e. on M: (a.S)(p)
o-(S(p)), and
and
is a complex number, then the following equa-
(S+T)(p) = S(p) S* (p) _ (S(p))*.
T(p), (ST)(p) = S(p)T(p), A similar result is stated
and 1 are separable in
for the case when
S
to which correspond in this fashion the functions
(21
T(.), and if
and
on M
[21,q. v.
The only point which offers any difficulty is the fact that every element of
has the stated form.
C2'
correspondence between the element
E fl where
and
T
Tr.v
{ITII 2
fp
n x n matrices over. Let
Of correspond to the matrix
((T.,,,,
)), where /(, V
is the operation of multiplication by k u v
on LOU)' The
means that Z II F, 1, T,
fµ II 2 < 11 T112 Zr Nfy. 112 E $ , or / F, I Y.p k',,,, (p)fd, (p) 2) dr(p) < fZ''A.,{ff, (p) 12 dr(p). As this equation remains valid when each
boundedness of
f
Now as shown above there is a natural
and appropriate
can be taken to be bounded and continuous on each mx .
k,,,,
for any
of
Q'
T
is multiplied by the characteristic function of a measurable set, it
results that n. a. on M, ZvI Z:p kpv (p)f (p) 12 IIT II 2 Z p I fP (p) 12. In particular, If f}, vanishes except for a
DECOMPOSITIONS OF OPERATOR ALGEBRAS. II finite set
F
R,,, and if
veG
,U s, and if for each ).1, f)A
of G
is
is continuous on each
any finite set of indices we have
'peF kyv (p)f,, (p)I 2
I
65
<
IITII2 ZNEF If (p)I 2, n. e.
As both sides of this inequality are continuous on each
Rx , and as the
complement of a null set in a finite perfect measure is dense in the space, the inequality is valid for every
p e R, and since
is an arbitrary
G
finite set of indices we can conclude that
I Z' y, F kr,v p E R.
((k,,,,,,
Hence there exists a (bounded) operator
(p)))
T(p)f.(p) =
seen that for any f e 14 being as above, - so that converges to
all
f
in 1-f ,
(Tf).(p), for
f
(fi}
<
11TH ,
It is readily
as above.
there is a sequence p, fi(p)
(p)
in t, for nearly all
(Tf).(p) = T(p)f.(p)
ed strongly measurable function on
hP
n. e. on
fi
in 14- , each
T(p)f'(p) _ (Tfi).(p), - and such that
in 14 ; for nearly all
(Tfi).(p) --)(Tf)(p) f
.t with matrix
on
T(p)
(relative to the obvious basis), with IIT(p)II
and such that
with
IITII2 z µEF If)A (P) 12 for all
(P)fd,, (p) 12
p.
M, and
in
oC
fi
, and
It follows that for T(.)
to the operators on Z .
is a bound-
66
I. E. Segal
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2.
G. W. Mackey, A theorem of Stone and von Neumann.
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D. Maharam, On homogeneous measure algebras.
Proc. Nat. Acad. Soi.
28(1942) 108-111. 4.
N. H. McCoy, Rings and ideals.
Baltimore, 1948.
5.
F. J. Murray and J. von Neumann, On rings of operators.
Ann. Math.
37(1936) 116-229. 6.
H. Nakano, Unitarinvarianten hypermaximale normale Operatoren.
Ann.
Math. 42(1941) 657-664. , Unitarinvarianten In ailgemeinen euklidisaher Raum.
7.
Math.
Ann. 118(1941/43) 112-133. 8.
A. I. Plessner and V. A. Rokhlin, Spectral theory of linear operators. Uspekhi. Mat. Nauk (N.S.) 1(1946) 71-191.
II (in Russian).
Cf. Math. Rev.
9(1948), p. 43. 9.
I. E. Segal, Decompositions of operator algebras. I.
Sybmitted to
Memoirs Amer. Math. Soc. , Equivalences of measure spaces.
10.
Amer. Jour. Math.
73(1951) 275-313.
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11.
Ann. Math.
48(1947) 930-948.
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groups. 13.
Duke Math. Jour. 18(1951) 221-265.
J. von Neumann, Uber Funktionen von Funktionaloperatoren.
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32(1931) 191-226. 14.
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