Decompositions of Operator Algebras I and II (Memoirs of the American Mathematical Society)

Number 9 'S

11H Mil

jar

TPHTOE MH

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 (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.


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
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

Upj (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

_
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) , 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
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)


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
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
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 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