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Quotients of finite W∗-algebras
JOURNAL
OF FUNCTIONAL
ANALYSIS
Quotients
9,
322-335
of Finite
(1972)
W*-Algebras+
JP~RGEN VESTERSTRDM Mutematisk
Institut,
Arhus
Communicated
Universitet,
Arhw
C, Denmark
by J. Dixmier
Received February 10, 1971
The main result is the following: Let M be a finite and o-finite W*-algebra and J a uniformly closed two-sided ideal of M. If (i) 1 is an intersection of maximal ideals, (ii) the center of M/J is a W*-algebra, (iii) the center of M/J is o-finite, then M/J is a W*-algebra. Conversely, if M/J is a W*-algebra, then the conditions (i) and (ii) are satisfied. It is shown by examples that if M is abelian, condition (ii) is not fulfilled in general. However, there are examples where (ii) is fulfilled without J being o-weakly closed. Finally it is proved that if M satisfies some mild conditions there exists an ideal J,, (resp. X, JE) satisfying the following condition (a) (resp. (b), (c)): (a) ] satisfies (ii) and (iii), but not (i); (b) 1 satisfies (i), but not (ii) and (iii); (c) J satisfies (i), (ii), and (iii), and is not o-weakly closed.
INTRODUCTION
AND NOTATION
In the algebraic reduction theory of finite IV*-algebras [4 and 151, it is shown that if J is a maximal ideal of a finite W*-algebra n/r, then the quotient C*-algebra Ml J is a finite factor. In this paper we study the natural generalization of this result, namely, what are the conditions that the quotient C*-algebra iVl/ J of a W*-algebra by a uniformly closed ideal J be a W*-algebra ? As far as properly infinite W*-algebras go, the answer has been given by Takemoto [14 and 131 provided M is representable on a separable space. His result states that M/J is a W*-algebra [if and] only if J is u-weakly closed. In the finite case, however, the picture is different. Since a maximal ideal J is u-weakly closed if and only if J n 2 is an isolated point in the maximal ideal space of 2, the center of M, the results of Wright [15] and Feldman [4] show that there are ideals J for which M/J is a W*-algebra without J being u-weakly closed. sity
+ The paper covers of Pennsylvania
part of the author’s under supervision
doctoral dissertation, of S. Sakai.
322 0
1972 by Academic
Press,
Inc.
written
for the Univer-
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323
In this paper, we characterize the ideals for which the associated quotient C*-algebra is a IV*-algebra (Theorem 1.l and Theorem 1.2). This is done by generalizing the method of Sakai [12]. The algebra M operates on its center 2 by (x, z) -+ (xx)+, where # is the center valued trace. We study the module spanned by the above maps, and prove theorems analogous to theorems from measure theory, e.g., the theorems of Riesz-Fisher, Egoroff and Lusin. In proving the sufficiency we imbed (M/J) m ’ t o a W*-algebra and use a criterion of Sakai to conclude that the injection actually is a surjection. This criterion, however, only applies in the u-finite case, and our conditions only work in that case. Theorems 1.1 and 1.2 show, roughly speaking, that the ideals of M for which Ml J .is a IV*-algebra are in 1 - 1 correspondence with the ideals I of Z, for which Z/1 is a W*-algebra. The correspondence isgivenby J-+ Jn Z.Th us, the problem of finding the IV*-quotients is reduced to the abelian case. In a subsequent paper we shall investigate the abelian case. Once the abelian case is discussed we can analyze the nonabelian case. This will be done, too, in a subsequent paper. The notation is standard. @([w)denotes the complex (real) numbers. If E and F are Banach spaces, L(E, F) denotes the Banach space of bounded linear transformations from E to F. “Ideal” always means “two-sided ideal.”
1. THE
MAIN
THEOREM
In this section, M usually denotes a finite W*-algebra and J a uniformly closed ideal. The object of this section is to find conditions that the C*-quotient M/J be a IV*-algebra. We start by deriving some necessary conditions. PROPOSITION 1.l. Let @ : M -+ N be a *-homomorphism between W*-algebras. Suppose that Q(M) = N and that M is finite. Then N is finite.
Proof. Suppose that e E N is a projection with e N I. We wish to show that e = I. Since e N I, there is a partial isometry u E N such that U*U = I and UU* = e. By surjectivity of di there is an x E M with Q(X) = u. Consider the polar decomposition x = yh, where y is a partial isometry and h = (x*x)~/~.
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Then we have @(h) = @((x*x)1/2)
= @(x*x)l~z
= (u*u)lP
= I,
24.= CD(x) = @(yh) = CD(y) @P(h) = Q(y).
(1)
(2)
Since y is a partial isometry, yy* and y*y are equivalent projections in M, and I - yy* and I - y*y are equivalent, because M is finite. Since @ preserves equivalence of projections, it follows that I-uuu*=q1-yy*)N@(I-yy*y)=I-u*u=o and e = uu* = I. That proves the proposition. THEOREM 1.I. Let J be a uniformly closed two-sided ideal of the jinite W*-algebra M. Suppose that M/J is a W*-algebra. Then, the following holds:
(i) (ii)
J is an intersection of maximal ideals. The center of M/J is a W*-algebra.
Proof. (ii) is trivial. As to (i) we notice that M/J is strongly semisimple, that is, the intersection of the maximal ideals of M/J is (0) (see [12]). Taking inverse images under the canonical map M -+ Ml J we conclude that J is an intersection of maximal ideals. This proves the theorem. To give sufficient conditions we find it convenient to restrict ourselves to the case where M is a-finite. Moreover, if we add one extra condition, we have a set of sufficient conditions. THEOREM
1.2. Let M be a a-jinite and $nite
W*-algebra with
center 2. Let J C M be a unifOrmly closed ideal satisfying
(i) (ii) (iii)
J is an intersection of maximal ideals. The center of M/ J is a W*-algebra. The center of M/J is a-jinite.
Then M/J is a unite]
W*-algebra.
Remark 1.1. If we assume that M can be represented on a separable space, and if we further assumethe Continuum Hypothesis, condition (iii) becomes necessary, too (see [13, p. 155, 1. 9-221 for an argument where the Continuum Hypothesis is used implicitly). Thus, under these extra hypotheses, conditions (i), (ii), and (iii) are necessary and sufficient conditions for M/J to be a W*-algebra.
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The remaining part of this section is devoted to the proof of Theorem 1.2. M, 2, and J will satisfy the hypotheses of Theorem 2 throughout the proof. We begin by introducing some notation. Let T be the spectrum of 2. We identify 2 with C(T), its algebra of Gelfand transforms. Thus, the maximal ideals of Z are t E T,
“4( = {z E 2 j x(t) = 01, and the maximal
(3)
ideals of M are given by Aft = {x E M 1(x*x)“(t)
= 01,
t E T,
(4)
where x E M -+ x# E Z is the canonical center valued trace on M. Observe that A%‘~n Z = Mt (see [7 and 121). By assumption, J is of the form ~=t?s~~=jrtMl(x*x)#=OonS}=~,, and since (X*X)+ is continuous I = J n Z. Then
(5)
we may assume that S is closed. Let
Let x(t) (resp. x”) be the image of x under the canonical map M + M/At (resp. M + M/J). Since J = nfES A’* , the map x” ---t BteS x(t) from M/J to GtsS M/A& is a well-defined, injective *-homomorphism between C *-algebras, and therefore we have
The #-map factors (see [12]). So we define x E M and x” = 0, then defined map # : M/J -+
II5 !I = sup II4t)il. tES
(7)
to M/AY~ -+ Z/Xi x(t)+ = x”(t) for x# = 0 on T, and Z/I given by x” -+
N C (complex numbers) x E M and t E T. Next, if therefore there is a wellx# Is .
LEMMA 1.1. The canonical *-homomorphism and maps onto the center of Ml J.
Z/I + Ml J is 1 -
1
Proof. Since J n Z = I, it is well-defined and 1 -. 1. It is easy to verify that the range is contained in the center of M/J. Conversely, let x E M/J be central. Then, by applying the *-homomorphism x -+ olcs x(t) we see that x(t) commutes with y(t) for y E M and t E 5’. M/At is simple, and therefore x(t) is a scalar, say x(t) = A( for t E S.
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Hence x(t)+ = A(t), so h is continuous on S and is of the form h = 5 for some z E 2. Then, for t E S, we have x(t) = s(t)l; so x - x E J, and thus x” is the image of z + I under Z/I+ M/J. The lemma is proved. Henceforth we shall identify Z/I with the center of Ml J. In particular we shall use the notation z + f for the canonical map 2 -+ Z/1. The map # : M/J -+ Z/I is now a map from Ml J to its center, and it is easily seen, that it satisfies all the properties of a center valued trace except the normality condition, which as yet is meaningless, since Ml J is not equipped with a a-weak topology. However, once we have proved Ml J is a W*-algebra, # will be the trace. To prove that Ml J is a W*-algebra we are going to exhibit the predual of M/ J. To this end we study the following Banach Z-module as introduced in [ 121 by Sak ai in his proof of the special case, where J is maximal. For a, x E M let Q,(X) = (ux)*. Ga is a linear and bounded map from M to 2, and moreover @,(~.s) = .~@Jx) for a, x E M and x E 2. These properties hold for 9, the uniform closure of {Qa / a E n/r} in L(M, 2). The f o11owing proposition summarizes and generalizes results from [12]. PROPOSITION
(i)
1.2. Let @ E 2’. Then, the following holds:
If x E At , then Q(x) E At .
(ii) The induced map Q(t) : x(t) + @(x)(t) is a bounded linear functional on M/A$ . (iii) II @ II = sup{II @(t)lI I t E 0 (iv) There is a partial isometry u E M such that II @(t)l] = @(u)(t) for t E T; in particular t + 11Q(t)11is continuous. Proof. See [12]. W e remark, that (iv) is proved by applying the polar decomposition of g, 0 CD,where y is a linear, positive, normal, and faithful functional on Z. PROPOSITION
(i)
1.3. Let 0 E 9.
Then the following holds:
If x E J, then CD(X)E J.
(ii) The induced map 6 : M/J -+ Z/I is norm-bounded.
de$ned by 8,(g) = @F)
(iii) II 6 II = sup4I @(t)ll I t E S>. (iv) The set of all 6, @ E 2, f arms a closedsubsetof the bounded linear mappingsfrom M/J to Z/I.
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(i) f 011 ows from Proposition 2(i), since J = nfES 45Zl.
(ii) and (iii): Recall (Eq. (7)) that I( x” 11= supfGs(1x(t)l\. Now II m)li
= IIrnll
= y’sp I @(W
< sup I! Wll II4t)li t&Y
< SUP II WI tss
. SUP II @)ll ta
That proves (ii) and 1)6 11< supaGs11@(t)ll. To prove the reverse inequality and thus (iii), let u be the partial isometry of Proposition 2(iv), and we have SUP II W>ll
tss
= sup I W)(~)l = IImll tes
G II6 Il.
(iv): The map A: Q-6 is norm-diminishing by (iii) and Proposition l(iii). If we can prove that the induced map x : .Z/ker A -+ L(M/J, Z/1) is an isometry, we will have proved our assertion. It is well-known that 11x I\ < 11h 11< 1, so we must prove that )/6,I~>I)@+kerhI\for@~~.Forz~Zwedenotebyx@themap x -+ @(~a) = z@(x). Clearly, z@ E 3, and if x = 1 on S we have (1 - z)@ E ker A, that is, .z@E @ + ker A. Now, let x = 1 on S and 2: 3 0. Then,
Let E > 0 and R = {t E T 1/I @(t)ll > I\ 6 11+ c}. By Proposition 1.2(iv) t---t 11G(t)\] is continuous; so R is closed, and, by (iii), R is disjoint from S (and possibly empty). By Urysohn’s Lemma we can find x~ZwithO
II Wll
G II6 II + 69
and therefore
Since E > 0 was arbitrary, proposition is proved.
the desired inequality
follows, and the
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As yet we have used only hypothesis (i) of Theorem 1.2. Now we invoke conditions (ii) and (iii), which say that the center of M/J is a u-finite W*-algebra. By Lemma 1, it follows that Z/I is a u-finite W*-algebra. Since Z/I N C(S), S is a Stonean space, and there is a normal faithful measure on 5” [l J. The set E of functionaIs p 0 8, @E 9, is a subset of (M/J)*, whose closure we denote by F. PROPOSITION 1.4. (i) E, and hence F, is translation invariant, that is, if a”E M/J and f E E (resp. F), then x”-f(Z) and x +f(Z) are in E (resp. F).
(ii)
Iff(x”)
= 0 for all f E E, then LZ= 0.
(iii> II P 0 4 II = .L II @(t)ll h(t). (iv) For ,U0 & E E there is a d E Ml J with
/I 6 I/ < 1 and
P o w> = II P o 6 IIProof. (i): For @ E 9 we have ~(6(%) = p(p(Z)), where YE 9 is defined by Y(x) = @(ax), so E is left-invariant. A similar proof shows that E is right-invariant. (ii): Letf(x”) Then we have
= 0 f or allf E E. Consider in particular
f = p ox
.
p 0 @S(2) = p((a*a)#) = 0 and since p is faithful,
(x)#
= 0; so x” = 0.
(iii) and (iv): Let @ E Y. Then I P a ml
= / j, @cw 44)
j
so we have jl p o 6 11< Js 11@(t)ll dp(t) where the integral is welldefined by Proposition 1.2(iv). For the u of the same proposition we have
Since I( u”jl < (( u (I < 1, we have (I EL.0 & I( = p 0 6(G), and (iv) and the second half of (iii) follow. The proposition is proved.
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329
We can now define the Banach space that will turn out to be the predual of M/J. Let (M/J)** denote the bidual of (M/J), and let E” be the polar of E in (M/J)* * under the canonical duality w/l)*
x (M/J)***
PROPOSITION
normal faithful (M/J)
1.5. (i) trace.
(M/J)**/EO
is a IV*-algebra with a jkite,
(ii) The composition (M/J) -+ (M/J)** in (M/J)**/EO as a C*-algebra. (iii)
The predual of (M/])**/Eo
-+ (M/J)**/EO
imbeds
is F.
Proof. (i) It is a standard fact that (M/J)** is a IV*-algebra. Since E is invariant, E” is an ideal (see [12]), and E” is weakly closed, so (M/J)**/EO is a W*-algebra. Next, observe that every f E (M/J)* has a unique o-continuous extension to (M/J)**, and in particular the trace x” --L p(Z#) has an extension, which by continuity is positive and central, and it annihilates E” by definition. Thus it is well-defined on (M/J)**/EO, and we denote it by Tr. Clearly Tr is positive and central. Let B E (M/J)**/EO be positive with Tr(B) = 0, and let A E (M/J)** be a positive representative. Then for x E M we have / Tr(xA)12 < Tr(x*x) Tr(A*A)
= 0.
Consequently Tr(SB) = 0 for all x E (M/J). Since the unique extension of y” -+ p 0 G,,(y) = Tr(%y) is A -+ Tr(xA) we see that A is annihilated by E and therefore A E E” and B = 0. Therefore Tr is faithful. The normality of Tr is automatic. (ii): From Proposition 1.4(ii) it follows that E” n (M/J) = (01, so the composition is injective, and it is well-known that each of the maps is *-homomorphic. (iii): This is a standard fact. The proposition is proved. To finish the proof of Theorem 1.2 we shall prove that (M/J) is actually equal to (M/J)**/EO via the embedding of the preceding proposition. For this we must establish the following property. (*I
f E F there andf (4 = llf II. For every
is an f E M/J with (1R (1< 1
Notice that (*) has been proved if f E E (Proposition 1.4(iv)), and we obtain the general result as a consequence of a more detailed study of F, the closure of E. 58019/3-6
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PROPOSITION 1.6. Let f EF. Then there exist a sequence (@J in 9 and a family Y(t),,, such that
(9 P 0 &n-+f; (ii) (iii)
Qp,(t) + Y(t) a.e. [Jo]; x E M, then t--f Y(t)(x(t))
If
fW (iv)
is p-integrable
= 1, w>w>>
t -+ I/ Y(t)ll is p-integrable
and
44);
and
llfll = j, II WY
440
Proof. The proof is analogous to the proof of the Riesz-Fisher theorem. Since F is the closure of E, there is a sequence Qn in A? with TV0 G’, --+f. We may assume that -pdQ
f&.G By Proposition
< CO.
1.4(iii) we can write 0) - @&)ll G(t) g1 i’, II @?a+1
and therefore
there is a p-null
set NC
< 009
S such that
so (@Jt)) is a Cauchy sequence for t 4 N. Let, for t E S \ N, Y(t) = lim Q,(t), and 0 elsewhere. This proves (i) and (ii). For x E M we have Y(t)(x(t)) = 1im @,(t)(x(t)) for t $ N; so Y(t)(x(t)) is p-measurable and
G
2 j I @,+lww) p=n s
G f
e=n
j
s
-
@&>W)l
44)
II @,+1(t) - @p@)llII x II 44t) - 0
as
n+co.
QUOTIENTS
Consequently, t -+ Y(t)(x(t)) f(f)
= lim p 0 g(x)
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is p-integrable and
= lim j” Qn(t)(x(t))
c+(t)
=
f Y((t)(x(t))
dp(t)
J
which proves (iii). As above we see t + ]I Y(t)/1 and t + Ij Y(t) - @Jt)jl are measurable functions, and
as
jsII y(t)ll G(t) = liq j, II @P,(t)ll444
n- --t *;
= lim II p 0K I/ = Ilf II
by using Proposition 1.4(iii). The proposition is proved. The following lemma pursues the analogy with standard measure theory. Now we establish a kind of Lusin theorem for elements of F by applying an analog of Egoroff’s theorem. LEMMA 1.2. Let E > 0 and z”EZ/Iwithp(l-,zZ)<~anda@~9suchthat f(%)
f
E F. Then there exist a projection
= p 0 6(Z)
for
x E M.
Proof. Let @, , Y(t), and N be chosen as in Proposition 1.6, and define Ek,m = {t E S\ N j 11G,(t) - Y(t)11 < l/m for all Y > k. By construction (& m)k is increasing for fixed m and u,“=i Ek,m = S \ N. Choose for each’integer m an integer R, such that 4s \ &,.vJ Let A = nzE1 E,,,,,
-=cQm.
. Then,
If t E A and m is an integer, I( Q?(t) - Y(t)11 < l/m holds for r 3 K, , and therefore d>,(t) -+ Y(t) uniformly in t E A. By regularity of p
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we can find a compact set K C A with p(A \ K) < E, and, since ,A is normal, we have p(K) = p(R), where R is the interior of K. p(S \ K) < 2~. K is open and closed, since S is Stonean, so 1, is of the form ,?Ywhere 2: is a projection in 2 (see [l]). We now have
so the sequence (&J is Cauchy. limn+aoXj, tion 1.3(iv), and for x E M it follows that f(ET) = liip
= (z, exists by Proposi-
06’,(L%) = lillp(16n(32)) = p 0 6(a).
This proves the lemma. LEMMA 1.3. Let f EF and supposethat there are projections xi in Z/I and pi in 2, i = 1, 2, with f(zq
=
p 0 d$(n)
fOY
x E M,
i=
1,2.
Then, there is a @ E 2 such that
f(E) = p 06(a), where x = z1 v z2 . Proof.
Define @ = xr@r + (zZ - xlze) @)2E 64. Then P o @(a) = P@dW
+ P@d(% - %w))
= f (iq + f (a& - ,c$iq) = f (qa, + f, - @iJ) = f (23).
Q.E.D.
THEOREM 1.3. Let f EF. Then, there exists a sequence(Gm) in 9 together with a sequence(zJ of pairwise orthogonal projections in Z such that
f(2) = g p 0&Qi%,)
for x EM.
n=l
Proof. By Lemma 1.2, we can find for each integer n a projection - ’ in Z/I and elements Qn in 9 such that p( 1 - s:,‘) < l/n and &ZJ = /A 0 &(Z). By L emma 1.3, we may assume that (&‘) forms an increasing sequence. Let now e, = f, and e, = 2%’ - &, for n >, 2. Then (e,) is a family of pairwise orthogonal projections in Z/Z and C p(e,) = p(I); so ~~=r e, = I. Using [15], Lemma 3.4
QUOTIENTS OF FINITE W*-ALGEI~MS we find a sequence of pairwise f, = e, . We then have f@qJ
orthogonal
= f(E&‘qJ
Combining
this with
PROPOSITION andf(q Proof.
1.7.
= I’ w>w)
(t)
(.zn) in 2 with
= /.Lc’ QaQ.
On the other hand, letf(x) = Js Y(t)(x(t)) of Proposition 1.6. Then we have f(x)
projections
333
&(t)
(I9 be the decomposition
44)
we obtain the desired result.
Let f E F. Then there is a ZI E M/J
with Ij 5 // < 1
= IlflL Let
f
be written
as in the above theorem.
Then
llfll ~~IIP%AII.
we have (-l-t)
By Proposition 1.4(iv) we find Us E A!l such that /) ZI, 11< 1 and p 0 dj)n(5:n5in) = Ij p o Z,$,, I). Since the projections z, are pairwise orthogonal ZI = C Z,ZI, is strongly convergent and I( w II = sup 11ZI, (1 < 1. Moreover x = ~7~ = v”,P, , and
Since 11u 11< 1, the proposition
follows
from this and (i-t).
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End of proof of Theorem 2. Sakai has proved the following theorem in [12]. Let B be a W*-algebra with predual B, and A C B a C*-subalgebra. If B admits a faithful positive functional and if for every f E B, , there is anx E A with11 x 11< 1 andf (x) = /jf 11,then A = B. Propositions 1.4 and 1.6 show that (M/J) = (M/J)**/EO via the injection of Proposition 1.4. In particular (M/J) is a W*-algebra. Remark 1.2. It would improve Theorem 1.2 if we could drop the countability conditions. The assumption that M is u-finite is inessential and is merely used for convenience, namely in Proposition 2(iv), but the proof can be effectuated without this assumption, simply by performing the polar decomposition for each x,@, where (2,) is a family of pairwise orthogonal, u-finite projections (cf. [12]). However, the condition that Z/I be o-finite is essential for our proof, but can probably be avoided. Indeed, a possible way is the following. Decompose (Z/1) in to a direct sum @ e,(Z/I) of o-finite W*-algebras. Than (M/J) ear is a W*-algebra, but we do not necessarily have Of/J) N 0 @f/J) eE. This would hold if we could lift the family (e,) to a family of .pairwise orthogonal projections in 2 (cf. [15, Lemma 3.41). This, however, we do not know, and it is false if we do not assume that Z/I is a W*-algebra (see [S]).
2. REMARKS The condition (ii) of Theorem 1.2 makes it pertinent to study the abelian case. Let 2 be an infinite dimensional abelian W*-algebra acting on a separable space. One can show the existence of non o-closed ideals for which Z/I is a W*-algebra and which are not finite intersections of maximal ideals, using results from [6] and [lo]. Also we can exhibit ideals I for which Z/I is not a W*-algebra (cf. [9]). Using results from the abelian case and analyzing the continuity properties of the functions t + I] x(t)11 we study the nonabelian case. Let (i), (ii), and (iii) be the three conditions of Theorem 1.2. It can be proved that if M is a finite W*-algebra satisfying some fairly mild conditions there are uniformly closed ideals J, , Jb , and Jc such that (a) J, satisfies (ii) and (iii), but not (i). (b) Jb satisfies (i), but not (ii) and (iii). (c) Jc satisfies (i), (ii), and (iii) and is neither a-closed nor a finite intersection of maximal ideals. This shows that the conditions of Theorem 1.2 are independent.
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ACKNOWLEDGMENT I want to express my gratitude to C. Akeman, conversations and advice, and especially to M. to these problems.
S. Sakai Takesaki,
and E. Stormer who directed
for valuable my attention
REFERENCES 1. J. DIXMIER, Sur certains espaces consider& par M. H. Stone, Summa Brasil. Math. t. 2 fast. 11 (1951), 151-182. 2. J. DIXMIER, Point &pares dans le spectre d’une C*-algebre, Acta. Sci. Math. (Szeged). 22 (1961), 115-128. 3. J. DIXMIER, Ideal center of a C*-algebra, Duke Math. /. 35 (1968), 375-382. 4. J. FELDMAN, Embedding of AW*-algebras, Duke Math. J. 23 (1956), 303-307. 5. J. FELDMAN AND J. FELL, Separable representations of rings of operators, Ann. of Math. 65 (1957), 241-249. 6. A. GLEASON, Projective topological spaces, Illinois 1. IMath. 2 (1958), 482-489. 7. R. GODEMENT, Memoire sur la theorie des caracteres dans les groupes localement compacts unimodulaires, 1. Math. Pures Appl., 30 (1951), p. l-l 10. 8. I. KAPLANSKY, Normed algebras, Duke Math. J. 16 (1949), 399-418. 9. I. KAPLANSKY, Representations of separable algebras, Duke Math. J. 19 (1952). 219-222. 10. D. MAHARAM, On homogeneous measure algebras, Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 108-111. 11. Y. MISONOU, On a weakly central operator algebra, T6koku Math. 1. 4 (1952), 194-202. 12. S. SAKAI, “The theory of IV*-algebras,” Yale University, Press, New Haven, Conn., 1962. 13. H. TAKEMOTO, On the homomorphism of von Neumann algebra, T6hoku Math. J. 21 (1969), 152-157. 14. H. TAKEMOTO, Complement to “On the homomorphism of von Neumann algebra,” T6hoku Math. 1. 22 (1970), 210-211. 15. F. B. WRIGHT, A reduction theory for algebras of finite type, Ann. of Math. 60 (1954), 560-570.
OF FUNCTIONAL
ANALYSIS
Quotients
9,
322-335
of Finite
(1972)
W*-Algebras+
JP~RGEN VESTERSTRDM Mutematisk
Institut,
Arhus
Communicated
Universitet,
Arhw
C, Denmark
by J. Dixmier
Received February 10, 1971
The main result is the following: Let M be a finite and o-finite W*-algebra and J a uniformly closed two-sided ideal of M. If (i) 1 is an intersection of maximal ideals, (ii) the center of M/J is a W*-algebra, (iii) the center of M/J is o-finite, then M/J is a W*-algebra. Conversely, if M/J is a W*-algebra, then the conditions (i) and (ii) are satisfied. It is shown by examples that if M is abelian, condition (ii) is not fulfilled in general. However, there are examples where (ii) is fulfilled without J being o-weakly closed. Finally it is proved that if M satisfies some mild conditions there exists an ideal J,, (resp. X, JE) satisfying the following condition (a) (resp. (b), (c)): (a) ] satisfies (ii) and (iii), but not (i); (b) 1 satisfies (i), but not (ii) and (iii); (c) J satisfies (i), (ii), and (iii), and is not o-weakly closed.
INTRODUCTION
AND NOTATION
In the algebraic reduction theory of finite IV*-algebras [4 and 151, it is shown that if J is a maximal ideal of a finite W*-algebra n/r, then the quotient C*-algebra Ml J is a finite factor. In this paper we study the natural generalization of this result, namely, what are the conditions that the quotient C*-algebra iVl/ J of a W*-algebra by a uniformly closed ideal J be a W*-algebra ? As far as properly infinite W*-algebras go, the answer has been given by Takemoto [14 and 131 provided M is representable on a separable space. His result states that M/J is a W*-algebra [if and] only if J is u-weakly closed. In the finite case, however, the picture is different. Since a maximal ideal J is u-weakly closed if and only if J n 2 is an isolated point in the maximal ideal space of 2, the center of M, the results of Wright [15] and Feldman [4] show that there are ideals J for which M/J is a W*-algebra without J being u-weakly closed. sity
+ The paper covers of Pennsylvania
part of the author’s under supervision
doctoral dissertation, of S. Sakai.
322 0
1972 by Academic
Press,
Inc.
written
for the Univer-
QUOTIENTS
OF FINITE
W*-ALGEBRAS
323
In this paper, we characterize the ideals for which the associated quotient C*-algebra is a IV*-algebra (Theorem 1.l and Theorem 1.2). This is done by generalizing the method of Sakai [12]. The algebra M operates on its center 2 by (x, z) -+ (xx)+, where # is the center valued trace. We study the module spanned by the above maps, and prove theorems analogous to theorems from measure theory, e.g., the theorems of Riesz-Fisher, Egoroff and Lusin. In proving the sufficiency we imbed (M/J) m ’ t o a W*-algebra and use a criterion of Sakai to conclude that the injection actually is a surjection. This criterion, however, only applies in the u-finite case, and our conditions only work in that case. Theorems 1.1 and 1.2 show, roughly speaking, that the ideals of M for which Ml J .is a IV*-algebra are in 1 - 1 correspondence with the ideals I of Z, for which Z/1 is a W*-algebra. The correspondence isgivenby J-+ Jn Z.Th us, the problem of finding the IV*-quotients is reduced to the abelian case. In a subsequent paper we shall investigate the abelian case. Once the abelian case is discussed we can analyze the nonabelian case. This will be done, too, in a subsequent paper. The notation is standard. @([w)denotes the complex (real) numbers. If E and F are Banach spaces, L(E, F) denotes the Banach space of bounded linear transformations from E to F. “Ideal” always means “two-sided ideal.”
1. THE
MAIN
THEOREM
In this section, M usually denotes a finite W*-algebra and J a uniformly closed ideal. The object of this section is to find conditions that the C*-quotient M/J be a IV*-algebra. We start by deriving some necessary conditions. PROPOSITION 1.l. Let @ : M -+ N be a *-homomorphism between W*-algebras. Suppose that Q(M) = N and that M is finite. Then N is finite.
Proof. Suppose that e E N is a projection with e N I. We wish to show that e = I. Since e N I, there is a partial isometry u E N such that U*U = I and UU* = e. By surjectivity of di there is an x E M with Q(X) = u. Consider the polar decomposition x = yh, where y is a partial isometry and h = (x*x)~/~.
324
VESTERSTRBM
Then we have @(h) = @((x*x)1/2)
= @(x*x)l~z
= (u*u)lP
= I,
24.= CD(x) = @(yh) = CD(y) @P(h) = Q(y).
(1)
(2)
Since y is a partial isometry, yy* and y*y are equivalent projections in M, and I - yy* and I - y*y are equivalent, because M is finite. Since @ preserves equivalence of projections, it follows that I-uuu*=q1-yy*)N@(I-yy*y)=I-u*u=o and e = uu* = I. That proves the proposition. THEOREM 1.I. Let J be a uniformly closed two-sided ideal of the jinite W*-algebra M. Suppose that M/J is a W*-algebra. Then, the following holds:
(i) (ii)
J is an intersection of maximal ideals. The center of M/J is a W*-algebra.
Proof. (ii) is trivial. As to (i) we notice that M/J is strongly semisimple, that is, the intersection of the maximal ideals of M/J is (0) (see [12]). Taking inverse images under the canonical map M -+ Ml J we conclude that J is an intersection of maximal ideals. This proves the theorem. To give sufficient conditions we find it convenient to restrict ourselves to the case where M is a-finite. Moreover, if we add one extra condition, we have a set of sufficient conditions. THEOREM
1.2. Let M be a a-jinite and $nite
W*-algebra with
center 2. Let J C M be a unifOrmly closed ideal satisfying
(i) (ii) (iii)
J is an intersection of maximal ideals. The center of M/ J is a W*-algebra. The center of M/J is a-jinite.
Then M/J is a unite]
W*-algebra.
Remark 1.1. If we assume that M can be represented on a separable space, and if we further assumethe Continuum Hypothesis, condition (iii) becomes necessary, too (see [13, p. 155, 1. 9-221 for an argument where the Continuum Hypothesis is used implicitly). Thus, under these extra hypotheses, conditions (i), (ii), and (iii) are necessary and sufficient conditions for M/J to be a W*-algebra.
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325
W*-ALGEBRAS
The remaining part of this section is devoted to the proof of Theorem 1.2. M, 2, and J will satisfy the hypotheses of Theorem 2 throughout the proof. We begin by introducing some notation. Let T be the spectrum of 2. We identify 2 with C(T), its algebra of Gelfand transforms. Thus, the maximal ideals of Z are t E T,
“4( = {z E 2 j x(t) = 01, and the maximal
(3)
ideals of M are given by Aft = {x E M 1(x*x)“(t)
= 01,
t E T,
(4)
where x E M -+ x# E Z is the canonical center valued trace on M. Observe that A%‘~n Z = Mt (see [7 and 121). By assumption, J is of the form ~=t?s~~=jrtMl(x*x)#=OonS}=~,, and since (X*X)+ is continuous I = J n Z. Then
(5)
we may assume that S is closed. Let
Let x(t) (resp. x”) be the image of x under the canonical map M + M/At (resp. M + M/J). Since J = nfES A’* , the map x” ---t BteS x(t) from M/J to GtsS M/A& is a well-defined, injective *-homomorphism between C *-algebras, and therefore we have
The #-map factors (see [12]). So we define x E M and x” = 0, then defined map # : M/J -+
II5 !I = sup II4t)il. tES
(7)
to M/AY~ -+ Z/Xi x(t)+ = x”(t) for x# = 0 on T, and Z/I given by x” -+
N C (complex numbers) x E M and t E T. Next, if therefore there is a wellx# Is .
LEMMA 1.1. The canonical *-homomorphism and maps onto the center of Ml J.
Z/I + Ml J is 1 -
1
Proof. Since J n Z = I, it is well-defined and 1 -. 1. It is easy to verify that the range is contained in the center of M/J. Conversely, let x E M/J be central. Then, by applying the *-homomorphism x -+ olcs x(t) we see that x(t) commutes with y(t) for y E M and t E 5’. M/At is simple, and therefore x(t) is a scalar, say x(t) = A( for t E S.
326
VESTERSTRBM
Hence x(t)+ = A(t), so h is continuous on S and is of the form h = 5 for some z E 2. Then, for t E S, we have x(t) = s(t)l; so x - x E J, and thus x” is the image of z + I under Z/I+ M/J. The lemma is proved. Henceforth we shall identify Z/I with the center of Ml J. In particular we shall use the notation z + f for the canonical map 2 -+ Z/1. The map # : M/J -+ Z/I is now a map from Ml J to its center, and it is easily seen, that it satisfies all the properties of a center valued trace except the normality condition, which as yet is meaningless, since Ml J is not equipped with a a-weak topology. However, once we have proved Ml J is a W*-algebra, # will be the trace. To prove that Ml J is a W*-algebra we are going to exhibit the predual of M/ J. To this end we study the following Banach Z-module as introduced in [ 121 by Sak ai in his proof of the special case, where J is maximal. For a, x E M let Q,(X) = (ux)*. Ga is a linear and bounded map from M to 2, and moreover @,(~.s) = .~@Jx) for a, x E M and x E 2. These properties hold for 9, the uniform closure of {Qa / a E n/r} in L(M, 2). The f o11owing proposition summarizes and generalizes results from [12]. PROPOSITION
(i)
1.2. Let @ E 2’. Then, the following holds:
If x E At , then Q(x) E At .
(ii) The induced map Q(t) : x(t) + @(x)(t) is a bounded linear functional on M/A$ . (iii) II @ II = sup{II @(t)lI I t E 0 (iv) There is a partial isometry u E M such that II @(t)l] = @(u)(t) for t E T; in particular t + 11Q(t)11is continuous. Proof. See [12]. W e remark, that (iv) is proved by applying the polar decomposition of g, 0 CD,where y is a linear, positive, normal, and faithful functional on Z. PROPOSITION
(i)
1.3. Let 0 E 9.
Then the following holds:
If x E J, then CD(X)E J.
(ii) The induced map 6 : M/J -+ Z/I is norm-bounded.
de$ned by 8,(g) = @F)
(iii) II 6 II = sup4I @(t)ll I t E S>. (iv) The set of all 6, @ E 2, f arms a closedsubsetof the bounded linear mappingsfrom M/J to Z/I.
QUOTIENTS
Proof.
OF FINITE
327
W*-ALGEBRAS
(i) f 011 ows from Proposition 2(i), since J = nfES 45Zl.
(ii) and (iii): Recall (Eq. (7)) that I( x” 11= supfGs(1x(t)l\. Now II m)li
= IIrnll
= y’sp I @(W
< sup I! Wll II4t)li t&Y
< SUP II WI tss
. SUP II @)ll ta
That proves (ii) and 1)6 11< supaGs11@(t)ll. To prove the reverse inequality and thus (iii), let u be the partial isometry of Proposition 2(iv), and we have SUP II W>ll
tss
= sup I W)(~)l = IImll tes
G II6 Il.
(iv): The map A: Q-6 is norm-diminishing by (iii) and Proposition l(iii). If we can prove that the induced map x : .Z/ker A -+ L(M/J, Z/1) is an isometry, we will have proved our assertion. It is well-known that 11x I\ < 11h 11< 1, so we must prove that )/6,I~>I)@+kerhI\for@~~.Forz~Zwedenotebyx@themap x -+ @(~a) = z@(x). Clearly, z@ E 3, and if x = 1 on S we have (1 - z)@ E ker A, that is, .z@E @ + ker A. Now, let x = 1 on S and 2: 3 0. Then,
Let E > 0 and R = {t E T 1/I @(t)ll > I\ 6 11+ c}. By Proposition 1.2(iv) t---t 11G(t)\] is continuous; so R is closed, and, by (iii), R is disjoint from S (and possibly empty). By Urysohn’s Lemma we can find x~ZwithO
II Wll
G II6 II + 69
and therefore
Since E > 0 was arbitrary, proposition is proved.
the desired inequality
follows, and the
328
VESTERSTR0M
As yet we have used only hypothesis (i) of Theorem 1.2. Now we invoke conditions (ii) and (iii), which say that the center of M/J is a u-finite W*-algebra. By Lemma 1, it follows that Z/I is a u-finite W*-algebra. Since Z/I N C(S), S is a Stonean space, and there is a normal faithful measure on 5” [l J. The set E of functionaIs p 0 8, @E 9, is a subset of (M/J)*, whose closure we denote by F. PROPOSITION 1.4. (i) E, and hence F, is translation invariant, that is, if a”E M/J and f E E (resp. F), then x”-f(Z) and x +f(Z) are in E (resp. F).
(ii)
Iff(x”)
= 0 for all f E E, then LZ= 0.
(iii> II P 0 4 II = .L II @(t)ll h(t). (iv) For ,U0 & E E there is a d E Ml J with
/I 6 I/ < 1 and
P o w> = II P o 6 IIProof. (i): For @ E 9 we have ~(6(%) = p(p(Z)), where YE 9 is defined by Y(x) = @(ax), so E is left-invariant. A similar proof shows that E is right-invariant. (ii): Letf(x”) Then we have
= 0 f or allf E E. Consider in particular
f = p ox
.
p 0 @S(2) = p((a*a)#) = 0 and since p is faithful,
(x)#
= 0; so x” = 0.
(iii) and (iv): Let @ E Y. Then I P a ml
= / j, @cw 44)
j
so we have jl p o 6 11< Js 11@(t)ll dp(t) where the integral is welldefined by Proposition 1.2(iv). For the u of the same proposition we have
Since I( u”jl < (( u (I < 1, we have (I EL.0 & I( = p 0 6(G), and (iv) and the second half of (iii) follow. The proposition is proved.
Quom~ms
0~ FINITE
W*-ALGEBRAS
329
We can now define the Banach space that will turn out to be the predual of M/J. Let (M/J)** denote the bidual of (M/J), and let E” be the polar of E in (M/J)* * under the canonical duality w/l)*
x (M/J)***
PROPOSITION
normal faithful (M/J)
1.5. (i) trace.
(M/J)**/EO
is a IV*-algebra with a jkite,
(ii) The composition (M/J) -+ (M/J)** in (M/J)**/EO as a C*-algebra. (iii)
The predual of (M/])**/Eo
-+ (M/J)**/EO
imbeds
is F.
Proof. (i) It is a standard fact that (M/J)** is a IV*-algebra. Since E is invariant, E” is an ideal (see [12]), and E” is weakly closed, so (M/J)**/EO is a W*-algebra. Next, observe that every f E (M/J)* has a unique o-continuous extension to (M/J)**, and in particular the trace x” --L p(Z#) has an extension, which by continuity is positive and central, and it annihilates E” by definition. Thus it is well-defined on (M/J)**/EO, and we denote it by Tr. Clearly Tr is positive and central. Let B E (M/J)**/EO be positive with Tr(B) = 0, and let A E (M/J)** be a positive representative. Then for x E M we have / Tr(xA)12 < Tr(x*x) Tr(A*A)
= 0.
Consequently Tr(SB) = 0 for all x E (M/J). Since the unique extension of y” -+ p 0 G,,(y) = Tr(%y) is A -+ Tr(xA) we see that A is annihilated by E and therefore A E E” and B = 0. Therefore Tr is faithful. The normality of Tr is automatic. (ii): From Proposition 1.4(ii) it follows that E” n (M/J) = (01, so the composition is injective, and it is well-known that each of the maps is *-homomorphic. (iii): This is a standard fact. The proposition is proved. To finish the proof of Theorem 1.2 we shall prove that (M/J) is actually equal to (M/J)**/EO via the embedding of the preceding proposition. For this we must establish the following property. (*I
f E F there andf (4 = llf II. For every
is an f E M/J with (1R (1< 1
Notice that (*) has been proved if f E E (Proposition 1.4(iv)), and we obtain the general result as a consequence of a more detailed study of F, the closure of E. 58019/3-6
330
VESTERSTR0M
PROPOSITION 1.6. Let f EF. Then there exist a sequence (@J in 9 and a family Y(t),,, such that
(9 P 0 &n-+f; (ii) (iii)
Qp,(t) + Y(t) a.e. [Jo]; x E M, then t--f Y(t)(x(t))
If
fW (iv)
is p-integrable
= 1, w>w>>
t -+ I/ Y(t)ll is p-integrable
and
44);
and
llfll = j, II WY
440
Proof. The proof is analogous to the proof of the Riesz-Fisher theorem. Since F is the closure of E, there is a sequence Qn in A? with TV0 G’, --+f. We may assume that -pdQ
f&.G By Proposition
< CO.
1.4(iii) we can write 0) - @&)ll G(t) g1 i’, II @?a+1
and therefore
there is a p-null
set NC
< 009
S such that
so (@Jt)) is a Cauchy sequence for t 4 N. Let, for t E S \ N, Y(t) = lim Q,(t), and 0 elsewhere. This proves (i) and (ii). For x E M we have Y(t)(x(t)) = 1im @,(t)(x(t)) for t $ N; so Y(t)(x(t)) is p-measurable and
G
2 j I @,+lww) p=n s
G f
e=n
j
s
-
@&>W)l
44)
II @,+1(t) - @p@)llII x II 44t) - 0
as
n+co.
QUOTIENTS
Consequently, t -+ Y(t)(x(t)) f(f)
= lim p 0 g(x)
OF FINITE
331
W*-ALGEBRAS
is p-integrable and
= lim j” Qn(t)(x(t))
c+(t)
=
f Y((t)(x(t))
dp(t)
J
which proves (iii). As above we see t + ]I Y(t)/1 and t + Ij Y(t) - @Jt)jl are measurable functions, and
as
jsII y(t)ll G(t) = liq j, II @P,(t)ll444
n- --t *;
= lim II p 0K I/ = Ilf II
by using Proposition 1.4(iii). The proposition is proved. The following lemma pursues the analogy with standard measure theory. Now we establish a kind of Lusin theorem for elements of F by applying an analog of Egoroff’s theorem. LEMMA 1.2. Let E > 0 and z”EZ/Iwithp(l-,zZ)<~anda@~9suchthat f(%)
f
E F. Then there exist a projection
= p 0 6(Z)
for
x E M.
Proof. Let @, , Y(t), and N be chosen as in Proposition 1.6, and define Ek,m = {t E S\ N j 11G,(t) - Y(t)11 < l/m for all Y > k. By construction (& m)k is increasing for fixed m and u,“=i Ek,m = S \ N. Choose for each’integer m an integer R, such that 4s \ &,.vJ Let A = nzE1 E,,,,,
-=cQm.
. Then,
If t E A and m is an integer, I( Q?(t) - Y(t)11 < l/m holds for r 3 K, , and therefore d>,(t) -+ Y(t) uniformly in t E A. By regularity of p
332
VESTERSTRPlM
we can find a compact set K C A with p(A \ K) < E, and, since ,A is normal, we have p(K) = p(R), where R is the interior of K. p(S \ K) < 2~. K is open and closed, since S is Stonean, so 1, is of the form ,?Ywhere 2: is a projection in 2 (see [l]). We now have
so the sequence (&J is Cauchy. limn+aoXj, tion 1.3(iv), and for x E M it follows that f(ET) = liip
= (z, exists by Proposi-
06’,(L%) = lillp(16n(32)) = p 0 6(a).
This proves the lemma. LEMMA 1.3. Let f EF and supposethat there are projections xi in Z/I and pi in 2, i = 1, 2, with f(zq
=
p 0 d$(n)
fOY
x E M,
i=
1,2.
Then, there is a @ E 2 such that
f(E) = p 06(a), where x = z1 v z2 . Proof.
Define @ = xr@r + (zZ - xlze) @)2E 64. Then P o @(a) = P@dW
+ P@d(% - %w))
= f (iq + f (a& - ,c$iq) = f (qa, + f, - @iJ) = f (23).
Q.E.D.
THEOREM 1.3. Let f EF. Then, there exists a sequence(Gm) in 9 together with a sequence(zJ of pairwise orthogonal projections in Z such that
f(2) = g p 0&Qi%,)
for x EM.
n=l
Proof. By Lemma 1.2, we can find for each integer n a projection - ’ in Z/I and elements Qn in 9 such that p( 1 - s:,‘) < l/n and &ZJ = /A 0 &(Z). By L emma 1.3, we may assume that (&‘) forms an increasing sequence. Let now e, = f, and e, = 2%’ - &, for n >, 2. Then (e,) is a family of pairwise orthogonal projections in Z/Z and C p(e,) = p(I); so ~~=r e, = I. Using [15], Lemma 3.4
QUOTIENTS OF FINITE W*-ALGEI~MS we find a sequence of pairwise f, = e, . We then have f@qJ
orthogonal
= f(E&‘qJ
Combining
this with
PROPOSITION andf(q Proof.
1.7.
= I’ w>w)
(t)
(.zn) in 2 with
= /.Lc’ QaQ.
On the other hand, letf(x) = Js Y(t)(x(t)) of Proposition 1.6. Then we have f(x)
projections
333
&(t)
(I9 be the decomposition
44)
we obtain the desired result.
Let f E F. Then there is a ZI E M/J
with Ij 5 // < 1
= IlflL Let
f
be written
as in the above theorem.
Then
llfll ~~IIP%AII.
we have (-l-t)
By Proposition 1.4(iv) we find Us E A!l such that /) ZI, 11< 1 and p 0 dj)n(5:n5in) = Ij p o Z,$,, I). Since the projections z, are pairwise orthogonal ZI = C Z,ZI, is strongly convergent and I( w II = sup 11ZI, (1 < 1. Moreover x = ~7~ = v”,P, , and
Since 11u 11< 1, the proposition
follows
from this and (i-t).
334
VESTERSTR0M
End of proof of Theorem 2. Sakai has proved the following theorem in [12]. Let B be a W*-algebra with predual B, and A C B a C*-subalgebra. If B admits a faithful positive functional and if for every f E B, , there is anx E A with11 x 11< 1 andf (x) = /jf 11,then A = B. Propositions 1.4 and 1.6 show that (M/J) = (M/J)**/EO via the injection of Proposition 1.4. In particular (M/J) is a W*-algebra. Remark 1.2. It would improve Theorem 1.2 if we could drop the countability conditions. The assumption that M is u-finite is inessential and is merely used for convenience, namely in Proposition 2(iv), but the proof can be effectuated without this assumption, simply by performing the polar decomposition for each x,@, where (2,) is a family of pairwise orthogonal, u-finite projections (cf. [12]). However, the condition that Z/I be o-finite is essential for our proof, but can probably be avoided. Indeed, a possible way is the following. Decompose (Z/1) in to a direct sum @ e,(Z/I) of o-finite W*-algebras. Than (M/J) ear is a W*-algebra, but we do not necessarily have Of/J) N 0 @f/J) eE. This would hold if we could lift the family (e,) to a family of .pairwise orthogonal projections in 2 (cf. [15, Lemma 3.41). This, however, we do not know, and it is false if we do not assume that Z/I is a W*-algebra (see [S]).
2. REMARKS The condition (ii) of Theorem 1.2 makes it pertinent to study the abelian case. Let 2 be an infinite dimensional abelian W*-algebra acting on a separable space. One can show the existence of non o-closed ideals for which Z/I is a W*-algebra and which are not finite intersections of maximal ideals, using results from [6] and [lo]. Also we can exhibit ideals I for which Z/I is not a W*-algebra (cf. [9]). Using results from the abelian case and analyzing the continuity properties of the functions t + I] x(t)11 we study the nonabelian case. Let (i), (ii), and (iii) be the three conditions of Theorem 1.2. It can be proved that if M is a finite W*-algebra satisfying some fairly mild conditions there are uniformly closed ideals J, , Jb , and Jc such that (a) J, satisfies (ii) and (iii), but not (i). (b) Jb satisfies (i), but not (ii) and (iii). (c) Jc satisfies (i), (ii), and (iii) and is neither a-closed nor a finite intersection of maximal ideals. This shows that the conditions of Theorem 1.2 are independent.
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0~
FINITE
335
W*-ALGEBRAS
ACKNOWLEDGMENT I want to express my gratitude to C. Akeman, conversations and advice, and especially to M. to these problems.
S. Sakai Takesaki,
and E. Stormer who directed
for valuable my attention
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