NON-COMMUTATIVE EXTENSION OF LUSIN'S THEOREM. (Received May 22, 1967)

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1 Tohoku Math, Journ, Vol. 19, No. 3, NON-COMMUTATIVE EXTENSION OF LUSIN'S THEOREM KAZUYUKI SAITO (Received May 22, 1967) 1. Introduction. Many important theorems in measure theory have been extended to operator algebras by many authors, especially, Dixmier [1], Dye Segal. Considered as non-commutative extensions, those are interesting themselves provide powerful tools in the further investigations of operator algebras. The purpose of this paper is to extend Lusin's theorem which is an important tool in measure theory into general operator algebra. Before going into discussions, the author wishes to express his hearty thanks to Prof. M. Fukamiya Dr. M. Takesakifor their many valuable suggestions in the presentation of this paper. 2. Notations Definitions. Let M be a W*-algebra, namely, C*-algebra with a dual structure as a Banach space,m* be the predual of M, that is, the Banach space of all bounded normal functionals on M M*+, the positive part of M*, that is, the set of all functionals ƒó in M* such that ƒó(x*x) 0 for all x ƒó M. We may consider the s*-topology, that is, the topology defined by a family of semi-norms {ƒ ƒó,ƒó*p; ƒó M*, where ap(x) = q}(x x)12, a* (x) =ƒó(xx*)1/2 for all x M}, the s-topology is that defined by a family of semi-norms {ƒ ƒó;ƒó M*+}.In [4, p. 1.64] Sakai shows that whenever M is represented as a weakly closed algebra of operators on some Hilbert space, the weak*-topology of M coincides with the weakoperator topology on the bounded sets of M. It follows from this that the topology coincides with the strong *-operator topology on bounded sets of M, the s-topology coincides with the strong operator topology on bounded sets of M. 3. Main theorems. The following theorem corresponds to the Egoroff theorem in the Lebesgue integration. THEOREM 1. (Density theorem) Let M be a W *-algebra M* the predual of M, moreover, let 'p be any positive functional in M*. Let N be any set in S (the unit sphere of M)), which is adherent to an element a in S in the s*-topology. Then for any positive number ƒã, a projection e in M, there exist a projection e0 in M a sequence {ai} i=1 ¼N such that e e0, ƒó(e-e0) <ƒã lira aaie0-ae0 a=0. In particular, for any sequence

2 NON-COMMUTATIVE EXTENSION OF LUSIN'S THEOREM 333 {an } n=1 in S, which converges to a in s*-topology, there exist a projection e0 M a countable subsequence {ani} n=1 of {an } n=1 such that ƒó(e-e0) <ƒã, e e0, aanie0-ae0 a 0 (as i ). As a non-commutative extension of Lusin's theorem in the usual Lebesgue integration, we have THEOREM 2. Let N be a C*-algebra with the identity 1 acting on some Hilbert space, M the weak closure o f N a be any element in M e any projection in M. Then, for any pxsitive functionalƒó in M* any positive numbers ƒã ƒâ(<1), there exist a projection f in M, f c e, b in N such that ƒó(e-f )< ƒã, af =bf ab a (1+ƒÂ) aaf a. [5]. Moreover, if a is an hermitian element, then b may be chosen hermitian element such that ab a 2(1 + ƒâ) aaf a ab a aa a + ƒâ, moreover, i f a is a unitary operator, then b may be chosen unitary such that ab -1 a < aa -1 a+ƒâ. 4. Proof of Theorem 1. It is sufficient to prove only the case a = 0 e=1. By the assumptiont, here is a net {aƒæ}ƒæ ƒæ in N which is convergent to 0 for s*-topology, bƒæ= a*ƒæaƒæ converges to0 for s-topology a b a 1. Then we can choose a family of projections {eƒæ}ƒæ ƒæin M such that lim eƒæ =1 for s-topology abƒæeƒæ a 1 for eachƒæ. In fact, let x be the characteristic function of the interval (-1,1), we defineeƒæ = x(bƒæ) for each ƒæ. Then we have b2o 1- eƒæ, we see that the left member of the inequality converges to 0, for s-topology, so eƒæ converges to 1, for s-topology. It is immediate that bƒæeƒæ a 1 for eachƒæ. Then, for a given ƒã > 0, there exists an index ƒæ1 such a that ƒó(1-eƒæ1) <ƒã/2, a bƒæleƒæ1 a 1, so that aaƒæleƒæ1 a = abƒæ1eƒæ1 a1/2 1. Consider the family {aƒæeƒæ1;ƒæ ƒæ1}, then aƒæeƒæl converges to 0 for s*-topology Again denoting as b'ƒæ=eƒæ1bƒæeƒæ1( eƒæ1meƒæ l) by the same way (but for 2-2), we can choose a projection eƒæ2 in eƒæ1meƒæl suchthat ƒó(eƒæ so that aaƒæ2eƒæ2 a = aeƒæ2bƒæ2eƒæ2 a, 1/2 <1/2. l -eƒæ2) <ƒã/22, ab'ƒæ2eƒæ2 a 2-2,. By the mathematical induction, we can choose a decreasing sequence of projections [eƒæi} (ƒæi ª)in M such that (eƒæi-1 - eƒæi) <ƒã/2 ƒói (eƒæ=1) for each i,

3 334 K. SAIT0 aaƒæieƒæi a 1/i. completes the proof. For the proof of theorem 2, we need some lemmas. LEMMA 1. Let N be a C*-algebra with the identity acting on some Hilbert space M the weak closure of N (Notethat M is a W*-algebra.). Let ƒóp be any functional in M*+. Then for any element a M, any projection e in M any positive numbers ƒã, ƒâ, we can choose a1 N a projection f M such that f e, a(a - a1) f a<ƒâ, aa1 a aae a ƒó (e- f) <ƒã PROOF. Since the unit sphere of N is adherent to the element ae in the s*-topology by [2, theorem 1] (We may assume aae a=1 without loss of generality.), it is clear from theorem 1. generality. 5. Proof of Theorem 2. Case (i). General case. We may consider the a ae a =1 without loss of We can take a positive functional ƒóo in M*+ such that ƒó0((ae)* (ae))1/2 1-ƒÂ ƒó0(1) =1. Put ƒó =ƒó +ƒó0. Then, by lemma 1, we can choose a1 in N a projection e1 in M such that a(a-a1)e1 a<ƒâ/2, a a1 a I a ae a =1 ƒó (e-e1) < min(ƒã/2, ƒâ2/2), e1 e. For (a-a1)e1 in M, again by the same lemma, there exist a projection e2 in M e2 e1 N such that (e1-e2) <min(ƒã/2,ƒâ2/22), ƒó (a - a{ a1)e1- a2} e2 a= a(a - a1-a2)e2 a <ƒâ/22 a a2 a a(a - a1)e1 a. By the mathematical induction, we can choose a decreasing sequence of projections {e1} i=1 in M {a1} i = 1 in N such that

4 NON-COMMUTATIVE EXTENSION OF LUSIN'S THEOREM 335 ƒó(ek-l-ek)<min(ƒã/2k,ƒâ2/2k) (where e0=e). Since we have If we define b as then b N ab a 1+ƒÂ. Putting is a, projection in M for all k. Hence a(a-b)f a = 0, that is, af = bf. As ƒó(e-f)< min (ƒã,ƒâ2) <ƒã, we have ƒó(e-f) <ƒã aaf a (1-3ƒÂ) ab a; in fact, we have aaf a ƒó0((af)* (af))1/2 ƒó0((ae)*(ae))1/2 - ƒó0( o ae(e - f)}* oae(e -f)})1/2. Since ƒó0(e-f) ƒâ2 ƒó0( oae(e-f)) * [ae(e-f )} )1/2 ae aƒâ=ƒâ, we have a of a 1-2ƒÂ. Noting that aae a=1 1 + ƒâ ab a, we obtain aaf a 1-2ƒÂ (1-3ƒÂ) ab a. Case (ii): a is an hermitian element of M. Firstly we can choose an hermitian element a1 a projection e1 in M such that e1 e, a(a-a1)e1 a <ƒâ/22, aa1 a aa a, aa1 a 2 aae a, ƒó(e-e1) <min(ƒã/2,ƒâ2/2). Case (ii,a) 2 aae a aa a:as {x;x N, ax a a a a,x is hermitian} is adherent to the element a for s-topology, there exists a net (as) such that ae converges to a for s-topology. Hence aƒæe converges to ae for s*-topology. By theorem 1, there exist a projection e1(e1 e), an hermitian element a1 in N such that a(a-a1)e1 a<ƒâ/22, aa1 a aa a 2 aae a, ƒó(e-e1) < min (ƒã/2,ƒâ2/2). Case (ii, b) aa a 2 aae a:as{x ; x N ax a ac0 a, x is hermitian} is adherent to the element ca for s-topology (where co=eae+(1-e)ae+ea(1-e) note that c0e = ae), by [2] our lemma 1, there are an hermitian element a1 N a projection e1(e1 e) in M such that a(a- a1)e1 a ƒâ/22, aa1 a ac0 a, ac0 a 2 aae a, ƒó(e-e1) < min(ƒã/2, ƒâ2/2). Hence we can choose an hermitian element a1 (in N) a projection e1 in M such that a(a-a1)e a<ƒâ/22, aai a aa a a1 a 2 aae a. a Putting c1= e1(a-a1)e1+(1-e1)(a-a1)e1 + e1(a- a1)(1-e1),c1 becomes an hermitian element of M such that (a-a1)e1= c1e1 ac a 2 a(a-a1)e1 a. For c1, by the same reason, there exist an hermitian element a2 in N, a projection e2 in M, e2 e1 such that aa2 a a c1 a, a(c1-- a2)e2 a<ƒâ/23 ƒó(e1 -e2) < min(ƒã/22, ƒâ2/22). Since (c1- a2)e2 =(c1-a2)e1e2=c1e1e2-a2e2 = (a-a1-a2)e2 we have that a(a - a1- a2)e2 a ƒâ/2, hence by the same way as in case (i), we can choose a decreasing sequence of projections in M, a sequence of hermitian elements in N a sequence of hermitian elements in M such that,

5 336 K. SAITO a(ci-1-ai)ei a<ƒð/2i+1 where ƒó(ei-1-ei) < min(ƒã/2i,ƒâ2/21), (e0=e). Now we define b (in N), then b E N, Further, putting f=inf ei, f is a projection inm as aj)e1, we have for all i. It follows a (a - b)f a = 0, that is, af = b f. By the same way as in case (i), we have aaf a aae a (1-2ƒÂ). Noting that ab a <2 aae a+ƒâ, we have of a ae a(1-2ƒâ) 2 ab a(1-3ƒâ). Case (iii):a = u u is a unitary element in M. We need the following lemma, which we prove by making use of an argument of Riesz-Nagy [3; p. 266 Theorem]. LEMMA 2. Let M be a W *-algebra, e be a projection in M, moreover, w be a unitary in M such that a(1- w)e a <1/8, then we can choose a unitary v in M as follows: ve = we, a 1-v a 7 a(1-w)e a. PROOF. Putting wew* = f, we have ae- f a= ae - wew * a= ae - we + we - wew* a = a(1- w)e + w{(1 - w)e}* a 2 a(1- w)e a <1/4. Next, putting a=1 + (1- e)(e -f)(1 - e) u = (1- f )a-1/2(1- e), we have u* u =1- e, uu*=1 - f. In fact, it follows from our hypothesis that a(1- e)(e - f)(1- e) a <1,

6 NON-COMMUTATIVE EXTENSION OF LUSIN'S THEOREM 337 that consequently the hermitian element a=1+(1-e)(e-f)(1-e) is strictly positive. Hence a-1 a-1/2=(a-1)1/2 exist. Consider the elements u=(1-f) a-1/2(1-e). u*=(1-e)a-1/2(1-f). Since we obviously have that 1-e commutes with a, we also have(1-e)a-1/2 = a-1/2(1-e), since furthermore (1-e)(1-f)2 (1-e)=(1-e)(1-f)(1-e)=(1--e)+(1-e)(e-f)(1--e)=(1-e)a, it follows that u*u = (1-e)a-1/2(1-f)2a-1/2(1-e) = a-1/2(1- e)aa-1/2 =(1-e)a-1/2aa-1/2=1-e. Moreover uu* is a projection majorized by 1-f. If ƒì Û(uu*)(we assume that N acts on some Hilbert space ),ƒì (1-f), that is (ƒì,un) = 0 for all. Then u*ƒì = 0; hence (1-e) (1-f)ƒÌ= a1/2a-1/2(1-e)(1-f) = a1/2(1-e)a-1/2(1-f) = a1/2u*ƒì=0, consequently (e-f)(1-f)ƒì=(1 -f)ƒì. In view of the hypothesis e-f a<1, this equation is possible only if (1-f)ƒÌ= 0, that is uu* a =1-f Then we have (1-u)(1-e) a= a(1-e) - (1-f)a)-1/2(1-e) = a(1-e) - (e-f)a-1/2(1-e) - a-1/2(1-e) a = (1-a-1/2)(1-e) - (e-f)a-1/2(1 - e) a (1-a-1/2) a+ aa-1/2 a ae-f a. a Noting that if x <1/4 (where x is a real number), then (x+1)-1/2-1 x (x+1)1/2<2, we have a(1-u)(1-e) a ae-f a+2 ae-f a 6 a(1 w)e a. - Putting v = we + u(1-e), v is a unitary in M such that ve = we. Combining the above estimations, we have inequalities 1-v a = a1-we-u(1-e) a= ae+1-e-we-u(1-e) a a a(1-w)e+(1-u)(1-e) a 7 a(1-w)e a. Thus the lemma follows. LEMMA 3. Let N be a C*-algebra with the identity acting on some Hilbert space, M be the weak closure of N (Observe that M becomes a W* - algebra.). Let ƒó be any positive functional in M+*. Then for any unitary u in M, any projection e in M, any positive number ƒã,ƒâ, we can choose unitary v in N a projection f in M, f e such that (u-v)f a<ƒâ, a1-v a a1-u a ƒó(e-f)<ƒã. a PROOF.By [2], [v;v is unitary in N; a1-v a a1-u a}is s*-dense in

7 338 K. SAITO {w; w is unitary in M, a1- w a11- u a}. Hence, it is clear from lemma 1. LEMMA 4. Let {ui} i=1 be a family of unitary operators on some Hilbert space such that for uniform operator topology the limit is also a unitary operator. PROOF. Since, for each pair of positive integers n we have By the hypothesis, the left side of the above inequality converges to o as exists for uniform operator topology the limit is also a unitary. The lemma follows. Proof of case (iii). We may assume e =1 withoutloss of generality. For any unitary element u in M, any positive functional cp in M*, any given ƒã> 0 1 >ƒâ> 0, we can choose families of unitary elements {uj} i=1 in N, {uj} i=1 in M, a decreasing sequenceof projections {fj} i=1 in M such that ƒó(fi-1-fi) <ƒã/2i(f0=1) for each i, a 1-ui a1-ui-1 a 7 a(1-u*i-lui-2)fi-1 a(i 2), (vo=u). In fact, by lemma 3, there exists a unitary element u1 in N a projection f 1 in M such that a1-u1 a a1-u a,ƒó(1-f1) <ƒã/2, au-u1)f1 a <ƒâ/8. For ul-1u(unitary in M), by lemma 2, there exists a unitary v1 in M such that v 1 f 1= u1-1uf1 a1-v1 a 7 a (1- u1-1u) f1 a <(7/8)ƒÂ <8. Then for unitary vein M, a projection fi in M, by lemma 3, there exist a unitary u2 in N a projection f 2 in M, f2 f1 suchthat a1-u2 a a1-u1 a,ƒó(fi -f2) <ƒã/22, a(v1 -u2)f2 a<ƒâ/8.2. For u2-1 v1, (unitary in M), by lemma 2, we can choose a unitary v2 in M such that v2f2, = u2-1v1f2 a1-v2 a 7 a(1- u2-1v1)f2 <ƒâ/2. Thus, by mathematical induction, we can choose families of unitary elements {ui} i=1 ¼N,{ui} i=1 M a decreasing sequence of projections {fi} i=1 in M as follows:

8 a1-ui a 7 a NON-COMMUTATIVE EXTENSION OF LUSIN'S THEOREM 339 ƒó(fi-1-fi) <ƒã/2i (f0=1) for each i, a 1-ui a a1-vi-1 a 7 a1-u*i-1vi-2)f2-l a (i 2). Putting f = inf fi, we have that Moreover, as 1-u*i-1vi-2)f2-l a<(7/8)(ƒâ/2i-2) <ƒâ/2i-2, it follows from Lemma 4 that exists for uniform operator topology (we denote it by a, u is a unitary in N). Moreover, we have hence, auf - uf a=0, that is, uf = uf, for all n. Therefore, a1- u a< a1- u a+ 2ƒÂ. This completes the proof of theorem 2.

9 340 K. SAITO REFERENCES [1] J. DIXMIER: Formes lineaires sur un anneau d'operateurs, Bull. Soc. Math. France, 81 (1953), [2 ] I. KAPLANSKY : A theorem on rings of operators, Pacific J. Math., 1(1951), [3] RIESz-NAGY : Lecons d'analyse Fonctionnelles, 3e ed., Gauthier-Villars Paris, et Akad, Sc. Hongre, Budapest, (1955). [4] S. SAKAI : The theory of W*-algebras, Lecture note, Yale Univ., (1962). [5] M. TOMITA: Spectral theory of operator algebras 1, Math. J. Okayama Univ., 9(1959), MATHEMATICAL INSTITUTE, TOHOKU UNIVERSITY, SENDAI, JAPAN.