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The Sequential Closedness of σ-Complete Boolean Algebras of Projections
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
208, 364]371 Ž1997.
AY975300
The Sequential Closedness of s-Complete Boolean Algebras of Projections W. J. Ricker School of Mathematics, Uni¨ ersity of New South Wales, Sydney, NSW, 2052, Australia Submitted by John Hor¨ ath ´ Received March 4, 1996
In the Banach space setting s-complete Boolean algebras of projections Žbriefly, B.a.. or, equivalently, their representation as spectral measures, were intensively studied by W. Bade w1, 2x. Such objects are a natural extension of the fundamental notion of the resolution of the identity of a normal operator in Hilbert space. Since the concepts involved are of a topological and order theoretic nature there is no difficulty in extending the study of such objects to the setting of locally convex Hausdorff spaces Žbriefly, lcHs. w10, 12x. If the underlying lcHs X is a separable Frechet ´ space, then it is known that such a B.a. M is necessarily a closed subset of LŽ X . with respect to the strong operator topology ts ; see w8, Proposition 3; 9, Theorem 5Ži.x. Here LŽ X . denotes the space of all continuous linear operators of X into itself. There are many sufficient conditions known on both X and M which guarantee that if M is s-complete, then it is necessarily a ts-closed subset of LŽ X .; see w7]9x, for example. Despite the quite general criteria referred to above it is easy to exhibit examples where M fails to be ts-closed in LŽ X .. For instance, let X denote the non-separable Hilbert space l 2 Žw0, 1x.. Let S denote the family of Borel subsets of w0, 1x. Define, for each E g S, the selfadjoint projection P Ž E . g LŽ X . by P Ž E . x s x E x for each x g X. It is straightforward to verify that M s P Ž E .; E g S4 is a s-complete B.a. To see that M is not ts-closed, let F : w0, 1x be any non-Borel set and let F Ž F . denote the family of all finite subsets of F directed by inclusion. Then P Ž E .; E g F Ž F .4 is a net in M which is ts-convergent to the selfadjoint projection Q g LŽ X . of multiplication by x F . However, Q f M . Despite the fact that M is not ts-closed it is routine to verify that M is sequentially ts-closed in LŽ X .. That is, if Pn4`ns1 : M is any sequence which is ts-convergent in LŽ X ., say to the element T g LŽ X ., then actually T g M . 364 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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Dr. Susumu Okada posed the question of whether it is always the case that a Bade s-complete B.a. M : LŽ X . is sequentially ts-closed. The aim of this note is to show, under quite mild completeness restrictions on LŽ X ., that the answer is yes in the case that the s-complete B.a. M is purely atomic. An example is given showing that the answer is no in general.
1. PRELIMINARIES Let X be a lcHs and M : LŽ X . be a B.a. always assumed to have as unit the identity operator I on X. Then M is said to be s-complete Žin the sense of Bade. if it is s-complete as an abstract B.a. and if, for every countable family Bn4 : M we have ŽHn Bn .Ž X . s Fn BnŽ X . and ŽEn Bn .Ž X . s sp Dn BnŽ X .4 , the closed subspace of X generated by Dn BnŽ X .; see w1x, for example. Let L s Ž X . and L w Ž X . denote LŽ X . equipped with the strong and weak operator topology, respectively. Given a set N : LŽ X . we denote the closure of N in L s Ž X ., respectively L w Ž X ., by N s, respectively, N w . A B.a. M : LŽ X . is said to have the s-monotone property if lim n Bn exists in L w Ž X . and is an element of M whenever Bn4 : M is a monotonic sequence. Of course, the partial order in M is range inclusion, i.e., P F Q iff P Ž X . : QŽ X .. LEMMA 1.1.
Let X be a lcHs and M : LŽ X . be a B.a.
Ži. If M has the s-monotone property, then M is s-complete. Žii. If M is equicontinuous Ž in LŽ X .., then M has the s-monotone property iff M is s-complete. For Ži. we refer to w7, Lemma 3.3x and Žii. follows from Proposition 4.1 and Corollary 4.6.1 of w7x. Let L be a topological Hausdorff space and Z : L. Then w Z x denotes the set of all elements in L each of which is the limit of some sequence of points from Z. A set Z : L is called sequentially closed if Z s w Z x. The sequential closure of a set Z : L is the smallest sequentially closed subset of L which contains Z. It is always equipped with the relative topology from L. If M : LŽ X . is a B.a., then w M x L s Ž X . denotes the sequential closure Žin L s Ž X .. of the linear span of M . It is clear that w M x L s Ž X . is a linear subspace of LŽ X .. The closure, in L s Ž X ., of the linear span of M is denoted by ² M :s . A compact, convex subset K of the locally convex algebra L w Ž X . is called a spectral carrier w4, Definition 2.1x, if K is commutative, closed under composition in LŽ X ., and ŽT q S y TS . g K whenever T, S g K.
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LEMMA 1.2. Let X be a lcHs and M : LŽ X . be a B.a. with the s-monotone property. If either L s Ž X . or w M x L s Ž X . or ² M :s is quasicomplete, then the closed con¨ ex hull coŽ M . is a spectral carrier in L w Ž X .. Proof. The s-monotone property ensures that there exists a spectral measure P: S ª L s Ž X ., defined on a s-algebra of sets S of some set V, such that its range P Ž S . s M w7, Proposition 4.6x. To say P is a spectral measure means that P is countably additive, P Ž V . s I, and P is multiplicative Ži.e., P Ž E l F . s P Ž E . P Ž F . for all E, F g S .. Since L w Ž X . is the lcHs L s Ž X . equipped with its weak topology and P is s-additive whether considered as being L s Ž X .-valued or w M x L s Ž X .-valued, or ² M :svalued, it follows from the quasicompleteness assumption that K s coŽ M . s coŽ P Ž S .. is a compact Žand convex. subset of L w Ž X . w11x. The other properties required to establish that K is a spectral carrier are easy consequences of the B.a. properties of M ; see the proof of w7, Proposition 3.15x. 2. ATOMIC BOOLEAN ALGEBRAS OF PROJECTIONS Let X be a lcHs and M : LŽ X . be a B.a. A non-zero projection B g M is called an atom if, whenever D g M satisfies D F B, then either D s 0 or D s B. Then M is called Žpurely. atomic if there exists a family of atoms Ba 4a g A in M such that, for every B g M there exists a subset B˜ : A such that Ý a g B˜ Ba s B. The summability of the family Ba 4a g B˜ is meant as the ts-limit in LŽ X . of the net of partial sums over all finite subsets of B˜ Ždirected by inclusion.. The family of atoms Ba 4a g A , necessarily pairwise disjoint and satisfying Ý a g A Ba s I, is said to generate M . The collection of all projections in LŽ X . is denoted by PŽ X .. PROPOSITION 2.1. Let X be a lcHs and M : LŽ X . be an atomic B.a., generated by the atoms Ba 4a g A , and ha¨ ing the s-monotone property. Suppose that at least one of the spaces L s Ž X . or w M x L s Ž X . or ² M :s is quasicomplete. Then, for e¨ ery subset F : A the series Ý a g F Ba con¨ erges in L s Ž X . to an element of PŽ X .. Proof. Fix an arbitrary subset F : A. Let F Ž F . denote the collection of all finite subsets of F, directed by inclusion. The pairwise disjointness of Ba 4a g A implies that Ý a g E Ba s Ea g E Ba belongs to M for each E g F Ž F .. Accordingly, Ý a g E Ba ; E g F Ž F .4 is an upwards directed system of elements from M : K s coŽ M .. Since K is a spectral carrier Žcf. Lemma 1.2. it follows that the limit lim E g F Ž F . Ý a g E Ba exists in L w Ž X . and is an element BF g PŽ X . w4, Theorems 2.10 and 2.11x. Then w7, Proposition 4.21x implies that the limit BF s lim E g F Ž F . Ý a g E Ba also exists in L s Ž X ..
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Remark. It is straightforward to produce examples where w M x L s Ž X . or ² M :s is quasicomplete, but L s Ž X . itself is not quasicomplete. Indeed, let X s l 1 equipped with its weak-) topology s Ž l 1 , c 0 .. Then X is quasicomplete but L s Ž X . is not even sequentially complete. Let V s N and S s 2 N . Then P: S ª L s Ž X . defined by P Ž E . w s x E w Žcoordinate-wise multiplication., for all w g X and E g S, is a spectral measure and so M s P Ž S . is a B.a. with the s-monotone property. It turns out that w M x L Ž X . is quasicomplete. For details of this example we refer to w7, s Example 2.3x. We now have the main result of the section. THEOREM. Let X be a lcHs and M : LŽ X . be an atomic B.a. ha¨ ing the s-monotone property. Suppose that at least one of the spaces L s Ž X . or w M x L Ž X . or ² M :s is quasicomplete. Then M is a sequentially closed subset of s L s Ž X .. If X is any quasicomplete, barrelled lcHs then it is a simple application of the Banach]Steinhaus theorem to deduce that L s Ž X . is necessarily quasicomplete. In particular, this includes all Frechet spaces and hence, all ´ Banach spaces. Since every s-complete B.a. acting on a Frechet space is ´ necessarily equicontinuous w12, Proposition 1.2x, the above remarks together with the Theorem and Lemma 1.1 give the following COROLLARY. Let X be a Frechet lcHs. Then e¨ ery s-complete, atomic ´ B.a. M : LŽ X . is a sequentially closed subset of L s Ž X .. The example in the introduction shows that this Corollary cannot be improved by replacing sequentially closed with closed. Proof of Theorem. Let Ba 4a g A be a generating family of atoms for M . By Proposition 2.1, the collection of sets S : 2 A given by S s F : A; Ý a g F Ba g M 4 is well defined. The claim is that S is a s-algebra of subsets of A. Clearly f , A g S. If F g S and F c denotes the complement of F in A, then Ý a g F c Ba q Ý a g F Ba s I shows that Ý a g F c Ba s I y B with B s Ý a g F Ba g M . Since M is a B.a. we have Ý a g F c Ba g M , i.e., F c g S. Suppose that F1 , F2 g S are disjoint. It is a routine calculation from the definition of the convergence of the series Ý a g F1 j F 2 Ba Žwhich converges to some projection in L s Ž X . by Proposition 2.1. that Ý a g F1 j F 2 Ba s Ý a g F1 Ba q Ý a g F 2 Ba . Since both B1 s Ý a g F1 Ba and B2 s Ý a g F 2 Ba belong to M and are disjoint it follows that B1 q B2 s B1 k B2 g M , i.e., F1 j F2 g S.
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Let E1 , E2 g S. If F Ž Ej . denotes the family of finite subsets of Ej , directed by inclusion, for j g 1, 24 , then
ŽÝa g E
1
Ba . ? Ž Ý b g E 2 Bb . s Ž Ý a g E1 Ba . lim F g F Ž E 2 . Ý b g F Bb .
Using continuity of Ý a g E1 Ba and the fact that each sum Ý b g F Bb for F g F Ž E2 . is finite, gives
ŽÝa g E
1
Ba . ? Ž Ý b g E 2 Bb . s lim F g F Ž E 2 . Ý b g F Ž Ý a g E1 Ba . Bb .
Ž 1.
But, for b g F and F g F Ž E2 . fixed, the continuity of Bb and the fact that lim G g F Ž E1 . ŽÝ a g G Ba . s Ý a g E1 Ba imply that
ŽÝa g E
1
Ba . Bb s lim G g F Ž E1 . Ý a g G Ba Bb ,
b g F.
Since Ba Bb s 0 if a / b and Ba Bb s Bb if a s b it follows that Ý b g F Ž Ý a g E1 Ba . Bb s Ý b g F l E1 Bb ,
F g F Ž E2 . .
This shows that the limit in Ž1. is taken with respect to F Ž E2 . l E1 , i.e., with respect to F Ž E1 l E2 .. Accordingly,
ŽÝa g E
1
Ba . ? Ž Ý b g E 2 Bb . s lim F g F Ž E1 l E 2 . Ýg g F Bg s Ýg g E1 l E 2 Bg .
Ž 2. Since the left-hand side of Ž2. is a product of two elements from M Žhence, is in M . it follows that Ýg g E1 l E 2 Bg g M , i.e., E1 l E2 g S. If now F1 , F2 g S are arbitrary, then the identity F1 j F2 s Ž F1c l F2 . j Ž F1 l F2c . j Ž F1 l F2 . , with the three terms in the union pairwise disjoint, implies that F1 j F2 g S. Accordingly, S is an algebra of subsets of A. Finally, let En4`ns1 be a sequence of pairwise disjoint elements from S. Then Fn s Dnks1 Ek , for n s 1, 2, . . . , is an increasing sequence of elements from S. Accordingly, Ý a g F n Ba 4`ns1 is an increasing sequence in M . Since M has the s-monotone property it follows that lim nª` Ý a g F n Ba s B exists in L w Ž X . and belongs to M . By w7, Proposition 4.21x this convergence is also valid in L s Ž X .. It is routine to check that lim nª` Ý a g F n Ba s Ý b g F Bb where F s D`ns1 Fn s D`ns1 En and hence, D`ns 1 En g S. This shows that S is indeed a s-algebra of subsets of A. Suppose now that Pn4`ns1 : M is a sequence such that Pn ª P in L s Ž X .. The aim is to show that P g M . Let EŽ n. g S be sets such that Pn s Ý a g EŽ n. Ba , for each n g N. Define E s D`Ns 1 ŽF`nsN EŽ n... Then E g S and so Q s Ý a g E Ba g M . We show that P s Q.
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Fix x g X. If a g E, then there exists Na such that a g EŽ n. for all n G Na . Accordingly, Ba x occurs in each series Ý b g EŽ n. Bb x for all n G Na . It follows that Pn Ba x s Ba x for all n G Na and hence, Ba x s lim n Pn Ba x. That is, Ba x s PBa x,
a g E.
Ž 3.
Adding up the terms in Ž3. and using continuity of P gives Qx s Ý a g E Ba x s Ý a g E PBa x s P Ž Ý a g E Ba x . s PQx. Since x g X was arbitrary it follows that Q s PQ.
Ž 4.
Suppose now that a f E. Then there is an increasing sequence nŽ1. Ž n 2. - ??? such that a f EŽ nŽ j .. for all j G 1. So, if x g Ba Ž X ., then PnŽ j. x s Ý b g EŽ nŽ j.. Bb x s 0,
j g N.
Hence, the subsequence PnŽ j. x 4`js1 of Pn x 4`ns1 converges to zero from which it follows that Px s 0,
x g Ba Ž X . , a f E.
Ž 5.
But, E c g S and I y Q s Ý a f E Ba g M showing that
Ž I y Q . x s x,
x g Ž I y Q . Ž X . s sp Da f E Ba Ž X . 4 .
Ž 6.
Applying P to Ž6., and using Ž5. and the continuity of P shows that P Ž I y Q . x s Px s 0,
x g Ž I y Q. Ž X . .
So, for x g X arbitrary it follows that P Ž I y Q. x s P Ž I y Q. Ž I y Q. x s 0 since y s Ž I y Q . x g Ž I y Q .Ž X .. This shows that P Ž I y Q . s 0 which, combined with Ž4., implies that P s Q. COUNTEREXAMPLE. Let X` s L`Žw0, 1x. equipped with its weak-) topology s Ž L` , L1 ., in which case X` is a quasicomplete lcHs. For p g w1, `. let X p denote L p Žw0, 1x. equipped with its weak topology s Ž L p , Lq ., where q satisfies py1 q qy1 s 1. Then X 1 is a sequentially complete lcHs and X p is a quasicomplete lcHs for 1 - p - `. If S denotes the family of all Borel subsets of V s w0, 1x, then it is routine to verify that Pp : S ª L s Ž X p . defined by Pp Ž E .: f ¬ x E f, for f g X p and E g S, is a
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spectral measure for all p g w0, `x. Accordingly, Mp s Pp Ž S . is a B.a. with the s-monotone property w7, Lemma 3.1x. The claim is that Mp is not a sequentially closed subset of L s Ž X p .. ny 1 Let I jn s Ž j2yn , Ž j q 1.2yn x for 0 F j - 2 n, and put AŽ n. s D2js1 I2njy1 for n g N. It is shown in w5, Example 4x that x AŽ n. ª 12 | in X` , where | denotes the function on V which is constantly 1. Fix p g w1, `x. Given f g X p and g g X pX s X q , the function w s fg g X 1 s X`X . Accordingly, if l denotes the Lebesgue measure, then lim n ª` ² x AŽ n. , w : s lim nª`
1
H0
x AŽ n. w d l s
11
H0
2
|w d l s ² 12 |, w :
since x AŽ n. ª 12 | in X` . That is, ² Pp Ž AŽ n.. f, g : ª ² 12 |f, g :, as n ª `, for every f g X p and g g X pX , which shows that Pp Ž AŽ n.. ª 12 Pp Ž V . s 12 I in L s Ž X p .. But, 12 I f Mp showing that Mp is not sequentially closed in L s Ž X p .. The B.a.’s Mp , 1 F p F `, are not equicontinuous w6, Proposition 4x. This raises the following QUESTION 1. Is an equicontinuous B.a. M with the s-monotone property necessarily sequentially closed in L s Ž X .? Or, e¨ en more specific, does there exist a s-complete B.a. M acting in a Banach Ž or Hilbert!. space which is not a sequentially closed subset of L s Ž X .? In the class of examples above the sequential closedness of the B.a. Mp fails because the limit operator is not a projection. Of course, if M is equicontinuous, then the limit in L s Ž X . of a sequence of projections from M is always an element of PŽ X .. QUESTION 2. Is a B.a. M with the s-monotone property a sequentially closed subset of PŽ X . l L s Ž X .? We show that the B.a.’s Mp , 1 F p F `, do not answer Question 2. Suppose first that 1 F p - `. Let Z p denote the Banach space L p Žw0, 1x. equipped with its norm topology. Then X p s Ž Z p .s is just Z p equipped with its weak topology s Ž Z p , ZXp .. If T g LŽ X p ., then T is also continuous from Ž X p .m to Ž X p .m where m is the Mackey topology on X p . But, this is precisely the norm topology and so LŽ X p . s LŽ Z p . as vector spaces. Moreover, it is routine to check that L w Ž Z p . s L s Ž X p . as lcH-spaces. Clearly Mp still has the s-monotone property considered as a B.a. in LŽ Z p . and hence, by Lemma 1.1, it is s-complete as a B.a. in L s Ž Z p .. Since the Banach space Z p is separable Mp is actually a complete B.a. in L s Ž Z p . w3, XVII, Lemma 3.21x. Now, suppose that Q n4`ns1 : Mp is a sequence and V g PŽ X p . such that Q n ª V in L s Ž X p .. Then also Q n ª V
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in L w Ž Z p . from which it follows via w1, Theorem 3.2x that Q n ª V in L s Ž Z p .. But, the separability of Z p implies that Mp is actually a closed subset of L s Ž Z p .; see the introduction. Accordingly, V g Mp . Now consider the case p s `. Observe that X` s Z1X with the weak-) topology s Ž Z1X , Z1 .. Suppose that Pn4`ns1 : M` s P`Ž S . is a sequence and W g PŽ X` . such that Pn ª W in L s Ž X` .. For each n g N, let Q n g M1 : LŽ Z1 . be the unique operator such that QXn s Pn . Then Q n4`ns1 is Cauchy in L w ŽŽ Z1 .s . s L s ŽŽ Z1 .s .. So, for each w g Z1 the sequential completeness of Ž Z1 .s guarantees the existence of Vw g Z1 such that Q n w ª Vw in Ž Z1 .s . Since sup5 Q n 5; n g N4 s 1 it follows that the linear map V: Z1 ª Z1 so defined belongs to LŽ Z1 .. It is straightforward to check that V X s W from which it follows that V g PŽ Z1 .. Accordingly, Q n ª V in L s ŽŽ Z1 .s . l PŽŽ Z1 .s . and so V g M1 by the previous paragraph. Since M1X s M` it follows that W g M` . ACKNOWLEDGMENT The author thanks Dr. Susumu Okada for valuable discussions on this topic.
REFERENCES 1. W. G. Bade, On Boolean algebras of projections and algebras of operators, Trans. Amer. Math. Soc. 80 Ž1955., 345]359. 2. W. G. Bade, A multiplicity theory for Boolean algebras of projections in Banach spaces, Trans. Amer. Math. Soc. 92 Ž1959., 508]530. 3. N. Dunford and J. T. Schwartz, ‘‘Linear Operators. III. Spectral Operators,’’ Interscience, New York, 1971. 4. C. K. Fong and L. Lam, On spectral theory and convexity, Trans. Amer. Math. Soc. 264 Ž1981., 59]75. 5. J. Hoffmann-Jørgensen, Vector measures, Math. Scand. 28 Ž1971., 5]32. 6. S. Okada and W. J. Ricker, Spectral measures which fail to be equicontinuous, Period. Math. Hungar. 28 Ž1994., 55]61. 7. S. Okada and W. J. Ricker, Boolean algebras of projections and ranges of spectral measures, Dissertationes Math., to appear. 8. W. J. Ricker, On Boolean algebras of projections and scalar-type spectral operators, Proc. Amer. Math. Soc. 87 Ž1983., 73]77. 9. W. J. Ricker, A concrete realization of the dual space of L1-spaces of certain vector and operator-valued measures, J. Austral. Math. Soc. Ser. A 42 Ž1987., 265]279. 10. H. H. Schaefer, Spectral measures in locally convex algebras, Acta Math. 107 Ž1962., 125]173. 11. I. Tweddle, Weak compactness in locally convex spaces, Glasgow Math. J. 9 Ž1968., 123]127. 12. B. Walsh, Structure of spectral measures on locally convex spaces, Trans. Amer. Math. Soc. 120 Ž1965., 295]326.
208, 364]371 Ž1997.
AY975300
The Sequential Closedness of s-Complete Boolean Algebras of Projections W. J. Ricker School of Mathematics, Uni¨ ersity of New South Wales, Sydney, NSW, 2052, Australia Submitted by John Hor¨ ath ´ Received March 4, 1996
In the Banach space setting s-complete Boolean algebras of projections Žbriefly, B.a.. or, equivalently, their representation as spectral measures, were intensively studied by W. Bade w1, 2x. Such objects are a natural extension of the fundamental notion of the resolution of the identity of a normal operator in Hilbert space. Since the concepts involved are of a topological and order theoretic nature there is no difficulty in extending the study of such objects to the setting of locally convex Hausdorff spaces Žbriefly, lcHs. w10, 12x. If the underlying lcHs X is a separable Frechet ´ space, then it is known that such a B.a. M is necessarily a closed subset of LŽ X . with respect to the strong operator topology ts ; see w8, Proposition 3; 9, Theorem 5Ži.x. Here LŽ X . denotes the space of all continuous linear operators of X into itself. There are many sufficient conditions known on both X and M which guarantee that if M is s-complete, then it is necessarily a ts-closed subset of LŽ X .; see w7]9x, for example. Despite the quite general criteria referred to above it is easy to exhibit examples where M fails to be ts-closed in LŽ X .. For instance, let X denote the non-separable Hilbert space l 2 Žw0, 1x.. Let S denote the family of Borel subsets of w0, 1x. Define, for each E g S, the selfadjoint projection P Ž E . g LŽ X . by P Ž E . x s x E x for each x g X. It is straightforward to verify that M s P Ž E .; E g S4 is a s-complete B.a. To see that M is not ts-closed, let F : w0, 1x be any non-Borel set and let F Ž F . denote the family of all finite subsets of F directed by inclusion. Then P Ž E .; E g F Ž F .4 is a net in M which is ts-convergent to the selfadjoint projection Q g LŽ X . of multiplication by x F . However, Q f M . Despite the fact that M is not ts-closed it is routine to verify that M is sequentially ts-closed in LŽ X .. That is, if Pn4`ns1 : M is any sequence which is ts-convergent in LŽ X ., say to the element T g LŽ X ., then actually T g M . 364 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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Dr. Susumu Okada posed the question of whether it is always the case that a Bade s-complete B.a. M : LŽ X . is sequentially ts-closed. The aim of this note is to show, under quite mild completeness restrictions on LŽ X ., that the answer is yes in the case that the s-complete B.a. M is purely atomic. An example is given showing that the answer is no in general.
1. PRELIMINARIES Let X be a lcHs and M : LŽ X . be a B.a. always assumed to have as unit the identity operator I on X. Then M is said to be s-complete Žin the sense of Bade. if it is s-complete as an abstract B.a. and if, for every countable family Bn4 : M we have ŽHn Bn .Ž X . s Fn BnŽ X . and ŽEn Bn .Ž X . s sp Dn BnŽ X .4 , the closed subspace of X generated by Dn BnŽ X .; see w1x, for example. Let L s Ž X . and L w Ž X . denote LŽ X . equipped with the strong and weak operator topology, respectively. Given a set N : LŽ X . we denote the closure of N in L s Ž X ., respectively L w Ž X ., by N s, respectively, N w . A B.a. M : LŽ X . is said to have the s-monotone property if lim n Bn exists in L w Ž X . and is an element of M whenever Bn4 : M is a monotonic sequence. Of course, the partial order in M is range inclusion, i.e., P F Q iff P Ž X . : QŽ X .. LEMMA 1.1.
Let X be a lcHs and M : LŽ X . be a B.a.
Ži. If M has the s-monotone property, then M is s-complete. Žii. If M is equicontinuous Ž in LŽ X .., then M has the s-monotone property iff M is s-complete. For Ži. we refer to w7, Lemma 3.3x and Žii. follows from Proposition 4.1 and Corollary 4.6.1 of w7x. Let L be a topological Hausdorff space and Z : L. Then w Z x denotes the set of all elements in L each of which is the limit of some sequence of points from Z. A set Z : L is called sequentially closed if Z s w Z x. The sequential closure of a set Z : L is the smallest sequentially closed subset of L which contains Z. It is always equipped with the relative topology from L. If M : LŽ X . is a B.a., then w M x L s Ž X . denotes the sequential closure Žin L s Ž X .. of the linear span of M . It is clear that w M x L s Ž X . is a linear subspace of LŽ X .. The closure, in L s Ž X ., of the linear span of M is denoted by ² M :s . A compact, convex subset K of the locally convex algebra L w Ž X . is called a spectral carrier w4, Definition 2.1x, if K is commutative, closed under composition in LŽ X ., and ŽT q S y TS . g K whenever T, S g K.
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LEMMA 1.2. Let X be a lcHs and M : LŽ X . be a B.a. with the s-monotone property. If either L s Ž X . or w M x L s Ž X . or ² M :s is quasicomplete, then the closed con¨ ex hull coŽ M . is a spectral carrier in L w Ž X .. Proof. The s-monotone property ensures that there exists a spectral measure P: S ª L s Ž X ., defined on a s-algebra of sets S of some set V, such that its range P Ž S . s M w7, Proposition 4.6x. To say P is a spectral measure means that P is countably additive, P Ž V . s I, and P is multiplicative Ži.e., P Ž E l F . s P Ž E . P Ž F . for all E, F g S .. Since L w Ž X . is the lcHs L s Ž X . equipped with its weak topology and P is s-additive whether considered as being L s Ž X .-valued or w M x L s Ž X .-valued, or ² M :svalued, it follows from the quasicompleteness assumption that K s coŽ M . s coŽ P Ž S .. is a compact Žand convex. subset of L w Ž X . w11x. The other properties required to establish that K is a spectral carrier are easy consequences of the B.a. properties of M ; see the proof of w7, Proposition 3.15x. 2. ATOMIC BOOLEAN ALGEBRAS OF PROJECTIONS Let X be a lcHs and M : LŽ X . be a B.a. A non-zero projection B g M is called an atom if, whenever D g M satisfies D F B, then either D s 0 or D s B. Then M is called Žpurely. atomic if there exists a family of atoms Ba 4a g A in M such that, for every B g M there exists a subset B˜ : A such that Ý a g B˜ Ba s B. The summability of the family Ba 4a g B˜ is meant as the ts-limit in LŽ X . of the net of partial sums over all finite subsets of B˜ Ždirected by inclusion.. The family of atoms Ba 4a g A , necessarily pairwise disjoint and satisfying Ý a g A Ba s I, is said to generate M . The collection of all projections in LŽ X . is denoted by PŽ X .. PROPOSITION 2.1. Let X be a lcHs and M : LŽ X . be an atomic B.a., generated by the atoms Ba 4a g A , and ha¨ ing the s-monotone property. Suppose that at least one of the spaces L s Ž X . or w M x L s Ž X . or ² M :s is quasicomplete. Then, for e¨ ery subset F : A the series Ý a g F Ba con¨ erges in L s Ž X . to an element of PŽ X .. Proof. Fix an arbitrary subset F : A. Let F Ž F . denote the collection of all finite subsets of F, directed by inclusion. The pairwise disjointness of Ba 4a g A implies that Ý a g E Ba s Ea g E Ba belongs to M for each E g F Ž F .. Accordingly, Ý a g E Ba ; E g F Ž F .4 is an upwards directed system of elements from M : K s coŽ M .. Since K is a spectral carrier Žcf. Lemma 1.2. it follows that the limit lim E g F Ž F . Ý a g E Ba exists in L w Ž X . and is an element BF g PŽ X . w4, Theorems 2.10 and 2.11x. Then w7, Proposition 4.21x implies that the limit BF s lim E g F Ž F . Ý a g E Ba also exists in L s Ž X ..
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Remark. It is straightforward to produce examples where w M x L s Ž X . or ² M :s is quasicomplete, but L s Ž X . itself is not quasicomplete. Indeed, let X s l 1 equipped with its weak-) topology s Ž l 1 , c 0 .. Then X is quasicomplete but L s Ž X . is not even sequentially complete. Let V s N and S s 2 N . Then P: S ª L s Ž X . defined by P Ž E . w s x E w Žcoordinate-wise multiplication., for all w g X and E g S, is a spectral measure and so M s P Ž S . is a B.a. with the s-monotone property. It turns out that w M x L Ž X . is quasicomplete. For details of this example we refer to w7, s Example 2.3x. We now have the main result of the section. THEOREM. Let X be a lcHs and M : LŽ X . be an atomic B.a. ha¨ ing the s-monotone property. Suppose that at least one of the spaces L s Ž X . or w M x L Ž X . or ² M :s is quasicomplete. Then M is a sequentially closed subset of s L s Ž X .. If X is any quasicomplete, barrelled lcHs then it is a simple application of the Banach]Steinhaus theorem to deduce that L s Ž X . is necessarily quasicomplete. In particular, this includes all Frechet spaces and hence, all ´ Banach spaces. Since every s-complete B.a. acting on a Frechet space is ´ necessarily equicontinuous w12, Proposition 1.2x, the above remarks together with the Theorem and Lemma 1.1 give the following COROLLARY. Let X be a Frechet lcHs. Then e¨ ery s-complete, atomic ´ B.a. M : LŽ X . is a sequentially closed subset of L s Ž X .. The example in the introduction shows that this Corollary cannot be improved by replacing sequentially closed with closed. Proof of Theorem. Let Ba 4a g A be a generating family of atoms for M . By Proposition 2.1, the collection of sets S : 2 A given by S s F : A; Ý a g F Ba g M 4 is well defined. The claim is that S is a s-algebra of subsets of A. Clearly f , A g S. If F g S and F c denotes the complement of F in A, then Ý a g F c Ba q Ý a g F Ba s I shows that Ý a g F c Ba s I y B with B s Ý a g F Ba g M . Since M is a B.a. we have Ý a g F c Ba g M , i.e., F c g S. Suppose that F1 , F2 g S are disjoint. It is a routine calculation from the definition of the convergence of the series Ý a g F1 j F 2 Ba Žwhich converges to some projection in L s Ž X . by Proposition 2.1. that Ý a g F1 j F 2 Ba s Ý a g F1 Ba q Ý a g F 2 Ba . Since both B1 s Ý a g F1 Ba and B2 s Ý a g F 2 Ba belong to M and are disjoint it follows that B1 q B2 s B1 k B2 g M , i.e., F1 j F2 g S.
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Let E1 , E2 g S. If F Ž Ej . denotes the family of finite subsets of Ej , directed by inclusion, for j g 1, 24 , then
ŽÝa g E
1
Ba . ? Ž Ý b g E 2 Bb . s Ž Ý a g E1 Ba . lim F g F Ž E 2 . Ý b g F Bb .
Using continuity of Ý a g E1 Ba and the fact that each sum Ý b g F Bb for F g F Ž E2 . is finite, gives
ŽÝa g E
1
Ba . ? Ž Ý b g E 2 Bb . s lim F g F Ž E 2 . Ý b g F Ž Ý a g E1 Ba . Bb .
Ž 1.
But, for b g F and F g F Ž E2 . fixed, the continuity of Bb and the fact that lim G g F Ž E1 . ŽÝ a g G Ba . s Ý a g E1 Ba imply that
ŽÝa g E
1
Ba . Bb s lim G g F Ž E1 . Ý a g G Ba Bb ,
b g F.
Since Ba Bb s 0 if a / b and Ba Bb s Bb if a s b it follows that Ý b g F Ž Ý a g E1 Ba . Bb s Ý b g F l E1 Bb ,
F g F Ž E2 . .
This shows that the limit in Ž1. is taken with respect to F Ž E2 . l E1 , i.e., with respect to F Ž E1 l E2 .. Accordingly,
ŽÝa g E
1
Ba . ? Ž Ý b g E 2 Bb . s lim F g F Ž E1 l E 2 . Ýg g F Bg s Ýg g E1 l E 2 Bg .
Ž 2. Since the left-hand side of Ž2. is a product of two elements from M Žhence, is in M . it follows that Ýg g E1 l E 2 Bg g M , i.e., E1 l E2 g S. If now F1 , F2 g S are arbitrary, then the identity F1 j F2 s Ž F1c l F2 . j Ž F1 l F2c . j Ž F1 l F2 . , with the three terms in the union pairwise disjoint, implies that F1 j F2 g S. Accordingly, S is an algebra of subsets of A. Finally, let En4`ns1 be a sequence of pairwise disjoint elements from S. Then Fn s Dnks1 Ek , for n s 1, 2, . . . , is an increasing sequence of elements from S. Accordingly, Ý a g F n Ba 4`ns1 is an increasing sequence in M . Since M has the s-monotone property it follows that lim nª` Ý a g F n Ba s B exists in L w Ž X . and belongs to M . By w7, Proposition 4.21x this convergence is also valid in L s Ž X .. It is routine to check that lim nª` Ý a g F n Ba s Ý b g F Bb where F s D`ns1 Fn s D`ns1 En and hence, D`ns 1 En g S. This shows that S is indeed a s-algebra of subsets of A. Suppose now that Pn4`ns1 : M is a sequence such that Pn ª P in L s Ž X .. The aim is to show that P g M . Let EŽ n. g S be sets such that Pn s Ý a g EŽ n. Ba , for each n g N. Define E s D`Ns 1 ŽF`nsN EŽ n... Then E g S and so Q s Ý a g E Ba g M . We show that P s Q.
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Fix x g X. If a g E, then there exists Na such that a g EŽ n. for all n G Na . Accordingly, Ba x occurs in each series Ý b g EŽ n. Bb x for all n G Na . It follows that Pn Ba x s Ba x for all n G Na and hence, Ba x s lim n Pn Ba x. That is, Ba x s PBa x,
a g E.
Ž 3.
Adding up the terms in Ž3. and using continuity of P gives Qx s Ý a g E Ba x s Ý a g E PBa x s P Ž Ý a g E Ba x . s PQx. Since x g X was arbitrary it follows that Q s PQ.
Ž 4.
Suppose now that a f E. Then there is an increasing sequence nŽ1. Ž n 2. - ??? such that a f EŽ nŽ j .. for all j G 1. So, if x g Ba Ž X ., then PnŽ j. x s Ý b g EŽ nŽ j.. Bb x s 0,
j g N.
Hence, the subsequence PnŽ j. x 4`js1 of Pn x 4`ns1 converges to zero from which it follows that Px s 0,
x g Ba Ž X . , a f E.
Ž 5.
But, E c g S and I y Q s Ý a f E Ba g M showing that
Ž I y Q . x s x,
x g Ž I y Q . Ž X . s sp Da f E Ba Ž X . 4 .
Ž 6.
Applying P to Ž6., and using Ž5. and the continuity of P shows that P Ž I y Q . x s Px s 0,
x g Ž I y Q. Ž X . .
So, for x g X arbitrary it follows that P Ž I y Q. x s P Ž I y Q. Ž I y Q. x s 0 since y s Ž I y Q . x g Ž I y Q .Ž X .. This shows that P Ž I y Q . s 0 which, combined with Ž4., implies that P s Q. COUNTEREXAMPLE. Let X` s L`Žw0, 1x. equipped with its weak-) topology s Ž L` , L1 ., in which case X` is a quasicomplete lcHs. For p g w1, `. let X p denote L p Žw0, 1x. equipped with its weak topology s Ž L p , Lq ., where q satisfies py1 q qy1 s 1. Then X 1 is a sequentially complete lcHs and X p is a quasicomplete lcHs for 1 - p - `. If S denotes the family of all Borel subsets of V s w0, 1x, then it is routine to verify that Pp : S ª L s Ž X p . defined by Pp Ž E .: f ¬ x E f, for f g X p and E g S, is a
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spectral measure for all p g w0, `x. Accordingly, Mp s Pp Ž S . is a B.a. with the s-monotone property w7, Lemma 3.1x. The claim is that Mp is not a sequentially closed subset of L s Ž X p .. ny 1 Let I jn s Ž j2yn , Ž j q 1.2yn x for 0 F j - 2 n, and put AŽ n. s D2js1 I2njy1 for n g N. It is shown in w5, Example 4x that x AŽ n. ª 12 | in X` , where | denotes the function on V which is constantly 1. Fix p g w1, `x. Given f g X p and g g X pX s X q , the function w s fg g X 1 s X`X . Accordingly, if l denotes the Lebesgue measure, then lim n ª` ² x AŽ n. , w : s lim nª`
1
H0
x AŽ n. w d l s
11
H0
2
|w d l s ² 12 |, w :
since x AŽ n. ª 12 | in X` . That is, ² Pp Ž AŽ n.. f, g : ª ² 12 |f, g :, as n ª `, for every f g X p and g g X pX , which shows that Pp Ž AŽ n.. ª 12 Pp Ž V . s 12 I in L s Ž X p .. But, 12 I f Mp showing that Mp is not sequentially closed in L s Ž X p .. The B.a.’s Mp , 1 F p F `, are not equicontinuous w6, Proposition 4x. This raises the following QUESTION 1. Is an equicontinuous B.a. M with the s-monotone property necessarily sequentially closed in L s Ž X .? Or, e¨ en more specific, does there exist a s-complete B.a. M acting in a Banach Ž or Hilbert!. space which is not a sequentially closed subset of L s Ž X .? In the class of examples above the sequential closedness of the B.a. Mp fails because the limit operator is not a projection. Of course, if M is equicontinuous, then the limit in L s Ž X . of a sequence of projections from M is always an element of PŽ X .. QUESTION 2. Is a B.a. M with the s-monotone property a sequentially closed subset of PŽ X . l L s Ž X .? We show that the B.a.’s Mp , 1 F p F `, do not answer Question 2. Suppose first that 1 F p - `. Let Z p denote the Banach space L p Žw0, 1x. equipped with its norm topology. Then X p s Ž Z p .s is just Z p equipped with its weak topology s Ž Z p , ZXp .. If T g LŽ X p ., then T is also continuous from Ž X p .m to Ž X p .m where m is the Mackey topology on X p . But, this is precisely the norm topology and so LŽ X p . s LŽ Z p . as vector spaces. Moreover, it is routine to check that L w Ž Z p . s L s Ž X p . as lcH-spaces. Clearly Mp still has the s-monotone property considered as a B.a. in LŽ Z p . and hence, by Lemma 1.1, it is s-complete as a B.a. in L s Ž Z p .. Since the Banach space Z p is separable Mp is actually a complete B.a. in L s Ž Z p . w3, XVII, Lemma 3.21x. Now, suppose that Q n4`ns1 : Mp is a sequence and V g PŽ X p . such that Q n ª V in L s Ž X p .. Then also Q n ª V
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in L w Ž Z p . from which it follows via w1, Theorem 3.2x that Q n ª V in L s Ž Z p .. But, the separability of Z p implies that Mp is actually a closed subset of L s Ž Z p .; see the introduction. Accordingly, V g Mp . Now consider the case p s `. Observe that X` s Z1X with the weak-) topology s Ž Z1X , Z1 .. Suppose that Pn4`ns1 : M` s P`Ž S . is a sequence and W g PŽ X` . such that Pn ª W in L s Ž X` .. For each n g N, let Q n g M1 : LŽ Z1 . be the unique operator such that QXn s Pn . Then Q n4`ns1 is Cauchy in L w ŽŽ Z1 .s . s L s ŽŽ Z1 .s .. So, for each w g Z1 the sequential completeness of Ž Z1 .s guarantees the existence of Vw g Z1 such that Q n w ª Vw in Ž Z1 .s . Since sup5 Q n 5; n g N4 s 1 it follows that the linear map V: Z1 ª Z1 so defined belongs to LŽ Z1 .. It is straightforward to check that V X s W from which it follows that V g PŽ Z1 .. Accordingly, Q n ª V in L s ŽŽ Z1 .s . l PŽŽ Z1 .s . and so V g M1 by the previous paragraph. Since M1X s M` it follows that W g M` . ACKNOWLEDGMENT The author thanks Dr. Susumu Okada for valuable discussions on this topic.
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