Criteria for Closedness of Spectral Measures and Completeness of Boolean Algebras of Projections

Journal of Mathematical Analysis and Applications 232, 197᎐221Ž. 1999 Article ID jmaa.1998.6262, available online at http:rrwww.idealibrary.com on

S. Okada

View metadata, citation and similar10 McDonaldpapers at core.ac.uk A¨e., Cooma, NSW 2630, Australia brought to you by CORE provided by Elsevier - Publisher Connector and

U W. J. Ricker

School of Mathematics, Uni¨ersity of New South Wales, Sydney, NSW, 2052, Australia

Submitted by John Hor¨ath´

Received April 13, 1998

The intimate connection between spectral measures and ␴-complete Boolean algebras of projections in Banach spaces was intensively investigated by W. Bade in the 1950s,w ‘‘Linear Operators III: Spectral Operators,’’ Wiley-Interscience, New York, 1971; Chap. XVIIx . It is well known that the most satisfactory situation occurs when the Boolean algebra is actually complete rather than just ␴-complete. This makes it desirable to have available criteriaŽ not just in Banach spaces but also in the nonnormable setting. which can be used to determine the completeness of Boolean algebras of projections. Such criteria, which should be effective and applicable in practice, as general as possible and apply to extensive classes of spaces, are presented in this article. A series of examples shows that these criteria are close to optimal. ᮊ 1999 Academic Press Key Words: closed spectral measures; complete Boolean algebra of projections.

Spectral measures are extensions to BanachŽ and more general locally convex. spaces of the fundamental notion of the resolution of the identity of a self-adjoint or normal operator in a Hilbert space. A detailed study of these objectsŽ. and the related class of spectral operators in the Banach space setting can be found inwx 7 . For the case of locally convex Hausdorff spacesŽ. briefly, lcHs we refer towx 7; XV, Section 16 for early work in this

* E-mail: [email protected].

197 0022-247Xr99 $30.00 Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved. 198 OKADA AND RICKER area andwx 2, 4, 5, 16, 18, 24 for other results. In the article wx 15 a detailed study was undertaken of certain important aspects of this class of measures. In particular, the topics dealt with there in detail were those of relative weak compactness of the range, connections with the theory of Bade completeŽ. and ␴-complete Boolean algebras of projections, and characterizations of the important subclass consisting of those spectral measures which are closed. Our aim is to continue and to extend further the investigation began inwx 15 . As noted inwx 15 , closed spectral measures enjoy important properties not characteristic to all spectral measures. For example, under very weak restrictions they always have relatively weakly compact range, their range is always a Bade complete Boolean algebra, under mild completeness 1 conditions on the underlying space X their associated L -space is topo- logically complete, if the measure is also equicontinuous then the closed operator algebras generated by the range of the spectral measure with respect to the strong operator topology and the topology of uniform convergence on the bounded sets of X coincide, and so on. Desirable features of this kind make it imperative to have available sufficient conditions, as general as possible, which can be used to determine the closedness of a given spectral measure. The aim of this article is to provide such criteria.

1. PRELIMINARIES

Given a lcHs Y let Ž.Y denote the family of all continuous seminorms X P on Y, and let Y denote the space of all continuous linear functionals on X Y. We write Y␴ to denote Y equipped with its weak topology ␴ ŽY, Y ..Bya ¨ector measure in Y is meant a ␴-additive map m: ⌺ ª Y whose domain ⌺ is a ␴-algebra of subsets of some nonempty set ⍀. The Orlicz᎐Pettis lemmawx 8; I, Theorem 1.3 implies that m is ␴-additive in Y iff it is X ␴-additive in Y␴ , that is, iff the set function ²m, y :: ⌺ ª ރ defined by X XX E ¬ ²Ž.mE, y :,forE g ⌺,is␴-additive for each y g Y . A ⌺-measurable function f: ⍀ ª ރ is called m-integrable if it is X XX integrable with respect to ²m, y :, for each y g Y , and if, for each g ⌺ E there is an element of Y, denoted by HE fdm, which satisfies X s X XXg ²HEEfdm, y :²H fd m, y : for every y Y . The linear space of all m-integrable functions is denoted by L Ž.m ; it always contains the space simŽ.⌺ of all ⌺-simple functions. Each q g P Ž.Y induces a seminorm qmŽ.in L Ž.m via the formula,

¬ <<<² X :< X g o g qmŽ.: f sup H fd m, y ; y Uq , f L Ž.m ,1 Ž. ½5⍀ SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 199 where <<␯ denotes the total variation measure of a measure ␯ : ⌺ ª ރ and o : X s y1 wx Uq Y is the polar of the closed q-unit ball Uq q Ž 0, 1. . The seminormsŽ. 1 , as q varies through PŽ.Y , define a lc-topology ␶ Ž.m in L Ž.m . Because ␶ Ž.m may not be Hausdorff we form the usual quotient space of Ž.m with respect to the closed subspace F qm Ž.y1 ŽÄ40.. L q g PŽY . The resulting lcHsŽŽ with topology again denoted by ␶ m..is denoted by 1 L Ž.m ; it can be identified with equivalence classes of functions from L Ž.m modulo m-null functions, where a function f g L Ž.m is m-null s g ⌺ g ⌺ whenever HE fdm 0 for all E . In particular, a set E is m-null iff mFŽ.s 0 for every F g ⌺ with F : E. All of these definitions and 1 wx further properties of L Ž.m can be found in 8 . ⌺ 1 ␹ g ⌺ : Let Ž.m be the subset of L Ž.m corresponding to Ä4E; E L Ž.m . Elements of ⌺Ž.m are identified with equivalence classes of ele- 1 ments from ⌺. The topology ␶ Ž.m of L Ž.m induces a topology on ⌺Ž.m by restrictionŽŽ..Ž.Ž. and again denoted by ␶ m . Then ⌺ m is a ␶ m -closed 1 subset of L Ž.m .If ŽŽ.Ž..⌺ m , ␶ m is a complete topological space, then m is called a closed measure,8;p.71.wx A measure m: ⌺ ª Y is called countably determined,wx 12 , if there exists X ϱ X g ⌺ a sequence Ä ynn4 s1 in Y such that a set E is m-null if it is X g ގ ²m, yn:-null for all n .

Remark 1.1. If ␳ is any lcH-topology in Y which is consistent with the X duality ŽY, Y ., then the Orlicz᎐Pettis lemma implies that m, considered as being Y␳-valuedŽ. in which case we write m␳ , is again additive. It is clear from the definition that m is countably determined iff m␳ is countably determined.

PROPOSITION 1.2Žwx 12; Proposition 1.2. . A countably determined ¨ector measure is a closed measure.

The converse of Proposition 1.2 is not true in general; see Example 2.22. Let ⌳ be a topological Hausdorff space and Z : ⌳. Then wxZ denotes the set of all elements in ⌳ each of which is the limit of some sequence of points from Z. A set Z : ⌳ is called sequentially closed if Z s wxZ . The sequential closure of a set Z : ⌳ is the smallest sequentially closed subset of ⌳ which contains Z. It is always equipped with the relative topology from ⌳. ⌺ ª wx Given a lcHs Y and a vector measure m: Y let Y m denote the sequential closureŽ. in Y of the linear hull of mŽ.⌺ s ÄmEŽ.; E g ⌺4. wx Then Y m is a vector subspace of Y; the point is that Y may be quite wx large whereas m only takes its values in a ‘‘smaller part’’ Y m of Y.We denote by mwY, mx the vector measure obtained when m is interpreted as wx taking its values in the lcHs Y m. 200 OKADA AND RICKER

PROPOSITION 1.3. Let m: ⌺ ª Ybea¨ector measure.

Ž.i m is a closed measure iff mwY; mx is a closed measure.

Ž.ii m is countably determined iff mwY; mx is countably determined. Proof. Ž.iIfm is a closed measure,then there is a localisable measure X XX ␭: ⌺ ª wx0, ϱ such that ²m, y : g ␭ for all y g Y ,wx 9; Corollary 13 . Let ␰ g wxX ᎐ wx Y m. By the Hahn Banach theoremŽ note that Y m may not be a ␰ g X ␰ wx closed subspace of Y . there exists Y which coincides with on Y m. ␰ s ␰ g ␭ wx Hence, ²:²:mwY; mx, m, . It follows from 8; IV, Theorem 7.3 that mwY; mx is a closed measure. A similar type of argument applies to the wx X converse because the restriction to Y m of any element of Y is an wxX element of Y m. ␰ ϱ : wxX Ž.ii Let mwY; mx be countably determined and let Ä4nns1 Y m have g ⌺ ␰ the property that E is mwY; mxw-null iff it is ²:m Y; mx, n -null for all g ގ g ގ ␰ g X n . For each n let n Y be a functional which coincides with ␰ wx g ⌺ nmon Y . Noting that E is m-null iff it is mwY; mx-null and that ␰ s ␰ g ގ g ⌺ ²:²:mwY; mx, nnm, , for each n , it follows that E is m-null iff ␰ g ގ it is ²:m, n -null for all n . Hence, m is countably determined. The wx converse follows along similar lines because the restriction to Y m of XXwx elements from Y are elements of Y m.

PROPOSITION 1.4. Let m: ⌺ ª Ybea¨ector measure. wxX ) ␴ wxX wx Ž.i If Ymm is separable for the weak- topology Ž Y , Y m., then m is countably determined. In particular, m is a closed measure. ␳ wx Ž.ii If there is a metrizable lcH-topology on Ym which is weaker wx than or equal to the Mackey topology on Y m, then m is countably determined. In particular, m is a closed measure. wx Proof. Ž.i By 12; Proposition 1.4 we have that mwY; mx is countably determined and hence, Proposition 1.3Ž. ii implies that m is countably determined. Proposition 1.2 then implies that m is also closed. Ž.ii Because m is ␴-additive for the Mackey topology in Y Žcf. ⌺ ª wx ␴ Remark 1.1.Ž.Ž the measure mwY, mx ␳ : Y m.␳ is also -additive. Then wx 12; Proposition 1.6 implies that Ž.mwY, mx ␳ is countably determined. Be- wxX : wxX cause Ž Y m.␳ Y m the measure mwY, mx is also countably determined and hence, so is m ŽŽcf. Proposition 1.3 ii.. . Proposition 1.2 then implies that m is also closed.

In Proposition 1.4Ž. ii the topology ␳ need not be consistent with the wx wxX duality Ž Y mm, Y .. For example, let Y denote the vector space of all functions ␸: ގ ª ރ such that Än g ގ; ␸Ž.n / 04 is finite. Define q: SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 201

ª w ϱ.Ž.Ž␸ s ϱ <␸Ž.< 2 .1r2 ␸ g ␰ g ϱ Y 0, by q Ýns1 n ,for Y. For each l define ª w ϱ ␸ s < ϱ ␸ ␰ < ␸ g q␰ : Y 0, .Ž.by q␰ Ýns1 Ž.n n for Y. The family of semi- j ␰ g ϱ norms Ä4q Äq␰ ; l 4 defines a lcH-topology in Y. Then the lcH-topology ␳ determined in Y by the seminorm q is metrizable and weaker than X the given topology in Y, but it is not consistent with the duality ŽY, Y .. wx wx Remarks 1.5 and 1.7 of 12 show that the use of Y m in Proposition 1.4, rather than Y itself, is a definite advantage in applications. For quasi-complete spaces over ޒ the following result can be found in wx26; Theorem 2.1 . An examination of the proof given there shows that the quasi-completenessŽ a standing assumption throughoutwx 26. is not needed for this particular result and that the result remains valid over ރ if convex hulls are replaced with balanced convexed hulls.

PROPOSITION 1.5. Let Y be a lcHs in which eëry bounded subset is metrizableŽ. for the relatië topology from Y . Let C be a closed, bounded, conëx subset of Y. If M s bcoŽ.C denotes the closed, balanced conëx hull ¨ s Dϱ of C, then there exists a norm on the ector subspace YMns1 nM of Y which induces the same topology on M as that induced by Y. The following consequence is needed in the sequel.

COROLLARY 1.5.1. Let Y be a lcHs in which e¨ery bounded subset is ¨ metrizableŽ. for the relati e topology from Y . Let Ý␣ g A y␣ be an uncondi- ¨ s : tionally con ergent series in Y such that F Ä4Ý␣ g F y␣ ; F A, F finite is a bounded subset of Y. Then at most countably many elements ofÄ4 y␣ ; ␣ g A are nonzero.

Proof. Let C s coŽ.F and M s bcoŽ.C . Proposition 1.5 implies that ␶ there is a norm on YM which induces on M the same topology as that of s 1 ␣ g s Y. Let z␣ 2 y␣␣, for each A, and z Ý g A z␣ . Then

y s lim z Ý z␣ 0,Ž. 2 g Ž.ž/ F F A ␣gF

where FŽ.A is the family of all finite subsets of A directed by inclusion. y g g Because Ž.z Ý␣ g F z␣ M, for each F F Ž.Ž.A , the limit 2 exists with respect to ␶ in M and hence, also with respect to the topology on M

induced by the norm from YM . That is, the series Ý␣ g A z␣ is also summable to z in YM from which it follows that at most countably many elements from Ä4z␣ ; ␣ g A hence, also from Ä4y␣ ; ␣ g A , are nonzero. ⌺ ª A vector measure m: Y has an associated integration map Im: 1 ª s g 1 L Ž.m Y given by IfmŽ. H⍀ fdm, for each f L Ž.m . It is clear from ␶ 1 the definition of Ž.m that Im is a continuousŽ. linear map from L Ž.m into Y. 202 OKADA AND RICKER

The vector space of all continuous linear operators of a lcHs X into itself is denoted by LXŽ.. The space LX Ž.equipped with the strong operator topologyŽ. i.e., the topology of pointwise convergence on X is denoted by LXsŽ.. It is a lcHs whose topology is generated by the family of ¬ g g g seminorms qx: T qTxŽ., with T LX Ž., for each x X and q PŽ.X . X ␰ ª ރ The dual space ŽŽ..LXssconsists of all linear functions : LXŽ. of the form,

n ␰ ¬ X g : T Ý ²:Tx jj, x , T LXsŽ.,3Ž. js1

n X n X g ގ for some finite subsets Ä4x jjs1 of X and Ä x jj4 s1 of X , and n .In ␴ X particular, the weak topology ŽŽ.ŽŽ..LXss, LX .Ž.of LX sis precisely the weak operator topology. The space LXŽ.equipped with the weak operator topology is denoted by LXwŽ.. ⌺ ª Because LXsŽ.is a lcHs we may consider vector measures P: ¨ LXsŽ., which are usually referred to as operator- alued measures. For each x g X,let Px: ⌺ ª X denote the X-valued vector measure E ¬ PExŽ., for E g ⌺.If PŽ.⌺ s Ä PEŽ.; E g ⌺4 is an equicontinuous part of LXŽ., then P is said to be an equicontinuous measure. An operator-valued ⌺ ª measure P: LXsŽ.is called a spectral measure if it is multiplicative, i.e., PEŽ.Ž.Ž.l F s PEPF for all E, F g ⌺ and PŽ.Ž⍀ s I the identity operator in X .Ž. For such a measure P aset E g ⌺ is P-null iff PE.s 0. Moreover, the integrability of functions with respect to spectral measures is simpler than for more general operator-valued measures. Indeed, a X ⌺-measurable function f: ⍀ ª ރ is P-integrable iff it is ² Px, x :- g XXg g integrable, for each x X and x X , and there exists Tf LXŽ.such X s X g XXg w that ²Txf , x :²H⍀ fd Px, x :, for each x X and x X ; 13; Lemma x s 1.2 . In this case Tf H⍀ fdP. It is always the case that simŽ.⌺ : L Ž.P . Under further assumptions L Ž.P contains elements other than just those from sim Ž.⌺ . Indeed, if ⌺ LXsŽ.is sequentially complete, then all bounded -measurable functions are P-integrable,wx 8; II, Lemma 3.1 . It even suffices for the smaller w xwx subspace LXsPŽ. to be sequentially complete, 11; Proposition 2.2 . Alternatively, if P is also equicontinuous, then all bounded ⌺-measurable functions will be P-integrable whenever X is sequentially complete,w 27; p. x 300 ; this is weaker than requiring LXsŽ.to be sequentially complete. It wx g also suffices for X Px to be sequentially complete, for each x X, rather than X itself. ⌺ ª The multiplicativity of a spectral measure P: LXsŽ.implies that the pointwise product fg of two P-integrable functions f and g is again SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 203

P-integrable,wx 14; Corollary 2.1 . Moreover,

HHHHHfg dP s PEŽ.и fdP gdPs PEŽ.и gdP fdP, E g ⌺. E ž/⍀⍀ ž/⍀ ⍀ Ž.4

The formulaŽ. 4 implies that IP is an algebra homomorphism. Sometimes more is true. wx⌺ ª PROPOSITION 1.6Ž 15; Lemma 1.10.Ž. . Let P: Ls X be an equicontinuous spectral measure. Then the integration map IP is a bicontinuous linear 1 1 : and algebra isomorphism of L Ž.P onto its range IPsŽL Ž..P LX Ž .. In particular, IP is a bicontinuous topological and Boolean algebra isomorphism of ⌺Ž.m onto P Ž.⌺ . Vector measures m in a lcHs X of the special type m s Px, where P: ⌺ ª g LXsŽ.is a spectral measure and x X, have properties not enjoyed by all vector measures. ⌺ ª LEMMA 1.7. Let X be a lcHs and let P: LsŽ. X be a spectral measure. Ž.i Each P-null set is Px-null, for e¨ery x g X. Ž.ii If x g X, then a set E g ⌺ is Px-null if P Ž E . x s 0. Ž.iii If P is a closed measure, then also each X-¨alued measure Px, for x g X, is closed. Ž.iv Let x g X. A ⌺-measurable function f is Px-integrable iff it is X XX ² Px, x :-integrable, for each x g X , and there exists x⍀ g X such that

X X XX ² x⍀ , x :²s fd Px, x :, x g X . H⍀

In this case,

s g ⌺ H fdŽ. Px PEx Ž.⍀ , E .5Ž. E

In particular, aPx-integrable function f is Px-null iff H⍀ fdŽ. Px s 0. Proof. Lemma 1.7 forms part ofwx 3; Proposition 1.7 where there is a standing hypothesis that X is quasi-complete and LXsŽ.is sequentially complete. However, an examination of the proof given inwx 3 shows that these completeness hypotheses are not needed for deducing the conclu- sions stated in Lemma 1.7. 204 OKADA AND RICKER

Remark 1.8. The converse ofŽ. iii in Lemma 1.7 fails. Let X be the 2 Ž.nonseparable Hilbert space l Žwx0, 1.Ž and, for each E g ⌺ the Borel ␴ ⍀ s wx g ¬ ␹ -algebra in 0, 1.Ž.Ž. , let PE LX be the projection x E x,for x g X. Because X is metrizable each measure Px: ⌺ ª X is closed; see Proposition 1.4Ž. ii . But, P is not a closed measure,wx 15; Example 3.2 .

2. SUFFICIENT CONDITIONS FOR CLOSEDNESS

As noted previously closed spectral measures have many desirable features not shared by other spectral measures. Proposition 3.16 ofwx 15 gives several equivalent characterizations of closedness of spectral measures. The aim of this section is to exhibit sufficient conditions, as general as possible, which can be used to establish the closedness of spectral measures. The example in Remark 1.8 shows that nonclosed spectral measures exist, even in Hilbert spaces. The spectral measure of Example 2.4 is also nonclosed; this can be argued along the lines ofw 15; Example 3.7x , which itself is another example of a nonclosed spectral measure. We begin with a simple but quite effective criterion. Recall that a nonzero projection PEŽ.is called an atom of the spectral measure P: ⌺ ª g ⌺ l s _ s LXsŽ.if, for each F , either PEŽ.Ž.F 0orPE F 0. ␴ ϱ Then P is called -atomic if there exists a sequence of atoms Ä PEŽ.nn4 s1, g ⌺ ⍀ s where the sets En can be chosen pairwise disjoint, so that Dϱϱs ns1 Ennor, equivalently I Ý s1 PEŽ.n with the series convergingŽ unconditionally.Ž. in LXs .

PROPOSITION 2.1Žwx 15; Corollary 3.5.1. . Let X be a lcHs and let P: ⌺ ª ␴ LXbeasŽ. -atomic spectral measure. Then P is a closed measure. 1 It was pointed out inwx 15; Section 2 that the Banach space l and all Frechet´ Montel spaces have the property that e¨ery spectral measure acting in such a space is necessarily ␴-atomicŽ. and equicontinuous and hence, is a closed measure. Further classes of Banach spaces with this property include the Grothendieck spaces with the Dunford᎐Pettis property,wx 21; Proposition 1 , and the hereditarily indecomposable Banach spaces,wx 25; Proposition 1 . This is also true for any ‘‘gestufter Raum’’ in the sense of Kothe,¨ wx 28 . The example in Remark 1.8 shows that Proposition 2.1 may fail if P has more than countably many atoms. Of course, there also exist purely atomic spectral measures with uncountably many atoms which are closed measures. Just consider again the example in Remark 1.8 where P is now considered to be defined on the ␴-algebra of all subsets ofwx 0, 1 , rather 2 than on the Borel sets. Because the underlying space X s l Žwx0, 1. in this example is nonseparable there arises the question of whether a purely SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 205 atomic spectral measure in a separable space can be closed and can have uncountably many atoms. This is indeed so. Let X s ރw0, 1x be the lcHs of Example 2.4. Then X is complete and separable. Consider the ‘‘same’’ spectral measure P as given in Example 2.4 but now having domain ⌺ consisting of all subsets of ⍀ s wx0, 1 . Then P is purely atomic, equicontin- uousŽ. as X is barrelled , and closed, yet P has uncountably many atoms. However, in separable spaces which are metrizable, or whose bounded sets are metrizable for a suitable topology, this cannot happen. ⌺ ª PROPOSITION 2.2. Let X be a separable lcHs and let P: LXbeasŽ. closed, purely atomic spectral measure. Either of the following conditions implies that P has at most countably many atoms. g ␳ wx Ž.i For each x X, there is a lcH-topology xPxin X , weaker than or equal to the gi¨en topologyŽ but, not necessarily compatible with the duality wx wxX ¨ wx ␳ Ž X Px, X Px.. such that e ery bounded subset ofŽ X Px, x. is metrizable. g ␳ wx Ž.ii For each x X, there is a lcH-topology xPin Xx which is wx wxX consistent with the dualityŽ X Px, X Px. and such that the bounded subsets wx ␳ of XPx are metrizable for the x-topology. Proof. The closedness of P implies that the range M s PŽ.⌺ is a Bade complete Boolean algebra Ž.briefly, B.a. of projections,wx 15; Proposition 3.4 . This means that M is complete as an abstract B.a.Ž where the partial order is given by range inclusion.Ž. and has the property that H␣ BX␣␣␣s F BX E s D : and Ž.␣ BX␣␣sp ŽBX␣ .for every family of elements Ä4B␣M, where H␣ B␣␣␣and E B denote the infimum and supremum, respectively, in the abstractly complete B.a M. In particular, if Ä4B␣ is any increasing g ª family of projections from M, then there exists B M such that B␣ B wx in LXsŽ.; see 15; Proposition 4.7 . So, let Ä4B␣␣g A be the family of all atoms in M, in which case it is a maximal disjoint system satisfying s ⌺ g ގ I ␣ B␣ in LXsnŽ.. Let Ä4x ; n be a countable dense set in X. Then ⌺ s g ގ ␣ Bx␣ nnx , n .6Ž. We now consider the two separate cases. g ގ ␳ Ž.iFixn . Because x is weaker than the given topology and ⌺ the range of any vector measure is a bounded set it follows that PxnŽ.is a ␳ -bounded subset of wxX .If F is any finite subset of A, then the xPnnx ␣ g s disjointness of elements in Ä4B␣ ; A implies that Ý␣ g F Bx␣ nnBx , s ⌺ where B Ý␣ g F B␣ is an element of PŽ.. Accordingly, all finite partial sums of the unconditionally convergent seriesŽ. 6 , which is also convergent for the ␳ -topology, belong to the ␳ -bounded set Px Ž.⌺ . Corollary 1.5.1 xxnnn ŽŽwith Y s wxX , ␳ .. implies the existence of a countable set A : A Pxnn x n s ␣ f ␣ fDϱ s such that Bx␣ nn0 for all A .If ns1 An, then Bx␣ n 0for 206 OKADA AND RICKER

g ގ s ϱϱD all n and so B␣ 0 by the density of Ä4xnns1. Because ns1 An is countable it follows that P is ␴-atomic. Ž.ii Fix n g ގ. The series Ž. 6 is also unconditionally ␳ -convergent x n in wxX by the Orlicz᎐Pettis lemma,w 8; I, Theorem 1.3x . Because the Pxn range of any vector measure is a bounded set and because ␳ is consistent X x n with the duality ŽwxX , wxX .Žit follows that Px ⌺.is a bounded subset Pxnn Px n of ŽwxX , ␳ .Ž. The proof can then be completed as in part i. . Pxnn x COROLLARY 2.2.1. Let X be a separable, metrizable lcHs and let P: ⌺ ª LsŽ. X be a spectral measure. Ž.i P is equicontinuous and closed. Ž.ii If P is purely atomic, then it is actually ␴-atomic. Proof. Because X is quasi-barrelled it follows that P is equicontinuous,wx 13; Proposition 2.5 . Moreover, the hypotheses on X ensure that P is countably determined,wx 12; Corollary 1.8 , and hence, P is a closed measure by Proposition 1.2. This establishesŽ. i . Part Ž ii . follows from Proposi- tion 2.2 and partŽ. i . We point out that Corollary 2.2.1Ž. ii is an extension to noncomplete separable, metrizable spaces of the known fact that a purely atomic Bade complete B.a. in a separable Frechet´ space can have at most countably many atoms,wx 27; p. 323 . Corollary 2.2.1Ž. i is also known for separable Frechet´ spaces,wx 19; Theorem 1 . We wish to improve on Proposition 2.2 and Corollary 2.2.1. A spectral ⌺ ª measure P: LXsŽ.is said to have a countable separating set if there g ގ s s exist vectors xn, n ,in X such that PEŽ. 0 whenever PExŽ.n 0 g ގ g ⌺ g ގ for all n , that is, a set E is P-null iff it is Pxn-null for all n ; see Lemma 1.7Ž. ii . ⌺ ª PROPOSITION 2.3. Let X be a lcHs and let P: LsŽ. X be a spectral measure. Ž.i If P is countably determined, then P has a countable separating set. ϱ g ގ Ž.ii If P has a countable separating setÄ4 xnns1 and, for each n , the ¨ector measure Px : ⌺ ª wxX is countably determined, then P is also nPxn countably determined. In particular, P is a closed measure. Ž.iii Assume that X is a metrizable lcHs. Then P is countably determined iff P has a countable separating set. ϱ : Proof. Ž.i It is clear from Ž 3 . that there exist countable sets Ä4xnns1 X X ϱ X X and Ä x 4 s : X such that E g ⌺ is P-null iff E is ² Px , x :-null for nn 1 X nnXX each n g ގ. Because Px-null sets are ² Px, x :-null, for all x g X ,it ϱ follows that Ä4xnns1 is a countable separating set. SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 207

g ގ ␰ Ž k. ϱ : Ž.ii By hypothesis, for each n , there exist elements Ä nk4 s1 wxX g ⌺ ␰ Ž k. g ގ X Px such that E is Pxn-null iff E is ² Pxn, n :-null, for all k . n ϱ g ⌺ Because Ä4xnns1 is countably separating it follows that E is P-null iff ␰ Žk.Žg ގ ␰ k. g X E is ²:Pxnn, -null, for all k, n ; here n X is any element agreeing with ␰ Žk. on wxX . Accordingly, P is countably determined. nPxn Proposition 1.2 implies that P is closed. Ž.iii Follows from partsŽ. i and Ž. ii and Proposition 1.4. Proposition 2.3Ž. ii is a generalization ofw 22; Theorem 5Ž. iix , where the hypotheses are significantly stricter: it is assumed there that X is separa- bleŽ. which implies the existence of a countable separating set and quasi- complete and that each cyclic space is metrizable for some weaker lcH- topologyŽ which implies that each measure Px: ⌺ ª X is countably determined by Proposition 1.4. . It is also assumed that P is equicontinuous. Propositions 1.3 and 1.4 show that Proposition 2.3Ž. ii has some generality. Indeed, the hypotheses of Proposition 2.3Ž. ii are satisfied whenever P has a countable separating set and, for each x g X, either one of the following three conditions is satisfied. wxX ␴ wxX wx Ž.a X Px is Ž X Px, X Px.-separable. wx␳ wx Ž.b X Px is metrizable for some lcH-topology x on X Px which is weaker thanŽ. or equal to the relative X-topology. wx ␳ Ž.c X Px is metrizable for some lcH-topology x consistent with the wx wxX duality Ž X Px, X Px.. ⌺ ª COROLLARY 2.3.1. Let X be a lcHs and let P: LsŽ. X be a spectral measure with a countable separating set. If P is purely atomic and, for each g wx ᎐ x X, the subspace XPx satisfies either of the conditions Ž.Ž.a c, then P has at most countably many atoms. Proof. Propositions 2.3Ž. ii , 1.3, and 1.4 imply that P is a closed measure. An examination of the proof of Proposition 2.2 then shows that the ϱ proof remains valid if the countable dense set Ä4xnns1 there is replaced by a countable separating set for P. ⌺ ª COROLLARY 2.3.2. Let X be a metrizable lcHs and let P: LXbesŽ. a spectral measure with a countable separating set. Ž.i P is closed and equicontinuous. Ž.ii If P is purely atomic, then it is actually ␴-atomic. Proof. As seen in the proof of Corollary 2.2.1 the metrizability of X ensures that P is equicontinuous. The closedness of P follows from Propositions 1.4 and 2.3Ž. ii . This establishes Ž. i . Part Ž. ii is immediate from Corollary 2.3.1. 208 OKADA AND RICKER

Because every spectral measure acting in a separable lcHs has a countable separating set it is clear that Corollaries 2.3.1 and 2.3.2 are general- izations of Propositions 2.2Ž. ii and Corollary 2.2.1, respectively. Examples are given later in this section to show that these extensions are genuine.

COROLLARY 2.3.3. Let X be the strict inducti¨e limit of a sequence ϱ Ä4Xkks1 of metrizable lcH-spacesŽ i.e., the topology of Xk is induced by that of s ⌺ ª Xkq1 and Xkk is closed in X q1, for k 1, 2, . . . .Žand P: LXbeas . spectral measure with a countable separating set. Ž.i P is equicontinuous and countably determined. In particular, Pisa closed measure. Ž.ii If P is purely atomic, then it is actually ␴-atomic. Proof. Ž.i Because X is quasi-barrelled,wx 10; p. 368 , P is equicontinu- ϱ g ގ ous. Let Ä4xnns1 be a countable separating set for P.Fix n . The range ⌺ ª of the vector measure Pxn: X is a bounded subset of X and so there g ގ ⌺ : wx is knŽ. such that PxnkŽ. X Žn., 10; p. 223 . Moreover, the relative ␳ topology that X induces on XkŽn. is the given metrizable topology kŽ n. in wx XkŽ n., 10; p. 222 . So, we may consider Pxn as a vector measure in ␳ s ␳ Ž.XkŽn., kŽn. . By Proposition 1.4 applied to Y Ž.XkŽn., kŽ n. it follows ␰ Žn. ϱ : X g ⌺ that there exists a sequence Ä rr4 s1 XkŽn. such that E is Pxn-null ␰ Žn. g ގ ␳ iff E is ² Pxnr, :Ž-null, for each r . Because XkŽn., kŽ n..has the ␰ Ž n. g X relative topology from X there exist elements r X whose restriction ␰ Žn. g ގ ϱ to XkŽn. coincide with rnn, for each r . Because Ä4x s1 is a countable separating set it is routine to check that E g ⌺ is P-null iff E is ␰ Žn. g ގ ²:Pxnr, -null for all r, n . This shows that P is countable determined and hence, is also a closed measureŽ. cf. Proposition 1.2 . Ž.ii By part Ž. i , P is a closed measure. Fix x g X. Then the relative ␯ wx topology xPof X restricted to X xhas the property that bounded wx ␯ subsets of X Pxare x-metrizable. This follows from the facts that wx bounded subsets of X Px are bounded in X, that a bounded subset of X g ގ ␯ is contained in Xkxfor some k , and the fact that restricted to l wx ␳ XkPxX is the metric topology kkin X . The conclusion then follows from Proposition 2.2. We note, in the notation of Corollary 2.3.3 that, for every x g X, there G wx: exists some k 1 such that the cyclic space X PxX k. Indeed, by the ⌺ : G proof of Corollary 2.3.3Ž. i we have PxŽ. Xk , for some k 1, and ⌺ : hence, also the linear span spÄ PxŽ.4 Xkk. Because X is closed in X and wx X induces on XkkPthe given topology of X it follows that X x s ⌺ : spÄ PxŽ.4 Xk . This property may fail if the increasing sequence Ä4XkkG1 fails to satisfy the hypothesis that Xkkis closed in X q1, for all k G 1. SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 209

To see this, let ރގ be the Frechet´ space consisting of the vector space of all ރ-valued functions on ގ equipped with the topology of pointwise ގ g ގ s ␰ g ރގ ϱ <<␰ k - ϱ convergence on . For each k ,let XknÄ ; Ý s1 n 4 be ރގ : equipped with the relative topology from . Then XkŽ1. XkŽ2. whenever - s Dϱ kŽ.1 k Ž.2 and the inclusion is proper. Let X ks1 Xk be equipped with the inductive limit topology. For each E g ⌺ s 2ގ define PEŽ.g s ␹ g g xл ⌺ LXŽ.by PEx Ž. Enx,for x X.Fix x X and let E in . Choose g ގ g g any k such that x Xkn, in which case also PEŽ. x Xkfor all g ގ ª n . Clearly PEŽ.nk x 0in X . Because the topology of X restricted ª to Xkkis the metric topology on X we have PEŽ.n x 0in X. This shows ␴ that P is -additive in LXsŽ.. Because the multiplicativity of P is clear we g s y2 see that P is a spectral measure. Let x X1 be the vector with xn n , g ގ ރގ for each n . Clearly each standard basis vector er of Žwith 1 in position r and 0 elsewhere.Ž belongs to the linear span of Px ⌺.and hence, wx g ގ ␰ g ␰ Žn. s n ␰ belongs to X Px, for each r . Given any X let Ýrs1 rre , g ގ g ގ ␰ g ␰ Žn. ª ␰ for n . Pick any k such that Xk . Then clearly for the metric topology in Xk ; because this is the relative topology from X it ␰ Žn. ª ␰ wxs wx follows that in X. Accordingly, X Px X and so X Px is not g ގ contained in Xj for any j . ⌺ ª g Given a spectral measure P: LXsŽ.and a vector x X, we recall that the cyclic space generated by x is the closure in X of the linear hull of Ä PExŽ.; E g ⌺4 and is denoted by PŽ.⌺ wxx . It is clear that always wx: ⌺ wx X Px PŽ.x ; this inclusion may be strict. w x EXAMPLE 2.4. Let X denote the space ރ 0, 1 of all ރ-valued functions defined on ⍀ s wx0, 1 equipped with the lcH-topology of pointwise convergence on ⍀. Let ⌺ denote the Borel subsets of ⍀. Define an equicontinu- ⌺ ª ␸ ¬ ␹␸ g ⌺ ous spectral measure P: LXsEŽ.by PE Ž.: , for each E and ␸ g X. Then the constant function | Ž.on ⍀ has the property that ⌺ wx| s w x PŽ. X. However, X P| is the proper subspace of X consisting of all Borel measurable functions. Under certain conditions the phenomenon of Example 2.4 cannot occur. ⌺ ª PROPOSITION 2.5. Let X be a lcHs and let P: LsŽ. X be an equicon- ¨ wx tinuous spectral measure. If x is a ector from X such that XPx is ⌺ ª wxs sequentially complete and Px: X is a closed measure, then X Px ⌺ wx w x PŽ.x . In particular, XPx is then actually a completeŽ. hence, closed 1 wxs ⌺ wx subspace of X and IPx is an isomorphism of L Ž.Px onto X Px PŽ.x . Proof. An examination of the proof ofwx 3; Proposition 2.1 shows that

the part of the proof which establishes that IPx is an isomorphism onto its range does not require either P to be a closed measure or X to be quasi-completeŽ a standing hypothesis inwx 3. . The only point to be checked is the use of Proposition 1.7Ž. iv fromwx 3 which is our formulaŽ. 5 . But, we 210 OKADA AND RICKER noted in Lemma 1.7Ž. iv that Ž. 5 is still valid without the completeness wx 1 ª assumptions of 3 . So, IPx: L Ž.Px X is an isomorphism onto its range. wx 1 By 23; Theorem 2 we know that L Ž.Px is complete and hence, the 1 wx range IPxŽ L Ž..Px is a complete subspace of X. Because X Px is the 1 wx wxs sequential closure of IPxŽ L Ž..Px , 12; p. 347 , it is clear that X Px 1 wx IPxŽ L Ž..Px . In particular, X Px is closed. Because the linear span of ⌺ wx ⌺ wxs w x PxŽ.is contained in X Px it follows that PŽ.x X Px. wx Remark 2.6.Ž. i In Example 2.4, the space X P| is certainly sequentially complete. However, the measure P|: ⌺ ª X is not closed. Ž.ii Proposition 2.5 is a considerable strengthening ofw 3; Proposition 2.1x because P itself is not assumed to be closedŽ. just Px and X is not wx assumed to be quasi-completeŽ just X Px is required to be sequentially complete, which does not imply X is sequentially completew 15; Example 2.5x. . Recall that x g X is called a cyclic ¨ector for the spectral measure P: ⌺ ª ⌺ wx LXsŽ.if the cyclic space PŽ.x equals X. The following result is a slight extension of Theorem 5Ž. iii inwx 22 where it is assumed that P is equicontinuous and that X is quasi-completeŽ a standing hypothesis throughoutwx 22. . An examination of the proof given inwx 22 shows that these conditions are not needed for that part of the proof. ⌺ ª PROPOSITION 2.7. Let X be a lcHs and let P: LsŽ. X be a spectral measure. If x g X is a cyclic ¨ector for P such that Px: ⌺ ª X is a closed measure, then P is also a closed measure. Proposition 2.7 has some generality because there are large classes of lcH-spaces in which every vector measure is automatically a closed measure,wx 20 . A useful property to look for when checking for the closedness of a ⌺ ª spectral measure P: LXsŽ.is the existence of a countable separating set. The following notion, which in practice is somewhat easier to determine, implies the existence of a countable separating set. Namely, P is called countably ⌺-dense if there exists a countable set ⌳ : X with the property that the linear hull of Ä PExŽ.; E g ⌺, x g ⌳4 is dense in X.Itis classical that this property implies the closedness of spectral measures in Banach spaces,wx 7; XVII, Lemma 3.21 . For the case of Frechet´ spaces we refer tow 22; Theorem 5Ž. ix . The next result shows that these classical cases guaranteeing the closedness of spectral measures can be considerably extended when countable ⌺-denseness is combined with the conditions Ž.quite general of Proposition 2.3 Ž. ii , Corollaries 2.3.1, and 2.3.3. ⌺ ª PROPOSITION 2.8. Let X be a lcHs and let P: LsŽ. X be a spectral measure which is countably ⌺-dense. Then P has a countable separating set. SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 211

Proof. Let ⌳ : X be a countable set such that the linear hull Z of Ä PExŽ.; E g ⌺, x g ⌳4 is dense in X. Then ⌳ is a countable separating set for P. To see this suppose that F g ⌺ satisfies PFxŽ.s 0 for all g ⌳ g s k ␣ ␣ g ރ g ⌺ x .If z Z then z Ý js1 jjjPEŽ. x for elements j, E j , g ⌳ F F and x j ,for1 j k. Then the multiplicativity of P implies that s k ␣ l s PFzŽ. Ý js1 jjPEŽ.Fx j0. The density of Z in X then implies that PFŽ.s 0.

Recall that a B.a. M : LXŽ.is called Bade ␴-complete if it is ␴-complete as an abstract B.a. and Ž.H BX␣ s F␣␣BX and ŽE␣␣BX . s D ; : spŽ ␣ BX␣␣4Äfor every countable family B 4M. A B.a. M LXŽ.is called countably decomposable if every pairwise disjoint family of elements from M is at most countable. PROPOSITION 2.9. Let X be a lcHs and let M : LŽ. X be a Bade ␴-complete B.a. which is countably decomposable. Then M is Bade complete. Proof. The countable decomposability implies that every set in M has a least upper bound which is also the least upper bound of a countable wx subset, 6, IV, Lemma 1.5 . So, M is complete as an abstract B.a. Let Ä4B␣ ␣ g A be a family of projections from M. Then there exist ␣ g ގ ␤ g ގ countable subsets Ä4Ä4nm; n and ; m ,of A, such that ϱ ϱ E␣ g B␣ s E s B␣␣and H g B␣ s H s B␤ . The Bade ␴-complete- A n 1 nmA m 1 ϱϱ ness of implies that Ž.E s BX␣ s spÄ4D s BX␣ . Accordingly, M n 1 nnn 1

ϱϱ spÄ4D␣ g BX␣␣: E g BX␣ s E s BX␣ s sp D s BX␣ A Ž.A Ä4Ä4n 1 nnn 1 : D spÄ4␣ g A BX␣ , E s D ␴ which shows that Ž.␣ g A BX␣␣spÄ4g A BX␣ . The Bade -complete- ϱ ϱ ness of also implies that ŽH s BX␤ . s F s BX␤ . It follows that M m 1 mmm 1

ϱ ϱ F␣ g BX␣ : F s BX␤ s H s BX␤␣s H g BX␣ A m 1 mmŽ.m 1 Ž.A : F ␣ g A BX␣ , H s F which shows that Ž.␣ g A BX␣␣g A BX␣ . Hence, M is Bade complete. ⌺ ª COROLLARY 2.9.1. Let X be a lcHs and let P: LsŽ. X be a spectral measure. If P is equicontinuous and if PŽ.⌺ is countably decomposable, then P is a closed measure.

Proof. The properties of a spectral measure ensure that M s PŽ.⌺ is a Bade ␴-complete B.a. So, Proposition 2.9 implies that M is Bade complete. It follows fromwx 15; Proposition 3.5 that P is a closed measure. 212 OKADA AND RICKER

LEMMA 2.10. Let X be a metrizable lcHs and let M : LŽ. X be a Bade ␴-complete B.a. with a countable separating set. Then M is equicontinuous, Bade complete, and countably decomposable. wx Proof. M is equicontinuous by 27; Proposition 1.2 and so M s PŽ.⌺ ⌺ ª wx for some spectral measure P: LXsŽ., 15; Proposition 4.1 . The ϱ hypotheses on M imply that P has a countable separating set, say Ä4xnns1. Then Corollary 2.3.2 implies that P is a closed measure and hence, M is Bade complete,wx 15; Proposition 3.4 . : ⌺ s Suppose Ä4B␣ M is a maximal disjoint system, in which case ␣␣B I ⌺ s g ގ in LXsŽ.. In particular, ␣␣Bxnnx for each n . The metrizability of s ␣ f X implies that there is a countable set An with Bx␣ nn0 for all A . ␣ fDϱ s s So, if ns1 An, then Bx␣ n 0 for all n and it follows that B␣ 0. ⌺ ª PROPOSITION 2.11. Let X be a metrizable lcHs and let P: LXbesŽ. a spectral measure. Then P has a countable separating set iff P is countably decomposable. In particular, e¨ery countably decomposable P is necessarily a closed measure. Proof. It is routine to check that the spectral measure properties of P imply that M s PŽ.⌺ is a Bade ␴-complete B.a. As seen in the proof of Corollary 2.2.1 the metrizability of X implies that P is equicontinuous. Suppose P has a countable separating set. Then P is countably decomposable by Lemma 2.10. Conversely, suppose that P is countably decomposable. Proposition 2.9 implies that M is Bade complete. Given x g X, its associated carrier s H g s projection Cxxcan then be defined by the formula C Ä4B M; Bx x ; the Bade completeness of M ensures that Cx exists and belongs to M. Let s Ä4 C Cx␣ be a maximal disjoint family of carrier projections. Countable ϱ decomposability of M implies that C is a countable family, say Ä4CxŽn. ns1. ϱ The claim is that Ä xnŽ.4ns1 is a countable separating set for M. To see this g s g ގ s suppose that B M satisfies BxŽ. n 0 for all n . Now, BCxŽ n. xŽ. n s g ގ y s BxŽ. n 0 for all n , which implies that ŽI BC .xŽ n. xn Ž.xn Ž.,for g ގ y G all n . By the definition of carrier projections we have Ž.I BCxŽ n. s g ގ Eϱ s CxŽn. and so BCxŽn. 0 for all n . But, by maximality ns1 CxŽn. I s Eϱ s and hence, B ns1 BCxŽn. which implies that B 0. So, P has a countable separating set. The statement concerning the closedness of P now follows from Corol- lary 2.3.2. Remark 2.12. The same proof as that of Proposition 2.11 shows that any spectral measureŽ. in an arbitrary lcHs X which is countably decomposable necessarily has a countable separating set; neither metrizabilty of X nor equicontinuity of P is needed in this case. For the converse implicationŽ. i.e., countable separating set « countably decomposable SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 213 the metrizability condition can be relaxed somewhat, e.g., replaced by conditions such asŽ. i and Ž ii . in Proposition 2.2 and the requirement that P be closed and equicontinuous. However, it is not possible to omit all such conditions completely. To see this let X s ރw0, 1x be the lcHs of Example 2.4. Consider the same spectral measure P as given in Example 2.4 but now having domain ⌺ consisting of all subsets of ⍀ s wx0, 1 . Then P is closed, equicontinuousŽ. as X is barrelled , and P certainly has a countable separating set as X is separable. However, because P has uncountably many atoms it cannot be countably decomposable. The following result is a slight extension of Proposition 2.11.

PROPOSITION 2.13. Let X be the strict inducti¨e limit of a sequence of ϱ ⌺ ª metrizable lcH-spacesÄ4 Xkks1 and let P: LsŽ. X be a spectral measure. Then P has a countable separating set iff it is countably decomposable. In particular, e¨ery countably decomposable P is necessarily a closed measure. Proof. Because X is quasi-barrelled P is equicontinuous. By Remark 2.12 it follows that if P is countably decomposable, then it has a countable separating set. ϱ Conversely, suppose that Ä4xnns1 is a countable separating set for P. Let Ä4B␣ : PŽ.⌺ be a maximal disjoint family in which case ⌺␣␣B s I in

LXsŽ.; note that P is a closed measure by Corollary 2.3.3. In particular, ⌺ s g ގ g ގ ␣ Bx␣ nnx for each n . Now fix n . Arguing as in the proof of ⌺ : Corollary 2.3.3 we have that PxnkŽ. X Žn., for some integer knŽ., and the relative X-topology in XkŽn. is precisely the metric topology of XkŽn.. ⌺ s Hence, the series ␣ Bx␣ nnx converges in the metrizable space XkŽ n. s ␣ f and so there exists a countable set An such that Bx␣ nn0 for all A . s ␣ fDϱ It follows that B␣ 0 for all ns1 An and hence, Ä4B␣ is a countable set.

Let M : LXŽ.be a B.a. Then an element x g X is called a separating ¨ector for M if B s 0 whenever B g M satisfies Bx s 0. We point out that Proposition 2.7 remains valid if x g X there is merely a separating vector instead of a cyclic vector. It is clear that a cyclic vector is always a separating vector but not conversely; consider the B.a. in X s ރ 3 generated by the projections,

100 000 s s P01000 and P 010. 000 001

It is also clear that if M has a separating vector, then it has a countable separating set. The following example shows that the converse is generally not true. 214 OKADA AND RICKER

EXAMPLE 2.14. Let X denote the normed linear subspace coo of the ␰ g Banach space coo, where elements c ohave only finitely many nonzero coordinates. For each E g ⌺ s 2ގ define a projection PEŽ.g LX Ž .by ␰ s ␹␰ ⌺ ª PEŽ.E Žcoordinatewise multiplication. . Then P: LXsŽ.is a ϱ : spectral measure and the family of standard basis vectors Ä4enns1 X, where en has a 1 in position n and has a 0 elsewhere, is a countable separating set. However, it is clear that P does not have a separating vector.

It is precisely the lack of completeness of coo in Example 2.14 which causes the phenomenon exhibited there. For Frechet´ spaces the following result shows that this cannot occur and that the criterion of Proposition 2.11 can be significantly simplified.

PROPOSITION 2.15. Let X be a Frechet´ space. Then a spectral measure P: ⌺ ª ¨ LsŽ. X is countably decomposable iff it has a separating ector. Proof. If P has a separating vector x, then it has a countable separating set, namely, Ä4x , and so P is countably decomposable by Proposition 2.11. Conversely, suppose that P is countably decomposable. Arguing as in ϱ : the proof of Proposition 2.11 there is a countable set Ä xnŽ.4ns1 X such ϱ that the carrier projections Ä4CxŽn. ns1 form a maximal disjoint family. Because X is metrizable there exist numbers ␣Ž.n ) 0 such that ␣ ϱ s Ä Ž.Ž.nxn4ns1 is a bounded subset of X. The property C␣ xxC , valid for g ␣ ) ϱ any x X and 0, shows that we may assume that Ä xnŽ.4ns1 itself is a bounded set. Then the completeness of X ensures that the series ϱ yj ␰ ␰ Ý js12 xjŽ.converges to some element of X,say . The claim is that is a separating vector for P. Suppose that B g PŽ.⌺ satisfies B␰ s 0. By disjointness of the carrier projections it follows, for each n g ގ, that

ϱϱ ␰ s yj s yj s yn CxŽn. ÝÝ2 CxjxŽn. Ž. 2 CCxjxŽn. xŽ j. Ž. 2 CxnxŽ n. Ž.. js1 js1

g ގ s nn␰ s ␰ s It follows, for each n , that BCxŽn. xŽ. n 2 BCxŽ n. 2 CBxŽ n. 0. Arguing as in the proof of Proposition 2.11 we have B s 0. This establishes that ␰ is a separating vector for P.

COROLLARY 2.15.1. E¨ery spectral measure in a separable Frechet´ space has a separating ¨ector. Proof. Separability implies that any spectral measure in such a space has a countable separating set and so is countably decomposable by Proposition 2.11. The conclusion then follows from Proposition 2.15. SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 215

For the Banach space setting Proposition 2.15 and Corollary 2.15.1 are knownwx 17; Lemma 2 . The next example shows that Proposition 2.13 does not have an extension along the lines of Proposition 2.11Ž to Proposition 2.15 and Corollary ϱ 2.15.1. . The reason is, for a given sequence Ä4xnns1 contained in an ␣ ) inductive limit space it is generally not possible to find numbers n 0 ␣ ϱ such that Ä4nnnx s1 is a bounded set; this property was crucial in the proof of Proposition 2.15. g ގ ރގ EXAMPLE 2.16. For each k ,let Xk be the vector subspace of consisting of those elements x: ގ ª ރ such that Än; xnŽ./ 04 : F - ϱ g ގ Ä41, 2, . . . , k . Fix any 1 p . For each k ,let Xk be equipped with p Ž. the norm from l so that Xk is a finite-dimensional Banach space . s Dϱ Then equip X ks1 Xk with the strict inductive limit topology, in which case X is separable, barrelled, and complete. Although X is not metrizable, its bounded sets are metrizable for the relative X-topology. Let ⌺ s ގ ⌺ ª s ␹ 2 and define a spectral measure P: LXsEŽ.by PEx Ž. x, for x g X and E g ⌺. The separability of X implies that P has a countable separating set. But it is clear that P does not have a separating vector. Our final criterion is the following result. For partŽ. i we refer tow 15; Proposition 3.9x . PartŽ. ii follows from the fact that ⌺ Ž.P is a ␶ Ž.P -closed 1 wx subset of L Ž.P ; see 15, Section 1 . ⌺ ª PROPOSITION 2.17. Let X be a lcHs and let P: LsŽ. X be a spectral measure. Either of the following conditions is sufficient for P to be a closed measure. X Ž.i There exists a localizable measure ␭: ⌺ ª wx0, ϱ such that² Px, x : X X g ␭, for each x g X and x g X . 1 Ž.ii The lcHs ŽL Ž.P , ␶ Ž..P is complete. w x Remark 2.18. We point out that if LXsPŽ. is sequentially complete 1 and if P is a closed measure, then necessarily L Ž.P is ␶ Ž.P -complete, wx23; Theorem 2 . However, it is not true in general that P closed implies 1 L Ž.P is ␶ Ž.P -complete. Indeed, let ⌺ denote the ␴-algebra of all Lebesgue measurable subsets of ⍀ s wx0, 1 and let X s simŽ.⌺ be the Ž normed . 1 wx subspace of the Banach space L Ž 0, 1.Ž. . For each E g ⌺ define PE␸ s ␹␸ ␸ g ⌺ ª Es,for X, in which case P: LXŽ.is a spectral measure; because 5 PEŽ.5 F 1 for all E g ⌺ it is clear that P is equicontinuous. It 1 s ⌺ turns out that L Ž.P sim Ž., as a vector space, and H⍀ fdP is the 1 1 operator in X of multiplication by f, for each f g L Ž.P .If L Ž.P is 1 complete, then so is IP Ž L Ž..P by Proposition 1.5 and it follows that w x s 1 wx wx LXsPPŽ. I Ž L Ž..P , 12; p. 347 . In particular, LXsPŽ. is complete 216 OKADA AND RICKER

ϱ 1 wx and so L Ž.P : L Ž.P , 11; Proposition 2.2 , which is a contradiction as 1 ϱ ϱ wx 1 L Ž.P s sim Ž.⌺ and L Ž.P s L Ž 0, 1. . Hence, L Ž.P cannot be com- pleteŽ. or even sequentially complete by the same argument . We now consider four examples which illustrate some of the various connections between cyclic vectors, separating vectors, separable spaces, countable ⌺-denseness and countable separating sets.

EXAMPLE 2.19. Cyclic vectors can exist in nonseparable spacesŽ even Hilbert spaces.Ž. . To see this, let ⍀, ⌺, ␮ be a nonseparable probability p measure. Then, for each 1 F p - ϱ, the space X s L Ž.␮ is a nonseparable Banach space. For each E g ⌺,let PEŽ.g LX Ž .be the operator in ␹ ⌺ ª X of multiplication by Es. Then P: LXŽ.is a spectral measure and the function | Ž.constantly equal to 1 on ⍀ is a cyclic vector for P.

EXAMPLE 2.20. We have seenŽ. cf. Corollary 2.15.1 that every spectral measure P in a separable Frechet´ space necessarily has a separating vector. Of course, it does not follow that P has a cyclic vector; see the comments prior to Example 2.14. Because cyclic vectors are separating vectors it is clear from Example 2.19 that separating vectors may also exist in nonseparable spaces. We now give an example in a nonseparable space where P has a separating vector but no cyclic vector. Let Ž.⍀, ⌺, ␮ be a nonseparable probability measure. For each n g ގ, s 2 ␮ let XnnL Ž.in which case X is a nonseparable Hilbert space. Let s [ϱ X ns1 Xn be the direct sum Hilbert space. Then X is nonseparable g s ϱ g and elements x X are sequences x Ž.xnns1, with xnnH , such that 55s ϱ 552 1r2 - ϱ g ⌺ g x ŽÝns1 xn . . For each E , define PEŽ.LX Ž .by s ␹ ϱ s ϱ ⌺ ª PExŽ. ŽEnnx .s1 for each x Ž.xnns1. Then P: LXsŽ.is a s y1| g ގ spectral measure. If xn n , for each n then it is routine to check s ϱ that x Ž.xnns1 is a separating vector for P. ␰ s ␰ ϱ g The claim is that P has no cyclic vectors. Suppose that Ž.nns1 X ␣ g ރ is a cyclic vector. Then, for any finite choice of elements j and k w k x ϱ E g ⌺,for1F j F k, we have Ý s ␣ PEŽ.␰ s ŽÝ s ␣␹ s ␰ . s . This jj1 jj j1 jEj nn1 shows that X is the closure of its vector subspace consisting of all ␸␰ ϱ ␸ g ⌺ elements of the form Ž.nns1 where simŽ.. Because each natural ␲ ª coordinate map nn: X X is a continuous surjection it follows that Xn ␸␰ ␸ g ⌺ g ގ is the closure of Ä n; simŽ.4, for each n . In particular, there g ⌺ ␮ s < ␰ < ) g exists a set Ennwith Ž.E 1 such that nŽ.w 0 for all w E n. s Fϱ ␮ s ␰ / Then E ns1 En satisfies Ž.E 1 and has the property that Ž.w 0 for every w g W. Now, the vector x s Ž.|, 0, 0, . . . belongs to X and so ␸ g ⌺ ␸␰ ϱ ª there exist elements kksimŽ.such that Žn .ns1 x as ª ϱ ␸␰ª | 2 ␮ ␸␰ª 2 ␮ k . In particular, k 1 in L Ž.and k 2 0in L Ž..By SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 217

␸␰ª | passing to a subsequence if necessary we may assume also that k 1 ␸␰ª ␮ ⍀ ␰ ␰ and k 2 0 pointwise -a.e. on . Because both 12and are nonzero on a set of full ␮-measure this is impossible. Accordingly, P has no cyclic vectors. It was noted in Proposition 2.8 that a spectral measure which is countably ⌺-dense necessarily have a countable separating set. The next example shows that these two notions are not equivalent, even in Hilbert spaces. We point out that Example 2.20 does not exhibit this phenomenon ⌺ g because P is countably -dense there. Indeed, if en X denotes the element which has | in position n and the zero function in all other g ⌺ g ގ coordinates, then it is easy to check that spÄ PEeŽ.n; E , n 4 is dense in X.

EXAMPLE 2.21. Let Y be any Banach space and let Ž.⍀, ⌺, ␮ be a wx p Lebesgue measure on ⍀ s 0, 1 . Fix 1 F p - ϱ and let X s L Ž.␮, Y be the Banach space of all Bochner ␮-integrable functions f: ⍀ ª Y such that

1rp 55s p ␮ - ϱ f X H fwŽ.Y d Ž.w ; ž/⍀ seewx 1; Chap. IV . For each E g ⌺ define PEŽ.g LX Ž .to be the g ¬ ␹ projection whose action on f X is given by PEfŽ.: w E Ž.Ž.wfw,for ␮-a.e. w g ⍀. ⌺ ª Ž.i The set function P: LsŽ. X is a spectral measure.Itis 5 5 p routine to check that P is multiplicative. Moreover, because PEfŽ. X s 5 5 p ␮ 5 и 5 p g 1 ␮ ¬ HE fwŽ.YY d Ž.w with fŽ. L Ž.it is clear that E PEfŽ. is ␴-additive, for each f g X; i.e., P is a spectral measure. ¨ g ␸ Ž.ii P has a separating ector. Fix any nonzero vector y Y. Let y: ⍀ ª Y denote the constant function which takes the value y at each point ⍀ ␸ g 55␸ s 55 g ⌺ ␸ of . Then yyX and XYy . Suppose that E and PEŽ. y s ¬ ␹ ␸ s ␹ s ␮ g ⍀ 0, that is, w EyŽ.w Ž.w E Ž.wy 0, for -a.e. w . Because / ␮ s s ␸ y 0 it follows that Ž.E 0 and so PEŽ. 0. Hence, y is a separating vector. Ž.iii P is a closed measure. This follows from Ž. ii and Proposition 2.11. Ž.iv X is nonseparable whene¨er Y is nonseparable. For each y g Y 55␸ s 55 ⌳ ª we have y XYy . Accordingly, the linear map : Y X given by ⌳ s ␸ g y y ,for y Y, is an isometry of Y onto a closed subspace of X. Because every closed subspace of a separable Banach space is itself separable the conclusion follows. 218 OKADA AND RICKER

Ž.v X can be chosen to be a nonseparable Hilbert space. Indeed, X is a Hilbert space whenever Y is a Hilbert space and p s 2. ByŽ. iv it then 2 suffices to choose Y nonseparable. For instance, p s 2 and Y s l Žwx0, 1 . will do. ⌺ ϱ Ž.vi P is not countably -dense if Y is nonseparable. Let Ä4fnns1 be g ގ ⍀ ª any countable set in X.Fix n . Because fn: Y is strongly ␮-measurable it follows as a consequence of the Pettis measurability wx theorem, 1; p. 42 , that there is a closed, separable subspace Yn of Y such g ␮ g ⍀ that fwnnŽ. Y for -a.e. w . Then the closed subspace Z of Y Dϱ ␮ generated by ns1 Yn is separable and there exists a -null set F such g g ގ f that fwnŽ. Z for every n and w F.Itfollowsthat w k ␣ x g f Ý js1 jjnPEŽ. fŽ j. Ž. w Z for each w F and any choice of finitely many ␣ g ރ g ⌺ g ގ s elements jj, E , and njŽ. . So, each element of W g ⌺ g ގ spÄ PEfŽ.n; E , n 4 takes all of its values in Z, except at points g g ϱ : w F. Suppose g W, in which case there exists a sequence Ä4gkks1 W ª ␮ ⍀ s 55␮ F such that gk g in X. Because Ž. 1 it follows that H⍀ fd 55 g ª 1 ␮ f X , for every f X, and hence, gk g in L Ž., Y . In particular, ␮ ª ␮ g ⌺ ␮ g g ⌺ HEkgd H Egd for all E . Because HEkgd Z for all E and g ގ wx ␮ g k , 1; p. 48 Corollary 8 , it follows that also HE gd Z for every E g ⌺. Thenwx 1; p. 49 Theorem 9 implies that gwŽ.g Z for ␮-a.e. w g ⍀. So, each element of W takes ␮-a.e. value in Z. Accordingly, each y g Y _ Z ␸ g ␸ f / has the property that yyX but W, that is, W X. Because the ϱ countable set Ä4fnns1 was arbitrary it follows that P is not countably ⌺-dense. Ž.vii If Y is nonseparable, then P has no cyclic ¨ectors. This is immediate fromŽ. vi because the existence of a cyclic vector always implies countable ⌺-denseness. Finally, we exhibit an example of a closed spectral measureŽ in Hilbert space. which satisfies none of the sufficient conditions of this section.

2 wx EXAMPLE 2.22. Let X be the nonseparable Hilbert space l Ž 0, 1. and let ⌺ be the family of all subsets of ⍀ s wx0, 1 . For each E g ⌺ define g s ␹ g PEŽ.LX Ž .by PEx Ž. E x,for x X. Then P is a spectral measure. Moreover, because counting measure ␭: ⌺ ª wx0, ϱ is a localizable mea- X XX sure satisfying ² Px, x : g ␭, for each x g X and x g X , it follows that P is a closed measure; see Proposition 2.17. However, because all elements x g X have the property that Äw g ⍀; xwŽ./ 04 is a countable set, it is clear that P has no cyclic vector, no separating vector and, indeed, no countable separating set. In particular, P cannot be countably ⌺-dense either. Finally, because P has no countable separating set it cannot be countably determinedŽŽ.. cf. Proposition 2.3 i . SPECTRAL MEASURES AND BOOLEAN ALGEBRAS 219

We conclude this section with a summary ofŽ. most of the essential results of this section in a format which should provide criteria general enough to decide about the closedness of most spectral measures likely to occur in practice. Not all of the criteria are independent of one another. ⌺ ª PROPOSITION 2.23. Let X be a lcHs and let P: LsŽ. X be a spectral measure. Each of the following conditions ensures that P is a closed measure.

Ž.i P is countably determined. 1 Ž.ii L Ž.Pis␶ Ž.P -complete. X Ž.iii There is a localizable measure ␭: ⌺ ª wx0, ϱ with² Px, x : g ␭, X X for x g X and x g X . Ž.iv Pis␴-atomic. Ž.v P is equicontinuous and countably decomposable. Ž.vi P has a countable separating set ⌳ : X such that each ëctor measure Px: ⌺ ª X, for x g ⌳, is countably determined. Ž.vii P is countably ⌺-dense and each ëctor measure Px: ⌺ ª X, for x g ⌳, is countably determined, where ⌳ : X is a countable set such that the linear hull ofÄ PŽ. E x; E g ⌺, x g ⌳4 is dense in X. Ž.viii P has a separating ëctor x and Px: ⌺ ª X is a closed measure. Ž.ix P has a separating ëctor x and Px: ⌺ ª X is countably determined. Ž.x P has a countable separating set and X satisfies either of the following four properties. Ž.I X is the strict inductië limit of a sequence of metrizable lcH-spaces. X X Ž.II Xis␴ Ž X , X .-separable. Ž.III There exists a metrizable lcH-topology on X which is weaker thanŽ. or equal to the giën topology. Ž.IV There exists a metrizable lcH-topology on X which is consistent X with the dualityŽ X, X .. Ž.xi P is countably ⌺-dense and X satisfies either of the properties Ž.I ᎐ ŽIV .in part Ž.x.

If X satisfies either of the properties Ž.I ᎐ ŽIV .in part Ž.x,then the conditions required of Px in each of Ž.Ž.vi ᎐ ix are automatically satisfied.

QUESTION.Ž. a Is the equicontinuity of P necessary in part Ž. v ? Ž.b Can the countably determined requirement of Px in each of Ž vi . , Ž.vii , and Ž. ix be replaced by closedness of Px? 220 OKADA AND RICKER

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