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

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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 Criteria for Closedness of Spectral Measures and Completeness of Boolean Algebras of Projections 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 mea- sures. 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 .
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