Criteria for closedness of spectral measures and completeness of Boolean algebras of projections
type: Item
instance of: scholarly article
title: Criteria for closedness of spectral measures and completeness of Boolean algebras of projections
zbMATH Open document ID: 0919.46032
author: S. Okada
author: Werner J. Ricker
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-08-26T00:00:00Z
review text: Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of some non-empty set. Let \(Y\) be a locally convex Hausdorff space with dual \(Y'\). A \(\sigma\)-additive measure \(m: \Sigma \to Y\) is called closed if the set of images of \(\Sigma\) under \(m\) is complete with respect to a family of metric induced by the total variation measures \(| \langle m(\cdot)y, y'\rangle| \) with \(y \in Y\), \(y' \in Y'\). It is known that \(m\) is closed if it is countably determined in the sense that there is a sequence \(y_n'\), \(n = 1,2, \ldots\), such that \(\langle m(E),y_n'\rangle = 0\) for all \(n\) implies \(m(E) = 0\). In the present paper the authors confine their attention to the case where \(Y\) is the space \(L_s(X)\) of all continuous linear maps on a locally convex Hausdorff space \(X\), endowed with the strong operator topology, and \(m\) is a spectral measure, that is, a multiplicative measure \(P: \Sigma \to L_s(X)\). They investigate various sufficient conditions for closedness of a spectral measure \(P\). Here are some of such sufficient conditions: (a) there is a numerical measure \(\lambda\) on \(\Sigma\) such that every measure \(\langle P(\cdot)x,x'\rangle\) is absolutely continuous with respect to \(\lambda\), (b) \(P\) has a separating vector \(x\) such that the vector measure \(P(\cdot)x: \Sigma \to X\) is countably determined, (c) \(P\) has a countable separating set of vectors and there is a metrizable locally convex topology on \(X\) which is weaker than the given topology. Here a sequence \(x_n\), \(n = 1,2, \ldots\), is separating for \(P\) if \(P(E)x_n = 0\) for all \(n\) implies \(P(E) = 0\).
reviewed by: T. Ando
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 28B05
zbMATH DE Number: 1278867
zbMATH Keywords: spectral measures
zbMATH Keywords: Boolean algebras of projections
zbMATH Keywords: separating vector
zbMATH Keywords: countable separating set of vectors
MaRDI profile type: Publication
full work available at URL: jmaa.1998.6262
OpenAlex ID: W1983089107
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DOI: 10.1006/JMAA.1998.6262
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description: scientific article; zbMATH DE number 1278867