 A Representation Theorem for Positive Functionals on Involution AlgebrasR. S. Bucy and G. Maltese A chapter in Contributions to Functional Analysis, 1966, pp 364-367 from Springer Abstract: Abstract D. Raikov has shown [6] that for a commutative Banach algebra A with symmetric involution, the set p of positive linear functionals on A having norm at most one is isometrically isomorphic to the set of positive measures (of norm at most one) defined on the maximal ideal space of A. Raikov’s proof of this theorem depends on the Gelfand theory of commutative Banach algebras and the Riesz-Markov Theorem (see also [8; p. 230]). Here we shall give a new and elementary proof of Raikov’s result by first proving a Radon-Nikodym type theorem for positive functionals (Theorem 1) and then showing directly that the extreme points of the compact convex set of positive linear functionals in the unit ball of A′ are exactly the set M of positive multiplicative linear functionals (Theorem 2). An application of the Krein-Milman Theorem makes possible the representation of every element of p as the centroid of a positive measure on M (Theorem 3) and uniqueness of this representation is a consequence of the Stone-Weierstrass Theorem. Keywords: Extreme Point; Maximal Ideal Space; Commutative Banach Algebra; Positive Functional; Positive Linear Functional (search for similar items in EconPapers) There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-85997-7_24 Ordering information: This item can be ordered from DOI: 10.1007/978-3-642-85997-7_24 Access Statistics for this chapter More chapters in Springer Books from Springer |
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