A Representation Theorem for Positive Functionals on Involution Algebras View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

1966

AUTHORS

R. S. Bucy , G. Maltese

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. More... »

PAGES

364-367

Book

TITLE

Contributions to Functional Analysis

ISBN

978-3-642-85999-1
978-3-642-85997-7

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-642-85997-7_24

DOI

http://dx.doi.org/10.1007/978-3-642-85997-7_24

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1039800684


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