results
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.
2022-01-01T19:28
positive measure
measures
positive multiplicative linear functionals
maximal ideal space
functionals
involution
Radon-Nikodym type theorem
space
Riesz-Markov Theorem
uniqueness
algebra
set M
convex sets
ball
Banach algebra A
elements
applications
theory
Gelfand theory
norms
Raikov’s proof
1966-01-01
positive linear functionals
positive functionals
elementary proof
algebra A
commutative Banach algebra A
unit ball
centroid
chapter
type theorem
point
A Representation Theorem for Positive Functionals on Involution Algebras
proof
set P
https://scigraph.springernature.com/explorer/license/
Banach algebra
linear functionals
en
false
Krein-Milman theorem
ideal space
commutative Banach algebras
compact convex sets
extreme points
consequences
Raikov
1966
Stone-Weierstrass theorem
Involution Algebras
representation theorem
364-367
theorem
set
symmetric involutions
representation
multiplicative linear functionals
Raikov’s result
chapters
https://doi.org/10.1007/978-3-642-85997-7_24
G.
Maltese
Springer Nature - SN SciGraph project
Springer Nature
Pure Mathematics
College Park, Maryland, USA
College Park, Maryland, USA
Mathematical Sciences
Bucy
R. S.
10.1007/978-3-642-85997-7_24
doi
978-3-642-85997-7
978-3-642-85999-1
Contributions to Functional Analysis
dimensions_id
pub.1039800684