Extreme Points of Positive Functionals and Spectral States on Real Banach Algebras

Extreme points of positive functionals and spectral states on real commutative Banach algebras are investigated and characterized as multiplicative functionals extending the well-known results from complex to real Banach algebras. As an application a new and short proof of the existence of the Shilov boundary of a real commutative Banach algebra with nonempty maximal ideal space is given.