Positive linear functionals on BP*-algebras

This is an English translation of the original article written in FrenchInternational audienceLet $A$ be a BP*-algebra with identity $e,P_{1}(A)$ be the set of all positive linear functionals f on $A$ such that $f(e)=1,$ and let $M_{s}(A)$ be the set of all nonzero hermitian multiplicative linear functionals on $A.$ We prove that $M_{s}(A)$ is the set of extreme points of $P_{1}(A).$ We also prove that, if $M_{s}(A)$ is equicontinuous, then every positive linear functional on $A$ is continuous. Finally, we give an example of a BP*-algebra whose topological dual is not included in the vector space generated by $P_{1}(A),$ which gives a negative answer to a question posed by M. A. Hennings