Positive linear functionals on BP*-algebras

TL;DR

This paper characterizes the extreme points of positive linear functionals on BP*-algebras, explores conditions for their continuity, and provides a counterexample related to the dual space structure.

Contribution

It proves that the set of hermitian multiplicative linear functionals are the extreme points of positive linear functionals on BP*-algebras and addresses a question about dual space inclusion.

Findings

M_s(A) is the set of extreme points of P_1(A).

Equicontinuity of M_s(A) implies all positive linear functionals are continuous.

Counterexample shows the dual space may not be generated by P_1(A).

Abstract

Let 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.