Extensions of hermitian linear functionals

Abstract We study, from a quite general point of view, the family all extensions positive hermitian linear functional $$\omega $$ ? , defined on dense *-subalgebra $${\mathfrak {A}}_0$$ A 0 topological *-algebra {A}}[\tau ]$$ [ ? ] with aim finding that behave regularly. The sole constraint we are dealing required to satisfy is their domain subspace $$\overline{G(\omega )}$$ G ( ) ¯ closure graph (these so-called slight extensions). main results two. first having characterized those elements {A}}$$ for which can find extension giving range possible values may assume these elements; second one proving existence maximal extensions. show as it apply in several contexts: Riemann integral, Infinite sums, and Dirac Delta.