Extensions of hermitian linear functionals

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