Abstract
We study, from a quite general point of view, the family of all extensions of a positive hermitian linear functional omega , defined on a dense *-subalgebra {mathfrak {A}}_0 of a topological *-algebra {mathfrak {A}}[tau ], 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 overline{G(omega )}, the closure of the graph of omega (these are the so-called slight extensions). The main results are two. The first is having characterized those elements of {mathfrak {A}} for which we can find a positive hermitian slight extension of omega , 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.
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