Extensions of Representable Positive Linear Functionals to Unitized Quasi *-Algebras: A New Method

In this paper we introduce a topological approach for extending a representable linear functional , defined on a topological quasi *-algebra without unit, to a representable linear functional defined on a quasi *-algebra with unit. In particular, we suppose that is continuous and the positive sesquilinear form , associated with , is closable and prove that the extension of the closure is an i.p.s. form. By we construct the desired extension.