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

A topological approach for extending a representable linear functional $\omega$, defined on a topological quasi *-algebra without unit, to a representable linear functional defined on a quasi *-algebra with unit, is introduced. In particular, we suppose that $\omega$ is continuous and the positive sesquilinear form $\vom$, associated to $\omega$, is closable and prove that the extension $\overline{\vom}^e$ of the closure $\overline{\vom}$ is an i.p.s. form. By $\overline{\vom}^e$ we construct the desired extension.