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

Abstract

In this paper we introduce 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. In particular, we suppose that ${\omega}$ is continuous and the positive sesquilinear form ${\varphi_\omega}$, associated with ${\omega}$, is closable and prove that the extension ${\overline{\varphi_\omega}^e}$ of the closure ${\overline{\varphi_\omega}}$ is an i.p.s. form. By ${\overline{\varphi_\omega}^e}$ we construct the desired extension.