Extensions of representable linear functionals to unitized quasi *-algebras

Abstract

This paper starts from noting that, under certain conditions, *-representability and extensibility to the unitized *-algebra of a positive linear functional, defined on a *-algebra without unit, are equivalent. Here some conditions for the equivalence of the same concepts for a hermitian linear functional defined on a quasi *-algebra $(\A,\Ao)$ without unit are given. The approach is twofold: on the one hand, conditions for the equivalence are exhibited by introducing a condition for the *- representability of the extension of a *-representable functional to the unitized quasi *-algebra, on the other hand a *-representable extension to the unitization of a hermitian linear functional by means of a closable invariant sesquilinear form is constructed.