Biweights on Partial *-Algebras

Abstract

Representations of *-algebras are generally obtained by the familiar Gelfand–Naimark–Segal (GNS) construction, starting from a positive linear functional or even a (quasi-) weight, that is, a functional that may take infinite values. In previous works, we have extended this construction to the case of a partial *-algebra, using invariant positive sesquilinear forms, which play the same role as positive linear functionals for *-algebras. In this paper we extend the construction further, by introducing and studying systematically the notion of biweight on a partial *-algebra. In particular, we characterize, through the associated GNS representation, the so-called approximately admissible biweights (i.e., limits of biweights with a bounded GNS representation).