GNS-like representations of a quasi *-algebra in Krein spaces

Abstract

In this paper unbounded representations of quasi *-algebras on an indefinite inner product space are introduced and studied. After introducing the partial *-algebra of unbounded closable operators sharing a common dense domain in a Krein space, the appropriate notion of *-representation in this setting is developed, and a GNS-type construction for quasi *-algebras based on $α$-positive sesquilinear forms is established. Furthermore, the notion of abstract conditional expectation is generalized to quasi *-algebras and how \emph{admissible} abstract conditional expectations $P$ give rise to representable $P$-forms is shown. The relation between $α$-positive forms and $P$-forms is also clarified. Several examples are provided.