Partial *-Algebras of Closable Operators. II. States and Representations of Partial *-Algebras

Abstract

This second paper on partial Op\*-algebras is devoted to the theory of representations. A new definition of invariant positive sesquilinear forms on partial \*-algebras is proposed, which enables to perform the familiar GNS construction. In order to get a better control of the corresponding representations, we introduce and study a restricted class of partial Op\*-algebras, called partial GW\*-algebras, which turn up naturally in a number of problems. As an example, we extend Powers' results about the standardness of GNS representations of abelian partial \*-algebras.