PARTIAL *-ALGEBRAS OF CLOSABLE OPERATORS: A REVIEW

Abstract

This paper reviews the theory of partial *-algebras of closable operators in Hilbert space (partial O*-algebras), with some emphasis on partial GW*-algebras. First we discuss the general properties and the various types of partial *-algebras and partial O*-algebras. Then we summarize the representation theory of partial *-algebras, including a generalized Gel’fand-Naimark-Segal construction; the main tool here is the notion of positive sesquilinear form, that we study in some detail (extendability, normality, order structure, …). Finally we turn to automorphisms and derivations of partial O*-algebras, and their mutual relationship. The central theme here is to find conditions that guarantee spatiality.