PARTIAL ASTERISK-ALGEBRAS OF CLOSABLE OPERATORS - A REVIEW

This paper reviews the theory of partial -algebras of closable operat ors in Hilbert space (partial O-algebras), with some emphasis on part ial GW-algebras. First we discuss the general properties and the vari ous types of partial -algebras and partial O*-algebras. Then we summa rize the representation theory of partial -algebras, including a gene ralized Gel'fand-Naimark-Segal construction; the main tool here is the notion of positive sesquilinear form, that we study in some detail (e xtendability, normality, order structure,...). Finally we turn to auto morphisms and derivations of partial O-algebras, and their mutual rel ationship. The central theme here is to find conditions that guarantee spatiality.