Partial *-algebras of closed operators and their commutants. II. Commutants and bicommutants
Jean-Pierre Antoine, Françoise Mathot, Camillo Trapanı · Annales De L Institut Henri Poincare-physique Theorique · 1987
In this second paper on partial Op*-algebras, we present a systematic analysis of commutants and bicommutants, both from the algebraic and the topological point of views, along the lines of the usual theory of W*- and Op*-algebras. In particular we obtain conditions for the validity of the following statements: given a family R of unbounded operators, its commutant is a partial Op*-algebra, and/or R is dense in its bicommutant for an appropriate topology. We introduce the class of symmetric partial Op*-algebras, which verify those conditions. Finally we compare the commutants of a partial Op*-algebra with those of its canonical extensions to larger domains On presente une etude systematique, tant algebrique que topologique, des commutants et bicommutants, dans la ligne de la theorie usuelle des W*- et des Op*-algebres. On obtient des conditions garantissant la validite des enonces suivants: etant donnee une famille R d'operateurs non bornes, son commutant est une Op*-algebre partielle, et/ou R est dense dans son bicommutant pour une topologie appropriee. On introduit la classe des Op*-algebres partielles symetriques qui verifient ces conditions. On compare les commutants d'une Op*-algebre partielle avec ceux de ces extensions canoniques a des domaines plus grands