Two theorems about topologies on countably generatedOp*-algebras

Abstract

The present paper deals with the topologization of unbounded operator algebras (Op.-algebras) in Hilbert space. We consider two possible topologies, the so-called uniform topology ~a introduced in [2] and the strong operator topology a ~. We characterize the countably generated [closed] Op.-algebras ~r for which z~ [resp. a ~] agrees with the strongest locally convex topology on ~'. Our main theorems contain some known result for concrete Op.-algebras ([2], [3], [4], [6]).In [5], theorem 1 was used in proving that for certain Op,-algebras ~ (for example, the Op .-algebra of all differential operators with polynomial coefficients on the Schwartz space Se (R~)) all linear functionals f on ~r are trace functionals, i.e. they can be given by f(a)=Trta, aE~], t an appropriate nuclear operator.

Spaces of Continuous Sesquilinear Forms Associated with Unbounded Operator Algebras

“…Then the strong-ôperat9r topology on 4 is equal to the finest locally convex topology OIl the vector space A (see e.g. [16]), so that is closed in L+ (0) with respect to the strong-Operator topology. Since…”