Abstract
In this paper we begin to study the order structure of topological -algebras of unbounded operators in Hilbert space with the investigation of the normality and the bounded decomposition property of the cones. We prove that for a large class of topological -algebras the normality of the wedge of positive elements is necessary and sufficient for a topological -algebra to be algebraically and topologically isomorphic to a -algebra of unbounded operators equipped with the uniform topology. From this theorem we obtain some corollaries, so for instance, well-known results of Lassner, Brooks and Grothendieck.
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