Bounded elements of C\(^\ast\)-inductive locally convex spaces (Q268192)

The paper under review is related to the study of algebras of unbounded linear operators. With that motivation, one studies the notion of bounded elements of a so-called \(C^*\)-inductive locally convex space, that is, a vector space which is the union of an upwards directed family of subspaces that are the underlying vector spaces of some \(C^*\)-algebras such that the inclusion maps between these subspaces are positive maps, in the sense that they preserve the cones of positive elements. A bounded element of such a space is by definition an element of the intersection of that family of subspaces, whose norms in these \(C^*\)-algebras are uniformly bounded. The main results of this paper provide characterizations of the bounded elements in terms of the order structure (Theorem 4.11) and in terms of representation theory (Theorem 4.8). The authors also investigate the notion of bounded element from the perspective of the \(C^*\)-inductive partial \(*\)-algebra structure.