Bounded elements of C*-inductive locally convex spaces

TL;DR

This paper introduces and explores the concept of bounded elements in C*-inductive locally convex spaces, focusing on their structure, order, and applications to spaces of continuous linear maps in rigged Hilbert spaces.

Contribution

It defines bounded elements in C*-inductive locally convex spaces and examines their properties through inductive structures and natural orderings, with applications to operator spaces.

Findings

Bounded elements are characterized via inductive structures.

The natural order of these spaces is linked to boundedness.

Application to continuous linear maps in rigged Hilbert spaces.

Abstract

The notion of bounded element of C*-inductive locally convex spaces (or C*-inductive partial *-algebras) is introduced and discussed in two ways: the first one takes into account the inductive structure provided by certain families of C*-algebras; the second one is linked to natural order of these spaces. A particular attention is devoted to the relevant instance provided by the space of continuous linear maps acting in a rigged Hilbert space.