The multiplier algebra of the noncommutative Schwartz space

TL;DR

This paper characterizes the multiplier algebra of the noncommutative Schwartz space, revealing its structure as a large *-algebra of unbounded operators and analyzing its properties relative to classical functional analysis tools.

Contribution

It provides a detailed description of the multiplier algebra of the noncommutative Schwartz space and examines its algebraic and topological properties.

Findings

The multiplier algebra is not a $\\mathcal{Q}$-algebra or $m$-convex.

Classical functional analysis tools remain applicable.

The algebra acts on a separable Hilbert space with the Schwartz space as domain.

Abstract

We describe the multiplier algebra of the noncommutative Schwartz space. This multiplier algebra can be seen as the largest ${}^*$-algebra of unbounded operators on a separable Hilbert space with the classical Schwartz space of rapidly decreasing functions as the domain. We show in particular that it is neither a $\mathcal{Q}$-algebra nor $m$-convex. On the other hand, we prove that classical tools of functional analysis, for example, the closed graph theorem, the open mapping theorem or the uniform boundedness principle, are still available.