Grothendieck's inequality in the noncommutative Schwartz space

TL;DR

This paper extends Grothendieck's inequality to the noncommutative Schwartz space, establishing optimal bounds and multiple reformulations within this algebra of smooth operators.

Contribution

It introduces the first optimal versions of Grothendieck's inequality for the noncommutative Schwartz space, a significant advancement in noncommutative functional analysis.

Findings

Established optimal inequalities for the noncommutative Schwartz space.

Reformulated the inequalities in multiple equivalent forms.

Demonstrated the inequalities hold in a non-optimal form via nuclearity.

Abstract

In the spirit of Grothendieck's famous inequality from the theory of Banach spaces, we study a sequence of inequalities for the noncommutative Schwartz space, a Fr\'echet algebra of smooth operators. These hold in non-optimal form by a simple nuclearity argument. We obtain optimal versions and reformulate the inequalities in several different ways.