The Soft Torus II: A Variational Analysis of Commutator Norms

TL;DR

This paper investigates the properties of Soft Tori C*-algebras, demonstrating their continuity and exploring how non-commuting unitaries can be perturbed to reduce their commutator norms, with specific results in finite dimensions.

Contribution

It provides a variational analysis of commutator norms in Soft Tori C*-algebras, including perturbation methods and characterization of local minima.

Findings

Soft Tori C*-algebras are continuous over [0,2].

Non-commuting unitaries can be weakly perturbed to lower commutator norms.

Characterization of local minima for commutator norms in finite dimensions.

Abstract

The field of C*-algebras over the interval [0,2] for which the fibers are the Soft Tori is shown to be continuous. This result is applied to show that any pair of non-commuting unitary operators can be perturbed (in a weak sense) in such a way to decrease the commutator norm. Perturbations in norm are also considered and a characterization is given for pairs of unitary operators which are local minimum points for the commutator norm in the finite dimensional case.