The Soft Torus: A Variational Analysis of Commutator Norms

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 noncommuting unitary operators can be pertubed (in a weak sense) in a way which decreases the commutator norm. Perturbations in the 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.