Local minima of commutator norms in finite factors (Q1280370)

Let \(\ell_2(\mathbb Z)\) be the Hilbert space of bilateral sequences, and \(u\) and \(v\) be operators in this space acting by the formulas \[ (u\xi)_n = e^{i\theta n}\xi_n,\qquad (v\xi)_n = \xi_{n-1},\tag{1} \] where \(\theta/\pi\in(0;1)\) and \((\xi_n)\in\ell_2(\mathbb Z)\). The operators (1) in a factor \(\mathcal A\) of type \(\text{II}_1\) are analogues of Voiculescu matrices in the finite dimensional case. The author discusses the cases when the local minimum of the commutator norm is attained (or not) at a pair of unitary operators (1).