Mass concentration in quasicommutators of Toeplitz matrices

Abstract

We consider the algebra generated by the principal finite sections of products of multidimensional block Toeplitz operators with bounded symbols. The matrices in this algebra may be regarded as higher-dimensional cubes of increasing side length. We prove that the matrices in the quasicommutator ideal of the algebra have asymptotically no mass in the interior. This observation yields another elementary proof of first-order Szegö-type results. We also determine the corners, edges, and subfaces of the cube that may carry mass asymptotically.