Compactness criteria in quasi-Banach symmetric operator spaces associated with a non-commutative torus

Abstract

We present two new compactness criteria in non-commutative quasi-Banach symmetric spaces associated to a finite von Neumann algebra, with focus on the non-commutative torus. The first result is novel, even in the commutative setting; while the second resembles the Kolmogorov–Riesz compactness theorem (see Theorem 4.1, Theorem 5.7, respectively). The work contributes to understanding a conjecture of Brudnyi, adapted here for the non-commutative torus.