Equivalence and Stability of Compactness in Operator Spaces over the Non-Commutative Torus

We investigate compactness in operator spaces over the non-commutative torus A_θ, applying the structure of non-commutative C^*-algebras as well as compact operators acting on Hilbert A_θ-modules, and provide a characterization of compactness in the framework of operator spaces. Key results include the equivalence between classical and complete compactness, and the stability of compactness under tensor products. Applications and examples of compact operators in operator spaces over the non-commutative torus A_θ are presented. We also discuss limitations and propose future research directions to extend these results to more general settings.