We extend functional analysis to quantum realms through the frameworks of operator algebras and operator spaces, emphasizing the structure and behavior of non-commutative mathematical objects. Central to our investigations are quantum approximation properties in C*-algebras and the study of convexity within non-commutative systems. Our research explores completely bounded maps, matricial ranges, and operator system structures, all of which play key roles in the analysis of quantum phenomena.
These investigations bridge abstract functional analysis with emerging themes in quantum information theory, providing tools for analyzing quantum channels, designing error correction mechanisms, and understanding the geometry of quantum state spaces. By focusing on the interplay between algebraic structure and geometric intuition, we aim to develop a rigorous yet versatile mathematical language for addressing foundational questions in quantum computation and operator theory.