Weighted noncommutative Banach function spaces

We review the concept of a weighted noncommutative Banach function space. This concept constitutes a generalisation of the by now well-known theory of noncommutative Banach function spaces associated with a semifinite von Neumann algebra. In this review we remind the reader of the quantum statistical problem which gave birth to this concept, we investigate the extent to which a weighted theory of measurable operators agrees with the standard theory, we explore competing methods for defining such spaces, before finally describing the monotone interpolation theory of such space