A representation theorem for local operator spaces

In this note, we generalize Ruan’s representation theorem and propose an Arveson-Hanh-Banach-Webster theorem for local operator spaces. Further, we investigate the decompos-ability of a complete contraction acting from a unital multinormed C*-algebra to a local operator System into a product of contractions and a unital contractive *-representation, and we study injectivity in both local operator space and local operator System contexts.