The Representation Theorem for Local Operator Spaces

In this note we propose a representation theorem for local operator spaces which extends Ruan's representation theorem for operator spaces, and Arveson–Hahn–Banach–Webster theorem for local operator systems. Further, we investigate the decomposition property of a complete contraction from a unital multinormed $C^*$-algebra into a local operator system as a product of contractions and unital contractive $*$-representation, and the injectivity in both local operator space and local operator system contexts.