We introduce a numerical radius operator space $(X, \mathcal{ W}_n)$. The conditions to be a numerical radius operator space are weaker than the Ruan's axiom for an operator space $(X, \mathcal{ O}_n)$. Let $w(\cdot)$ be the numerical radius norm on $\mathbb{ B}(\mathcal{ H})$. It is shown that if $X$ admits a norm $\mathcal{ W}_n(\cdot)$ on the matrix space $\mathbb{ M}_n(X)$ which satisfies the conditions, then there is a complete isometry, in the sense of the norms $\mathcal{ W}_n(\cdot)$ and $w_n(\cdot)$, from $(X, \mathcal{ W}_n)$ into $(\mathbb{ B}(\mathcal{ H}), w_n)$. We study the relationship between the operator space $(X, \mathcal{ O}_n)$ and the numerical radius operator space $(X, \mathcal{ W}_n)$. The category of operator spaces can be regarded as a subcategory of numerical radius operator spaces.
Categories: math.OA