Bilinear Ideals in Operator Spaces

We introduce a concept of bilinear ideal of jointly completely bounded mappings between operator spaces. In particular, we study the bilinear ideals $\mathcal{N}$ of completely nuclear, $\mathcal{I}$ of completely integral, $\mathcal{E}$ of completely extendible bilinear mappings, $\mathcal{M}\mathcal{B}$ multiplicatively bounded and its symmetrization $\mathcal{S}\mathcal{M}\mathcal{B}$. We prove some basic properties of them, one of which is the fact that $\mathcal{I}$ is naturally identified with the ideal of (linear) completely integral mappings on the injective operator space tensor product.