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 NN of completely nuclear, II of completely integral and EE of completely extendible bilinear mappings. We also consider the multiplicatively bounded bilinear mappings MBMB and its symmetrization SMBSMB. We prove some basic properties of them, one of which is the fact that II is naturally identified with the ideal of (linear) completely integral mappings on the injective operator space tensor product.