Bilinear ideals in operator spaces

Abstract

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