Completely bounded bilinear mappings

In the case of bilinear mappings between operator spaces one has to distinguish between two different notions of complete boundedness: on one hand we have the jointly completely bounded [BP91, Def. 5.3 (jointly completely bounded)] and, on the other hand, the completely bounded bilinear mappings [CS87, Def. 1.1]. The class of completely bounded bilinear maps is is contained in the first one. These notions are in perfect analogy to those of bounded bilinear forms on normed spaces. For completely bounded bilinear mappings, we have at our disposal similar representation and extension theorems⁴⁷ as in the case of completely bounded linear maps. There are two tensor products corresponding to the above two classes of bilinear mappings, namely the projective and the Haagerup tensor product. Depending on the class of bilinear maps, one uses different methods to define the amplification of a bilinear mapping $\Phi: X \times Y \rightarrow Z$.

Footnotes

... theorems⁴⁷

Extension theorems for completely bounded bilinear (and, more generally, multilinear) maps can be derived from the injectivity of the Haagerup tensor product .