Completely integral mappings are defined by the aid of completely nuclear mappings . A mapping is said to be completely integral, if there are a constant and a net of finite rank maps with converging to in the point norm topology 70.
The set of all these mappings forms the space of the completely integral mappings.
The infimum of all those constants satisfying the above condition is actually attained and denoted by . is a norm turning into a Banach space. The unit sphere of is merely the point norm closure of the unit sphere of .
One obtains the canonical operator space structure by defining the unit sphere of as the point norm closure of unit sphere of .
By definition we have ; for finite dimensional we have moreover [EJR98, Lemma 4.1]
Integral 71mappings are completely integral [ER94, 3.10].
The completely integral mappings enjoy also the -ideal property . Contrasting the situation of completely nuclear mappings they are local . In general one only has [EJR98].