Sometimes one considers an operator space norm on the algebraic tensor product of two fixed operator spaces. Then one usually demands that this norm and its dual norm are at least cross norms. Operator space tensor norms always have these properties.
An operator space norm on the algebraic tensor product of two operator spaces and is said to be a cross norm, if
For cross norms is completely isometric.
For an operator space norm on the algebraic tensor product of two fixed operator spaces and one usually asks for the following three properties (i)-(iii).32
Then the dual operator space norm is defined on the algebraic tensor product by the algebraic embedding
There is a smallest operator space norms among the operator space norms on , for which and the dual norm are cross norms. This is the injective operator space tensor norm . [BP91, Prop. 5.10].
There is a greatest operator space norm among the operator space norms on , for which and the dual norm are cross norms. This is the projective operator space tensor norm [BP91, Prop. 5.10].
On the algebraic tensor product one can compare the operator space norms for which and the dual norm are cross norms with the injective tensor norm and the projective tensor norm of normed spaces: