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:
![]() |
![]() |
||
![]() |
![]() |