The representations of two operator spaces in
and
in
yield a representation of the algebraic tensor product of
and
in
.
The operator space structure obtained in this way turns out to be independent of the
representations chosen.
It is called the
injectiveoperator space tensor product of
and
and is denoted by
[BP91, p. 285].
Hence, in the case of
-algebras, the injective
operator space tensor product and the minimal
-tensor product
coincide.33
By means of the
duality of tensor products
we obtain a formula
[BP91, Thm. 5.1]
for the injective operator space tensor norm of an element
which is representation free:
Interpreting, as is usual, the elements of the algebraic tensor product as finite rank operators we have the completely isometric embeddings [BP91, Cor. 5.2]
The injective operator space tensor norm is the least cross norm whose dual norm again is a cross norm.
The injective operator space tensor product is symmetric , associative and injective . But it is not projective [BP91, Cor. 5.2].
The injective norm is the dual norm of the
projective
operator space tensor norm
[BP91, Thm. 5.6];
but
the projective operator space tensor norm
is not in general the dual of the
injective operator space tensor norm even if one of the two spaces involved is
finite dimensional
[ER90a, p. 168],
[ER91, p. 264].