- Let
and
be C
-algebras in
.
Then on the algebraic tensor product
the Haagerup tensor norm is explicitly given by
where
,
,
.
The Haagerup norm of
equals the
cb-norm of the elementary operator
.
The Haagerup tensor product
is the completion of the
algebraic tensor product
with respect to the above norm.
The following more general definition in particular
yields a completely isometric embedding
.
-
We
have
where
,
,
.
In fact, one summand suffices
[BP91, Lemma 3.2].
-
For
elements
in the algebraic tensor product there is an
and elements
,
such that
The infimum occuring in the formula describing the norm in this case is actually a minimum
[ER91, Prop. 3.5].
-
The
Haagerup norm of an element
can also be expressed using a supremum:40
where
,
,
,
[ER91, Prop. 3.4].
-
From
the definition of the Haagerup norm one easily deduces that the
shuffle -map
is a complete contraction.
Hence the Haagerup tensor product enjoys property
of an operator space tensor product.
But the shuffle-map is not an isometry in general
as shown by the following example:41
Since the bilinear mapping
is contractive, it is
jointly completely
contractive.42
In fact, using
, we see that it is even
completely
contractive.
-
For
the row and column structure the
shuffle -map even is
a complete isometry.
We have the Lemma of Blecher and Paulsen
[BP91, Prop. 3.5]:
In many cases it suffices to prove a statement about the
Haagerup tensor product
on the first matrix level and then to deduce it for all matrix levels
using the above formula.43
Here we list some special cases of the Lemma of Blecher and Paulsen:
-
In
contrast to
, for
one obtains the finer operator space structure
of the trace class

.
For an operator space
we have
[Ble92b, Prop. 2.3]:
-
By
the very construction the bilinear mapping
,
is a complete contraction.
Hence its amplification,
the
tensor matrix product
also is a
complete contraction .
The linearization of the tensor matrix product gives the
complete contraction

.
- The
bilinear mapping44
is
completely contractive 45and gives rise to a complete contraction
-
Let
and
be Hilbert spaces.
Taking the Haagerup tensor product of the
column Hilbert space
and the
row Hilbert space
one obtains completely isometrically
the space of
compact operators
resp. of trace class46operators
[ER91, Cor. 4.4]:
This
example
also shows that the Haagerup tensor product is not symmetric.