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Tensor products of operator matrices

As usual we define the algebraic tensor product of operator matrices $ x = [x_{ij}] \in M_{p}(X)$, $ y = [y_{kl}] \in M_{q}(Y)$ by setting

$\displaystyle x \otimes y
%% [x_{ij}]_{i,j} \otimes [y_{kl}]_{k,l}
:=
[ x_{ij} \otimes [y_{kl}]_{k,l} ]_{i,j}
\in M_p(X \otimes M_q(Y)).
$

Here we have used the definition $ M_p(X) = M_p \otimes X$ and the associative law

$\displaystyle M_p(X) \otimes M_q(Y) = M_p \otimes (X \otimes M_q(Y)) = M_p(X \otimes M_q(Y)).$ (5)

In view of the next identification one should note that the shuffle-map is an algebraic isomorphism:

$\displaystyle X \otimes (M_q \otimes Y)
 \rightarrow 
 M_q \otimes (X \otimes Y),$ (6)

$\displaystyle x \otimes (\beta \otimes y)
\mapsto \beta \otimes (x \otimes y),
$

for $ \beta \in M_q$, $ x \in X$, $ y \in Y $. The shuffle-isomorphism at hand we obtain the identification:

$\displaystyle x \otimes y
=
[ x_{ij} \otimes [y_{kl}]_{k,l} ]_{i,j}
=
[ [x_{ij} \otimes y_{kl}]_{k,l} ]_{i,j}
\in M_p(M_q(X \otimes Y)).
$

Finally we use the usual72 identification $ M_p(M_q) = M_{pq}$

$\displaystyle [ [x_{ij} \otimes y_{kl}]_{k,l} ]_{i,j}
=
[ x_{ij} \otimes y_{kl} ]_{(i,k),(j,l)}
$

to obtain

$\displaystyle {}M_p(X) \otimes M_q(Y) = M_{pq}(X \otimes Y).$ (7)

We call this algebraic isomorphism the shuffle-isomorphism. One should note that for operator space tensor products the algebraic identifications ([*]) and ([*]) are only complete contractions:
$\displaystyle {\mathbb{M}}_p(X) \otimes_\alpha Y$ $\displaystyle \rightarrow$ $\displaystyle {\mathbb{M}}_p(X \otimes_\alpha Y),$ (8)
$\displaystyle X \otimes_\alpha {\mathbb{M}}_q(Y)$ $\displaystyle \rightarrow$ $\displaystyle {\mathbb{M}}_q(X \otimes_\alpha Y).$ (9)

In general these are not isometries even for $ p=1$ resp. $ q=1$. For an operator space tensor product the shuffle-map

$\displaystyle {\mathbb{M}}_p(X) \otimes_\alpha {\mathbb{M}}_q(Y)
 \rightarrow
 {\mathbb{M}}_{pq}(X \otimes_\alpha Y)$ (10)

in general is only completely contractive. In the case of the injective operator space tensor product this is of course a complete isometry. More generally, one considers the shuffle-map for rectangular matrices73:

$\displaystyle {\mathbb{M}}_{m,n}(X) \otimes_\alpha {\mathbb{M}}_{p,q}(Y)
\rightarrow
{\mathbb{M}}_{mp,nq}(X \otimes_\alpha Y).
$

Another example is provided by the Blecher-Paulsen equation .

Footnotes

... usual72
In the matrix so obtained $ (i,k)$ are the row indices and $ (j,l)$ are the column indices, where $ i,j = 1,\dots,p$ and $ k,l = 1,\dots,q$. The indices $ (i,k)$ resp. $ (j,l)$ are ordered lexicographically.
... matrices73
The shuffle-map

$\displaystyle (U \otimes X) \otimes (V \otimes Y)
\rightarrow
(U \otimes V) \otimes(X \otimes Y),
$

$\displaystyle (u \otimes x) \otimes (v \otimes y)
\mapsto (u \otimes v) \otimes (x \otimes y),
$

$ U$,$ V$,$ X$,$ Y$ operator spaces, has been studied for various combinations of operator space tensor products [EKR93, Chap. 4].

next up previous contents index
Next: Joint amplification of a Up: Tensor products Previous: Tensor products   Contents   Index
Prof. Gerd Wittstock 2001-01-07