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Operator modules

Let $ A_1, A_2 \subset B({\mathcal{H}})$ be $ C^*$-algebras with $ \mathrm{1\!\!\!\:l}_{{\mathcal{H}}} \in A_1, A_2$. A closed subspace $ X$ of $ B({\mathcal{H}})$ is called a concrete $ (A_1,A_2)$-operator module , if $ A_1X \subset X$ and $ XA_2 \subset X$. Whenever $ A_1=A_2=A$, we call $ X$ a concrete $ A$-operator bimodule (cf. [ER88, p. 137]).

In analogy to the operator space or the operator algebra situation, there is an abstract characterization of operator modules (cf. [Pop00, Déf. 4.1]):
Consider, as above, two unital $ C^*$-algebras $ A_1, A_2 \subset B({\mathcal{H}})$ with $ \mathrm{1\!\!\!\:l}_{{\mathcal{H}}} \in A_1, A_2$, and an (algebraic) $ (A_1,A_2)$-module $ X$. We call $ X$ an abstract $ (A_1,A_2)$-operator module, if it carries an operator space structure satisfying the following axioms (of Ruan type):

(R1) $ \Vert axb\Vert _m$ $ \leq$ $ \Vert a\Vert \Vert x\Vert _m \Vert b\Vert$
       
(R2) $ \left\Vert
\left( \begin{array}{cc}
x & 0 \\
0 & y \end{array} \right)
\right\Vert _{m+n}$ $ =$ $ {\rm {max}} \{ \Vert x\Vert _m, \Vert y\Vert _n \},$
where $ m,n \in {\mathbb{N}}$, $ a \in M_m(A_1)$, $ x \in M_m(X)$, $ y\in M_n(Y)$, $ b \in M_m(A_2)$.

For abstract operator modules holds a representation theorem of Ruan type (cf. [Pop00, Thm. 4.7]):
Let $ V$ be an abstract $ (A_1,A_2)$-operator module. Then there exist a Hilbert space $ \mathcal{K}$, a complete isometry $ \Theta: X \hookrightarrow B(\mathcal{K})$ and $ ^*$-representations $ \pi_1$, $ \pi_2$ of $ A_1$ resp. $ A_2$ in $ B(\mathcal{K})$ such that:

$\displaystyle \Theta(axb) = \pi_1(a) \Theta(x) \pi_2(b),
$

where $ x \in X$, $ a \in A_1$, $ b \in A_2$. In case $ A_1=A_2$, one can even choose $ \pi_1=\pi_2$.



Subsections
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Next: Basic examples of operator Up: Multiplicative Structures Previous: Multiplicative Structures   Contents   Index
Prof. Gerd Wittstock 2001-01-07