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Basic examples of operator modules

Let $ A$ be a unital $ C^*$-algebra, $ X$ a normed space, and $ Y$ an operator space. Then $ B(X, A)$ resp. $ \mathit{CB}(Y, A)$ are operator spaces via the identifications $ M_n(B(X, A))=B(X, M_n(A))$ resp. $ M_n(\mathit{CB}(Y, A))=\mathit{CB}(Y, M_n(A))$. These become $ A$-operator bimodules, when endowed with the natural module operations as follows ([ER88, p. 140]):

$\displaystyle (a \cdot \varphi \cdot b)(x) = a \varphi(x) b$      

for all $ a, b \in A$, $ \varphi \in B(X, A)$ resp. $ \mathit{CB}(Y, A)$, $ x \in X$ resp. $ x \in Y$.

In the category of operator modules, the morphisms are the completely bounded module homomorphisms . For these we have a representation and an extension theorem .
Representation theorem (cf. [Hof95, Kor. 1.4]):
Let $ {\mathcal{H}}$ be a Hilbert space, $ M$ a $ C^*$-algebra in $ B({\mathcal{H}})$, and $ A$, $ B$ $ C^*$-subalgebras of $ M$. Then the following hold true:

(a)
(cf. [Pau86, Thm. 7.4]) For each completely bounded $ (A,B)$-module homomorphism $ \Phi: M \rightarrow B({\mathcal{H}})$, there exist a Hilbert space $ \mathcal{K}$, a $ ^*$-representation $ \pi: M \rightarrow B(\mathcal{K})$ and linear operators $ v,w \in B({\mathcal{H}},\mathcal{K})$ sharing the following properties:
(a1)
$ \Phi(x)=v^*\pi(x)w$ for all $ x \in M$, i.e. $ (\mathcal{K};\pi;v^*;w)$ is a representation of $ \Phi$
(a2)
$ \Vert \Phi \Vert _{\mathrm{cb}} = \Vert v\Vert\Vert w\Vert$
(a3)
$ \overline{\mathrm{lin}}(\pi(M)v{\mathcal{H}}) = \overline{\mathrm{lin}}(\pi(M)w{\mathcal{H}}) = \mathcal{K}$
(a4)
$ v^*\pi(a) = av^*$ for all $ a\in A$ and $ \pi(b)w = wb$ for all $ b \in B$.
(b)
(cf. also [Smi91, Thm.3.1]) If, in addition, $ M \subset B({\mathcal{H}})$ is a von Neumann algebra and $ \Phi: M \rightarrow B({\mathcal{H}})$ is a normal completely bounded $ (A,B)$-module homomorphism, one can require the $ ^*$-representation $ \pi$ of part (a) to be normal. There exist families $ (a_i)_{i \in I}$ and $ (b_i)_{i \in I}$ in the commutant of $ A$ and $ B$, respectively, with the following properties (the sums are to be taken in the WOT topology):
(b1)
$ \Phi(x) = \sum_{i \in I} a_ixb_i$ for all $ x \in M$
(b2)
$ \sum_{i \in I} a_ia^*_i \in B({\mathcal{H}}),~\sum_{i \in I} b^*_ib_i \in B({\...
...}
a_ia^*_i\Vert^{\frac{1}{2}} \Vert \sum_{i \in I} b^*_ib_i \Vert^{\frac{1}{2}}$.
Extension theorem ([Wit84a, Thm. 3.1], cf. also [MN94, Thm. 3.4] and [Pau86, Thm. 7.2]):
Let $ F$ be an injective $ C^*$-algebra, and let $ A, B \subset F$ be two unital $ C^*$-subalgebras. Consider furthermore two $ (A,B)$-operator modules $ E_0$ and $ E$ with $ E_0 \subset E$. Then for each $ \phi_0 \in \mathit{CB}_{(A,B)} (E_0,F)$, there exists an extension $ \phi \in \mathit{CB}_{(A,B)} (E,F)$ with $ \phi\vert _{E_0} = \phi_0$ and $ \Vert\phi\Vert _{\mathrm{cb}} = \Vert \phi_0 \Vert _{\mathrm{cb}}$.

The decomposition theorem for completely bounded module homomorphisms can be found in the corresponding chapter.


next up previous contents index
Next: Completely bounded module homomorphisms Up: Operator modules Previous: Operator modules   Contents   Index
Prof. Gerd Wittstock 2001-01-07