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Tensor products

Let $ X$ be an operator space. We have complete isometries [ER91, Thm. 4.3 (a)(c)] [Ble92b, Prop. 2.3 (i)(ii)]:

$\displaystyle {\mathcal{C}}_\H\otimes_{h}X \stackrel{\mathrm{cb}}{=}{\mathcal{C}}_\H\stackrel{\scriptscriptstyle \vee}{\otimes}X$

and

$\displaystyle X \otimes_{h}{\mathcal{C}}_\H\stackrel{\mathrm{cb}}{=}X \stackrel{\scriptscriptstyle \wedge}{\otimes}{\mathcal{C}}_\H$.

Herein, $ \otimes_{h}$ is the Haagerup tensor product , $ \stackrel{\scriptscriptstyle \vee}{\otimes}$ the injective tensor product and $ \stackrel{\scriptscriptstyle \wedge}{\otimes}$ the projective tensor product .

For Hilbert spaces $ \H$ and $ \mathcal{K}$, we have complete isometries [ER91, Cor. 4.4.(a)] [Ble92b, Prop. 2.3(iv)]

$\displaystyle {\mathcal{C}}_\H\otimes_{h}{\mathcal{C}}_\mathcal{K}
\stackrel{\m...
...C}}_\mathcal{K}\stackrel{\mathrm{cb}}{=}{\mathcal{C}}_{\H\otimes_2 \mathcal{K}}$.$\displaystyle $



Prof. Gerd Wittstock 2001-01-07