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Columns and rows of an operator space

The space $ X^p$ of $ p$-tupels over an operator space $ X$ can be made into an operator space for instance by reading the $ p$-tupels as $ p\times 1$- or as $ 1\times p$-matrices. This leads to the frequently used columns and rows of an operator space $ X$:

$\displaystyle {C_p(X)} := {\mathbb{M}}_{p,1}(X)$   and$\displaystyle \quad {R_p(X)} := {\mathbb{M}}_{1,p}(X)$   .

The first matrix level of these spaces are

$\displaystyle M_1(C_p(X)) = M_{p,1}(X)$   and$\displaystyle \quad M_1(R_p(X)) = M_{1,p}(X)$   , respectively.

If $ X \neq \{0\}$, the spaces $ C_p(X)$ and $ R_p(X)$ are not completely isometric. In general even the first matrix levels $ M_{p,1}(X)$ and $ M_{1,p}(X)$ are not isometric.

$ {\mathcal{C}}_p :=C_p({\mathbb{C}})$ is called the p-dimensional column space and $ {\mathcal{R}}_p :=R_p({\mathbb{C}})$ the p-dimensional row space.

The first matrix levels of $ {\mathcal{C}}_p$ and $ {\mathcal{R}}_p$ are isometric to $ l_2^p$, but $ {\mathcal{C}}_p$ and $ {\mathcal{R}}_p$ are not completely isometric .



Prof. Gerd Wittstock 2001-01-07