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$ 1$-direct sums

Let $ I$ be an index set and $ X_i$ for each $ i\in I$ an operator space. Then there are an operator space $ X$ and complete contractions $ \iota_i:X_i\to X$ with the following universal mapping property: For each family of complete contractions $ \varphi_i:X_i\to Z$ there is exactly one complete contraction $ \varphi:X\to Z$ such that $ \varphi_i=\varphi\circ\iota_i$ for all $ i$. $ X$ is called $ 1$-direct sum of the $ X_i$ and is denoted by $ \bigoplus_1 (X_i\;\vert\;i\in I)$.The $ \iota_i$ are completely isometric.

One can construct a $ 1$-direct sum for instance as the closure of the sums of the images of the mappings $ X_i\hookrightarrow X_i^{**}\stackrel{\pi_i^*}{\to}
(\bigoplus_\infty (X_i^*\;\vert\;i\in I))^*$, where $ \pi_i$ is the projection from $ \bigoplus_\infty (X_i^*\;\vert\;i\in I)$ onto $ X_i^*$.

The equation

$\displaystyle (\bigoplus_1 (X_i\;\vert\;i\in I))^*=\bigoplus_\infty (X_i^*\;\vert\;i\in I)$

holds isometrically.



Prof. Gerd Wittstock 2001-01-07