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$ \infty$-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 $ \pi_i:X\to X_i$ with the following universal mapping property: For each family of complete contractions $ \varphi_i:Z\to X_i$ there is exactly one complete contraction $ \varphi:Z\to X$ such that $ \varphi_i=\pi_i\circ\varphi$ for all $ i$. $ X$ is called $ \infty$-direct sum of the $ X_i$ and is denoted by $ \bigoplus_\infty (X_i\;\vert\;i\in I)$. The $ \pi_i$ are complete quotient mappings.

One can construct a $ \infty$-direct sum for instance as the linear subspace $ X=\{(x_i)\in\prod_{i\in I}X_i\;\vert\;\sup\{\Vert x_i\Vert\;\vert\;i\in I\}<\infty\}$ of the cartesian product of the $ X_i$, the $ \pi_i$ being the projections on the components. We have $ M_n(X)=\{(x_i)\in\prod_{i\in I}M_n(X_i)\;\vert\;\sup\{\Vert x_i\Vert\;\vert\;i\in I\}<\infty\}$, and the operator space norm is given by $ \Vert(x_i)\Vert=\sup\{\Vert x_i\Vert\;\vert\;i\in I\}$.



Prof. Gerd Wittstock 2001-01-07