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Examples

matrix convexity
  1. The unit ball of an operator space $ X$, given by the family $ \mathrm{Ball}(X)_n=\{x\in M_n(X)\;\vert\;\Vert x\Vert\leqslant 1\}$ for all $ n\in{\mathbb{N}}$,61is absolutely matrix convex and closed.
  2. The set of matrix states of a unital $ C^*$-algebra $ A$, given by the family $ \mathrm{CS}(A)_n=
\{\varphi:A\to M_n\;\vert\;\varphi$    completely positive and unital$ \}$ for all $ n\in{\mathbb{N}}$, is matrix convex and weak-$ *$-compact.
  3. If $ A$ is a $ C^*$-algebra, the positive cones $ M_n(A)^+$ for all $ n\in{\mathbb{N}}$ form a closed matrix cone.



Footnotes

...,61
Also $ \mathrm{Ball}=\{x\in M(X)\;\vert\;\Vert x\Vert\leqslant 1\}$.


Prof. Gerd Wittstock 2001-01-07