This article shows the existence of a class of closed bounded matrix convexsets which do not have absolute extreme points. The sets we consider arenoncommutative sets, $K_X$, formed by taking matrix convex combinations of asingle tuple $X$. In the case that $X$ is a tuple of compact operators with nonontrivial finite dimensional reducing subspaces, $K_X$ is a closed boundedmatrix convex set with no absolute extreme points. A central goal in the theory of matrix convexity is to find a natural notionof an extreme point in the dimension free setting which is minimal with respectto spanning. Matrix extreme points are the strongest type of extreme pointknown to span matrix convex sets; however, they are not necessarily thesmallest set which does so. Absolute extreme points, a more restricted type ofextreme points that are closely related to Arveson's boundary, enjoy a strongnotion of minimality should they span. This result shows that matrix convexsets may fail to be spanned by their absolute extreme points.
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机译：本文显示了一类没有绝对极端点的封闭有界矩阵凸集。我们考虑的集合是非交换集合$ K_X $，是通过采用单个元组$ X $的矩阵凸组合形成的。在$ X $是具有非平凡有限维约化子空间的紧凑算子的元组的情况下，$ K_X $是没有绝对极端点的封闭有界矩阵凸集。矩阵凸度理论的中心目标是找到一个自然的概念，即在自由尺寸设置中相对于跨度最小的一个极端。矩阵极点是跨越矩阵凸集的最强类型的极点。但是，它们不一定是这样做的最小集合。绝对极限点是与Arveson边界密切相关的更受限制的极限点类型，它们在跨越时具有强烈的最小化概念。该结果表明矩阵凸集可能无法覆盖其绝对极点。
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