We explore the finite-dimensional part of the non-commutative Choquet boundary of an operator algebra. In other words, we seek finite-dimensional boundary representations. Such representations may fail to exist even when the underlying operator algebra is finite dimensional. Nevertheless, we exhibit mechanisms that detect when a given finite-dimensional representation lies in the Choquet boundary. Broadly speaking, our approach is topological and requires identifying isolated points in the spectrum of the $\textrm{C}^{\ast }$-envelope. This is accomplished by analyzing peaking representations and peaking projections, both of which being non-commutative versions of the classical notion of a peak point for a function algebra. We also connect this question with the residual finite dimensionality of the $\textrm{C}^{\ast }$-envelope and to a stronger property that we call $\textrm{C}^{\ast }$-liminality. Recent developments in matrix convexity allow us to identify a pivotal intermediate property, whereby every matrix state is locally finite dimensional.

中文翻译：

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非交换 Choquet 边界中的有限维度：峰值现象和 C*-Liminality

我们探索算子代数的非交换 Choquet 边界的有限维部分。换句话说，我们寻求有限维边界表示。即使底层算子代数是有限维的，这种表示也可能不存在。然而，我们展示了检测给定有限维表示何时位于 Choquet 边界的机制。从广义上讲，我们的方法是拓扑的，需要识别 $\textrm{C}^{\ast}$-包络谱中的孤立点。这是通过分析峰值表示和峰值投影来完成的，这两者都是函数代数峰值点的经典概念的非交换版本。我们还将这个问题与 $\textrm{C}^{\ast }$-envelope 的剩余有限维度以及我们称之为 $\textrm{C}^{\ast }$-liminality 的更强属性联系起来。矩阵凸性的最新发展使我们能够确定一个关键的中间属性，即每个矩阵状态都是局部有限维的。