论文标题
非交通choquet边界中的有限维度：峰值现象和$ \ mathrm {c}^*$ - liminatie
Finite-dimensionality in the non-commutative Choquet boundary: peaking phenomena and $\mathrm{C}^*$-liminality
论文作者
论文摘要
我们探索操作员代数的非交通choquet边界的有限尺寸部分。换句话说，我们寻求有限维度的边界表示。即使基础运算符代数是有限维度的，这些表示形式也可能无法存在。然而，我们表现出检测到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，需要在$ \ mathrm {c}^*$ - 信封中识别隔离点。这是通过分析峰值表示和峰值投影来实现的，这两者都是函数代数的峰值点的经典概念的非共同版本。我们还将此问题与$ \ mathrm {C}^*$ - 信封的残留有限维度联系起来，并与我们称为$ \ mathrm {c}^*$ - liminatie的更强属性。矩阵凸度的最新发展使我们能够确定关键的中间性质，每个矩阵状态都是局部有限的。
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 $\mathrm{C}^*$-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 $\mathrm{C}^*$-envelope and to a stronger property that we call $\mathrm{C}^*$-liminality. Recent developments in matrix convexity allow us to identify a pivotal intermediate property, whereby every matrix state is locally finite-dimensional.
