分块Toeplitz矩阵上的纯矩阵态

中文摘要 AI 辅助

设$\u2113_{n,m} = \u2113_n(M_m(\u2102))$表示所有分块Toeplitz矩阵构成的算子系统，其中矩阵形式为$ T = (( T_{i-j}))_{i,j=1}^n$，元素$T_k \u2208 M_m(\u2102) $，矩阵整体$T \u2208 M_{mn}(\u2102)$，其结构为以$T_0$为对角元、次对角元依次为$T_{\u00b11}, T_{\u00b12}, \u2026, T_{\u00b1(n-1)}$的分块Toeplitz阵列。\n本文刻画了从$\u2113_{n,m}$到$M_m(\u2102)$的所有纯幺正完全正（ucp）映射。通过Stinespring等距算子$V = (V_1, \u2026, V_n)^t \u2236 \u2102^m \u2192 \u2102^{mn}$以及矩阵值多项式$Q_V(z) = \u2211_{i=1}^n z^{n-i} V_i$，我们证明：映射$\u03c6$是纯的当且仅当它存在唯一的纯ucp延拓到$M_{mn}(\u2102)$，当且仅当$Q_V$的次数为$n-1$且所有根都位于单位圆周$\u216b$上。\n每个这样的纯映射$\u03c6$都会诱导出$C(\u216b, M_m(\u2102))$上的一个ucp映射$\u03a6_{Q_V}$，其定义为$\u03a6_{Q_V}(f) = \u222b_{\u216b} Q_V(z)^* f(z) Q_V(z)\u2009 dz$。设$\u212c_m$是从$C(\u216b, M_m(\u2102))$到$M_m(\u2102)$的所有ucp映射构成的紧凸集。赋予$\u212c_m$矩阵值Monge-Kantorovich度量$\u03c1$，通过分析$\u212c_m$的端点并结合点分裂引理，我们证明上述诱导映射在$\u212c_m$中是$\u03c1$-稠密的。由此可得，若$\u212c_{m,n}$表示满足上述条件的$Q_V$对应的归一化$\u03a6_{Q_V}$的集合，则当$n \u2192 \u221e$时，Hausdorff距离$d_H(\u212c_{m,n}, \u212c_m) \u2192 0$，这将单位圆周上正正则Borel测度的已知逼近结果推广到了矩阵值完全正映射的情形。

英文摘要

Let $\mathcal{T}_{n,m} = \mathcal T_n(M_m(\mathbb{C}))$ denote the operator system of all block Toeplitz matrices $ T = (( T_{i-j}))_{i,j=1}^n$ with entries $T_k \in M_m(\mathbb C) $ % T_{k} = \left[ t^{(k)}_{p-q} \right]_{p,q=1}^m. \[ T = \begin{pmatrix} T_0 & T_{-1} & \cdots & T_{-(n-1)} T_1 & T_0 & \cdots & T_{-(n-2)} \vdots & \vdots & \ddots & \vdots T_{n-1} & T_{n-2}& \cdots & T_0 \end{pmatrix} \in M_{mn}(\mathbb{C}). \] We characterize all pure unital completely positive ({\it{ucp}}) maps from $\mathcal{T}_{n,m}$ to $M_m(\mathbb{C})$. Working through the Stinespring isometry $V = (V_1, \ldots, V_n)^t \colon \mathbb{C}^m \to \mathbb{C}^{mn}$ and the matrix-valued polynomial $Q_V(z) = \sum_{i=1}^n z^{n-i} V_i$, we prove that $φ$ is pure if and only if it admits a unique pure \ucp extension to $M_{mn}(\mathbb{C})$ if and only if $Q_V$ has degree $n-1$ with all its roots on the unit circle $\T$. Every such pure $φ$ induces a \ucp map $Φ_{Q_V}$ on $C(\mathbb{T}, M_m(\mathbb{C}))$ given by \[ Φ_{Q_V}(f) = \int_{\mathbb{T}} Q_V(z)^* f(z) Q_V(z)\, dz. \] Let $\mathcal{Y}_m$ be the compact convex set of all \ucp maps from $C(\mathbb{T}, M_m(\mathbb{C}))$ to $M_m(\mathbb{C})$. Endowing $\mathcal{Y}_m$ with the matricial Monge-Kantorovich metric $ρ$, via an analysis of the extreme points of $\mathcal{Y}_m$ together with a point-splitting lemma, we show that the induced maps as above are $ρ$-dense in $\mathcal{Y}_m$. Consequently, if $\Bmn$ denotes the set of normalized $Φ_{Q_V}$ where $Q_V$ is as above, then the Hausdorff distance $d_H(\Bmn, \mathcal{Y}_m) \to 0$ as $n \to \infty$, extending known results of approximation of positive regular Borel measures on the unit circle to the setting of matrix-valued completely positive maps.

补充信息

备注

20 pages