通过纯态定义的希尔伯特 $C^*$-模上的 Schatten 范数
Schatten norms on Hilbert $C^*$-modules via pure states
- Ferdowsi University of Mashhad（马什哈德 Ferdowsi 大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在希尔伯特 $C^*$-模上通过纯态引入 Schatten 范数，证明其上下界不等式，并给出紧算子或交换代数情形下的完备化等式。
AI中文摘要：
设 $(\mathscr{E}, \langle \cdot, \cdot\rangle)$ 是 $C^*$-代数 $\mathfrak{A}$ 上的希尔伯特 $C^*$-模。$\mathscr{E}$ 上可共轭算子空间记为 $\mathcal{L}\left(\mathscr{E}\right)$。$\mathfrak{A}$ 上所有态和纯态的集合分别记为 $\mathcal{S}\left(\mathfrak{A}\right)$ 和 $\mathcal{P}\left( \mathfrak{A}\right)$。对于 $\tau\in\mathcal{S}\left( \mathfrak{A}\right)$，定义 $\mathcal{N}^{\mathscr{E}}_{\tau}:=\left\lbrace x\in\mathscr{E}:\tau\left( \left\langle x,x\right\rangle\right)=0 \right\rbrace$。${\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_{\tau}}$ 的希尔伯特完备化记为 $\mathscr{E}_{\tau}$。对于 $T\in\mathcal{L}(\mathscr{E})$，算子 $T_{\mathscr{E}_\tau}\in\mathbb{B}\left( \mathscr{E}_\tau\right)$ 定义为 $T_{\mathscr{E}_\tau}\left(x+\mathcal{N}^{\mathscr{E}}_{\tau}\right)=Tx+\mathcal{N}^{\mathscr{E}}_{\tau}$。本文证明，当 $\mathfrak{A}$ 是紧算子 $C^*$-代数或交换 $C^*$-代数时，$\mathscr{E}_{\tau}={\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_{\tau}}$。我们在希尔伯特 $C^*$-模的背景下引入一个量，记为 $\pi^{\mathscr{E}}_k(\cdot)$（$k\geq1$）。我们证明对每个 $T\in\mathcal{L}\left(\mathscr{E}\right)$，有 $\pi^{\mathscr{E}}_k(T)\leq\sup_{\tau\in\mathcal{P}\left( \mathfrak{A}\right)}\left\\|T_{\mathscr{E}_\tau}\right\\|_{\left(k\right)}\leq{\pi}^{\mathscr{E}^{\sharp}}_k(T_{\mathscr{E}^{\sharp}})$，其中空间 $\mathscr{E}^{\sharp}$ 是通过将 $\mathfrak{A}$ 嵌入其包络 von Neumann 代数 $\mathfrak{A}^{**}$ 而构造的 $\mathscr{E}$ 的扩张。
英文摘要：
Let $(\mathscr{E}, \langle \cdot, \cdot\rangle)$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathfrak{A}$. The space of adjointable operators on $\mathscr{E}$ is denoted by $\mathcal{L}\left(\mathscr{E}\right)$. The sets of all states and pure states on $\mathfrak{A}$ are denoted by $\mathcal{S}\left(\mathfrak{A}\right)$ and $\mathcal{P}\left( \mathfrak{A}\right)$, respectively. For $τ\in\mathcal{S}\left( \mathfrak{A}\right)$, let us define $\mathcal{N}^{\mathscr{E}}_τ:=\left\lbrace x\in\mathscr{E}:τ\left( \left\langle x,x\right\rangle\right)=0 \right\rbrace$. The Hilbert completion of ${\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_τ}$ is denoted by $\mathscr{E}_τ$. For $T\in\mathcal{L}(\mathscr{E})$, the operator $T_{\mathscr{E}_τ}\in\mathbb{B}\left( \mathscr{E}_τ\right)$, is defined by $T_{\mathscr{E}_τ}\left(x+\mathcal{N}^{\mathscr{E}}_τ\right)=Tx+\mathcal{N}^{\mathscr{E}}_τ$. In this paper, we show that $\mathscr{E}_τ={\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_τ}$ when $\mathfrak{A}$ either is a $C^*$-algebra of compact operators or is commutative. We introduce a quantity in the context of Hilbert $C^*$-modules, denoted by $π^{\mathscr{E}}_k(\cdot)$ for $k\geq1$. We prove that $π^{\mathscr{E}}_k(T)\leq\sup_{τ\in\mathcal{P}\left( \mathfrak{A}\right)}\left\|T_{\mathscr{E}_τ}\right\|_{\left(k\right)}\leqπ^{\mathscr{E}^{\sharp}}_k(T_{\mathscr{E}^{\sharp}})$ for every $T\in\mathcal{L}\left(\mathscr{E}\right)$, where the space $\mathscr{E}^{\sharp}$ is constructed as the extension of $\mathscr{E}$ by the embedding of $\mathfrak{A}$ into its enveloping von Neumann algebra $\mathfrak{A}^{**}$.
