Schatten norms on Hilbert $C^{\ast }$-modules via pure states

Abstract

Let $(\mathcal{E},⟨\cdot ,\cdot ⟩)$ be a Hilbert $C^{\ast }$-module over a $C^{\ast }$-algebra $\mathfrak{A}$. The space of adjointable operators on $\mathcal{E}$ is denoted by $\mathcal{L}\left(\mathcal{E}\right)$. 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 $\tau \in \mathcal{S}\left(\mathfrak{A}\right)$, let us define ${\mathcal{N}}_{\tau }^{\mathcal{E}}:=\left\{x\in \mathcal{E}:\tau \left(⟨x,x⟩\right)=0\right\}$. The Hilbert completion of $\mathcal{E}/{\mathcal{N}}_{\tau }^{\mathcal{E}}$ is denoted by ${\mathcal{E}}_{\tau }$. For $T\in \mathcal{L}(\mathcal{E})$, the operator $T_{{\mathcal{E}}_{\tau }}\in \mathbb{B}\left({\mathcal{E}}_{\tau }\right)$, is defined by $T_{{\mathcal{E}}_{\tau }}\left(x+{\mathcal{N}}_{\tau }^{\mathcal{E}}\right)=Tx+{\mathcal{N}}_{\tau }^{\mathcal{E}}$. In this paper, we show that ${\mathcal{E}}_{\tau }=\mathcal{E}/{\mathcal{N}}_{\tau }^{\mathcal{E}}$ when $\mathfrak{A}$ either is a $C^{\ast }$-algebra of compact operators or is commutative. We introduce a quantity in the context of Hilbert $C^{\ast }$-modules, denoted by ${\pi }_k^{\mathcal{E}}(\cdot )$ for $k\ge 1$. We prove that ${\pi }_k^{\mathcal{E}}(T)\le {\sup }_{\tau \in \mathcal{P}\left(\mathfrak{A}\right)}{\Vert T_{{\mathcal{E}}_{\tau }}\Vert }_{\left(k\right)}\le {\pi }_k^{{\mathcal{E}}^{♯}}(T_{{\mathcal{E}}^{♯}})$ for every $T\in \mathcal{L}\left(\mathcal{E}\right)$, where the space ${\mathcal{E}}^{♯}$ is constructed as the extension of $\mathcal{E}$ by the embedding of $\mathfrak{A}$ into its enveloping von Neumann algebra ${\mathfrak{A}}^{\ast \ast }$.

22 pages