Schatten norms on Hilbert $C^*$-modules via pure states

Abstract

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 $\mathfrak{A}^{**}$.