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 $\mathcal {L}(\mathscr {E})$. The sets of all states and pure states on $\mathfrak {A}$ are denoted by $\mathcal {S}(\mathfrak {A})$ and $\mathcal {P}( \mathfrak {A})$, respectively. For $\tau \in \mathcal {S}( \mathfrak {A})$, let us define $\mathcal {N}^{\mathscr {E}}_{\tau }:= \lbrace x\in \mathscr {E}:\tau ( \langle x,x\rangle )=0 \rbrace $. The Hilbert completion of $\mathscr {E}/{\mathcal {N}^{\mathscr {E}}_{\tau }}$ is denoted by $\mathscr {E}_{\tau }$. For $T\in \mathcal {L}(\mathscr {E})$, the operator $T_{\mathscr {E}_\tau }\in \mathbb {B}( \mathscr {E}_\tau )$, is defined by $ T_{\mathscr {E}_\tau }(x+\mathcal {N}^{\mathscr {E}}_{\tau })=Tx+\mathcal {N}^{\mathscr {E}}_{\tau }$. In this paper, we show that $\mathscr {E}_{\tau }={\mathscr {E}}/{\mathcal {N}^{\mathscr {E}}_{\tau }}$ 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 $\pi ^{\mathscr {E}}_k(\cdot )$ for $k\geq 1$. We prove that $\pi ^{\mathscr {E}}_k(T)\leq \sup _{\tau \in \mathcal {P}( \mathfrak {A})}\|T_{\mathscr {E}_\tau }\|_{(k)}\leq {\pi }^{\mathscr {E}^{\sharp }}_k(T_{\mathscr {E}^{\sharp }})$ for every $T\in \mathcal {L}(\mathscr {E})$, 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 $\mathfrak {A}^{**}$. Thus, the norm $\pi ^{\mathscr {E}}_k(\cdot )$ preserves the Schatten properties, as in Hilbert spaces, particularly when $\pi ^{\mathscr {E}}_k(T) = {\pi }^{\mathscr {E}^{\sharp }}_k(T_{\mathscr {E}^{\sharp }})$ for all $T \in \mathcal {L}(\mathscr {E})$. This equality holds for $\mathscr {E}$ over commutative $C^*$-algebras with a frame. Moreover, it holds for $\mathscr {E}$ over $C^*$-algebras of compact operators.