A separation theorem for Hilbert $W^*$-modules
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| Format: | Preprint |
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2024
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| _version_ | 1866915002210844672 |
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| author | Eskandari, Rasoul Moslehian, Mohammad Sal |
| author_facet | Eskandari, Rasoul Moslehian, Mohammad Sal |
| contents | Let $\mathscr E$ be a Hilbert $\mathscr A$-module over a $C^*$-algebra $\mathscr A$. For each positive linear functional $ω$ on $\mathscr A$, we consider the localization $\mathscr E_ω$ of $\mathscr E$, which is the completion of the quotient space $\mathscr E/\mathscr {N}_ω$, where $\mathscr N_ω=\{x\in \mathscr E:ω\langle x,x\rangle=0\}$. Let $\mathscr H$ and $\mathscr K$ be closed submodules of $\mathscr E$ such that $\mathscr H\cap \mathscr K$ is orthogonally complemented, and let $ω=\sum_{j=1}^{\infty}λ_jω_j$, where $λ_j>0$, $\sum_{j=1}^{\infty}λ_j=1$, and $ω_j$'s are positive linear functionals on $\mathscr A$. We prove that if $(\mathscr H\cap \mathscr K)_{ω_j}=\mathscr H_{ω_j}\cap \mathscr K_{ω_j}$ for each $j$, then \[ (\mathscr H\cap \mathscr K)_ω=\mathscr H_ω\cap \mathscr K_ω\,. \] Furthermore, let $\mathscr L$ be a closed submodule of a Hilbert $\mathscr A$-module $\mathscr E$ over a $W^*$-algebra $\mathscr A$. We pose the following separation problem: ``Does there exist a normal state $ω$ such that $ι_ω(\mathscr L)$ is not dense in $\mathscr E_ω$?'' In this paper, among other results, we give an affirmative answer to this problem, when $\mathscr E$ is a self-dual Hilbert $C^*$-module over a $W^*$-algebra $\mathscr A$ such that $\mathscr E\backslash \mathscr L$ has a nonempty interior with respect to the weak$^*$-topology. This is a step toward answering the above problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_04850 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A separation theorem for Hilbert $W^*$-modules Eskandari, Rasoul Moslehian, Mohammad Sal Operator Algebras Functional Analysis {Primary 46L08, Secondary 46L05, 46L10, 47C15 Let $\mathscr E$ be a Hilbert $\mathscr A$-module over a $C^*$-algebra $\mathscr A$. For each positive linear functional $ω$ on $\mathscr A$, we consider the localization $\mathscr E_ω$ of $\mathscr E$, which is the completion of the quotient space $\mathscr E/\mathscr {N}_ω$, where $\mathscr N_ω=\{x\in \mathscr E:ω\langle x,x\rangle=0\}$. Let $\mathscr H$ and $\mathscr K$ be closed submodules of $\mathscr E$ such that $\mathscr H\cap \mathscr K$ is orthogonally complemented, and let $ω=\sum_{j=1}^{\infty}λ_jω_j$, where $λ_j>0$, $\sum_{j=1}^{\infty}λ_j=1$, and $ω_j$'s are positive linear functionals on $\mathscr A$. We prove that if $(\mathscr H\cap \mathscr K)_{ω_j}=\mathscr H_{ω_j}\cap \mathscr K_{ω_j}$ for each $j$, then \[ (\mathscr H\cap \mathscr K)_ω=\mathscr H_ω\cap \mathscr K_ω\,. \] Furthermore, let $\mathscr L$ be a closed submodule of a Hilbert $\mathscr A$-module $\mathscr E$ over a $W^*$-algebra $\mathscr A$. We pose the following separation problem: ``Does there exist a normal state $ω$ such that $ι_ω(\mathscr L)$ is not dense in $\mathscr E_ω$?'' In this paper, among other results, we give an affirmative answer to this problem, when $\mathscr E$ is a self-dual Hilbert $C^*$-module over a $W^*$-algebra $\mathscr A$ such that $\mathscr E\backslash \mathscr L$ has a nonempty interior with respect to the weak$^*$-topology. This is a step toward answering the above problem. |
| title | A separation theorem for Hilbert $W^*$-modules |
| topic | Operator Algebras Functional Analysis {Primary 46L08, Secondary 46L05, 46L10, 47C15 |
| url | https://arxiv.org/abs/2405.04850 |