2024/05/08 by Rasoul Eskandari, Mohammad Sal Moslehian, Eskandari, Rasoul +1
Mathematics · #46L10 #47C15 #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Secondary 46L05 #Spectral Theory in Mathematical Physics #{Primary 46L08
paper · pdf · doi:10.48550/arxiv.2405.04850
openalex publication_date 2024/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Eω of \mathscr E, which is the completion of the quotient space \mathscr E/\mathscr Nω, where \mathscr Nω=\x∈ \mathscr E:ω⟨ x,x⟩=0\. Let \mathscr H and \mathscr K be closed submodules of \mathscr E such that \mathscr H∩ \mathscr K is orthogonally complemented, and let ω=∑j=1∞λjωj, where λj>0, ∑j=1∞λj=1, and ωj's are positive linear functionals on \mathscr A. We prove that if (\mathscr H∩ \mathscr K)ωj=\mathscr Hωj∩ \mathscr Kωj for each j, then (\mathscr H∩ \mathscr K)ω=\mathscr Hω∩ \mathscr Kω . 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 ιω(\mathscr L) is not dense in \mathscr Eω?'' 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.