Abstract
In this paper, we provide necessary conditions for the convergence of a net which is monotone and bounded with respect to the diamond order. Also, we show that each bounded sequence in a Hilbert \({\mathcal{A}}\)-module has a convergent subsequence. Moreover, we prove that a module map \(\Phi\) preserves the orthonormality of a basis whenever it is a surjective \(\varphi\)-morphism. Finally, we present sufficient conditions for a Hilbert \({\mathcal{A}}\)-module to be finite-dimensional.
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(Submitted by A. I. Volodin)
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Janfada, A.R., Farokhi-Ostad, J. Convergence of Bounded Sequences in Hilbert \(\boldsymbol{C}^{\mathbf{*}}\)-Modules. Lobachevskii J Math 43, 2493–2500 (2022). https://doi.org/10.1134/S1995080222120150
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DOI: https://doi.org/10.1134/S1995080222120150