Abstract
In this paper, we recall the concept of Segal Fréchet algebra in a Fréchet algebra \(({\mathcal {A}},p_{\ell })\) and show that in some cases, every continuous linear left multiplier on \(({\mathcal {A}},p_{\ell })\) is a continuous linear left multiplier of any Segal Fréchet algebra \(({\mathcal {B}},q_m)\) in \(({\mathcal {A}},p_{\ell })\). As the main result, we prove that if \({\mathcal {A}}\) is a commutative Fréchet Q-algebra with an approximate identity, \({\mathcal {A}}\) is semisimple if and only if \({\mathcal {B}}\) is semisimple.
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Acknowledgments
The authors express their sincere gratitude to Professor Hans Georg Feichtinger for his invaluable comments on the manuscript. We would like to thank the referee of the paper for his/her indispensable suggestions and comments on the manuscript and very careful reading of the paper. We also would like to thank the Banach algebra center of Excellence for Mathematics, University of Isfahan.
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Abtahi, F., Rahnama, S. & Rejali, A. Semisimple Segal Fréchet algebras. Period Math Hung 71, 146–154 (2015). https://doi.org/10.1007/s10998-015-0092-1
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DOI: https://doi.org/10.1007/s10998-015-0092-1