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Topologically Spectral Algebras and One-Side Topological Radicals

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Abstract

Topological spectral seminorm on topological algebras is studied. Main properties of topologically spectral algebras and topologically spectral subalgebras of topological algebras are described. Moreover, the descriptions of one-sided topological radicals of topological algebras are given.

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Notes

  1. That is, a topological algebra which topology is given by a family \(\{\Vert \cdot \Vert _{\alpha }: \alpha \in \Lambda \}\) of submultiplicative seminorms, i.e., \(\Vert xy\Vert _{\alpha }\le \Vert x\Vert _{\alpha }\Vert y\Vert _{\alpha }\) for each \(\alpha \in \Lambda \) and all \(x,y\in A\).

  2. A topological algebra A is a TQ algebra if \(\mathrm {Tqinv}(A)\) is open in A. Main properties of TQ algebras are given in [3] and [4].

  3. Here and later on, \(cl_{A}(U)\) denotes the closure of U in the topology of A.

  4. Here and later on, \(\mathrm {rad}(A)=\mathrm {rad}_{\ell }(A)\cap \mathrm {rad}_{r}(A)\).

  5. Here, \(\mathrm {Tqinv}(A)=\mathrm {Tqinv}_{\ell }(A)\cap \mathrm {Tqinv}_{r}(A)\).

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Acknowledgements

Research of the first author is in part supported by the institutional research funding IUT20–57 of the Estonian Ministry of Education and Research. The second author is supported by a scholarship of CONACyT Mexico.

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Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of Tartu, 2 Liivi St. room 614, Tartu, Estonia

    Mati Abel & Yuliana de Jesús Zárate-Rodríguez

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  1. Mati Abel
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  2. Yuliana de Jesús Zárate-Rodríguez
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Correspondence to Yuliana de Jesús Zárate-Rodríguez.

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Communicated by Hamid Reza Ebrahimi Vishki.

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Abel, M., de Jesús Zárate-Rodríguez, Y. Topologically Spectral Algebras and One-Side Topological Radicals. Bull. Iran. Math. Soc. 44, 305–327 (2018). https://doi.org/10.1007/s41980-018-0022-0

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  • DOI: https://doi.org/10.1007/s41980-018-0022-0

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