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TOPOLOGICALLY NOETHERIAN BANACH ALGEBRAS
Keywords:
Banach algebra, topologically Noetherian, chain conditions.DOI:
https://doi.org/10.17654/0972555523018Abstract
A Banach algebra $\mathfrak{A}$ is said to be topologically left Noetherian (TLN) if for any increasing chain of closed left ideals $I_1 \subset I_2 \subset \cdots$, there exists an $n \in \mathbb{N}$ such that $I_m=I_n$ for all $m \geq n$. We study some basic properties of this class of Banach algebras. In particular, we show that if $\mathfrak{A}$ has an essential socle, then it is of finite dimension.
Received: May 15, 2023
Accepted: July 5, 2023
References
A. Al-Moajil, The compactum and finite dimensionality in Banach algebras, Int. J. Math. Math. Sci. 5 (1982), 275-280.
A. Al-Ahmari, F. Aldosray and M. Mabrouk, Banach algebras satisfying certain chain conditions on closed ideals, Stud. Sci. Math. Hung. 57(3) (2020), 290-297.
B. Aupetit, Propriétés spectrales des algèbres de Banach, Springer-Verlag, Berlin New York, Vol. 735, 1979.
B. Aupetit, Inessential elements in Banach algebras, Bull. Lond. Math. Soc. 18(5) (1986), 493-497.
B. Aupetit, A Primer on Spectral Theory, Universitext, Springer-Verlag, 1991.
F. F. Bonsall and J. Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, Springer Berlin Heidelberg, 2012.
C.-Y. Chou, Notes on the separability of C*-algebras, Taiwan. J. Math. 16(2) (2012), 555-559.
S. Giotopoulos and M. Roumeliotis, Algebraic ideals of semiprime Banach algebras, Glasg. Math. J. 33(3) (1991), 359-363.
S. Grabiner, Finitely generated, Noetherian, and Artinian Banach modules, Indiana Univ. Math. J. 26(3) (1977), 413-425.
R. Grigorchuk, M. Musat and M. Rordam, Just-infinite C*-algebras, Comment. Math. Helv. 93(1) (2018), 157-201.
R. Prosser, On the Ideal Structure of Operator Algebras, Mem. Amer. Math. Soc., Vol. 45, 1963.
C. Rachid, A concept of finiteness in topological algebras, Contemp. Math. 427 (2007), 131-137.
C. E. Rickart, General theory of Banach algebras, University Series in Higher Mathematics, Van Nostrand, 1960.
W. Rudin, The closed ideals in an algebra of analytic functions, Canad. J. Math. 9 (1957), 426-434.
M. Sinclair and W. Tullo, Noetherian Banach algebras are finite dimensional, Math. Ann. 211(2) (1974), 151-153.
Sin-Ei Takahasi, Finite dimensionality in socle of Banach algebras, Int. J. Math. Math. Sci. 7(3) (1984), 519-522.
A. W. Tullo, Algebraic conditions on Banach algebras, Ph. D. Thesis, University of Stirling (United Kingdom), 1974.
J. T. White, Left ideals of Banach algebras and dual Banach algebras, Banach Algebras and Applications: Proceedings of the International Conference held at the University of Oulu, July 3-11, 2017, Mahmoud Filali, ed., De Gruyter, Berlin, Boston, 2020, pp. 227-254.
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