Geodesic
The kh-socle of a commutative semisimple Banach algebra
Mathematica Bohemica, Tome 145 (2020) no. 4, pp. 387-399
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Let $\mathcal {A}$ be a commutative complex semisimple Banach algebra. Denote by ${\rm kh}({\rm soc}(\mathcal {A}))$ the kernel of the hull of the socle of $\mathcal {A}$. In this work we give some new characterizations of this ideal in terms of minimal idempotents in $\mathcal {A}$. This allows us to show that a ``result'' from Riesz theory in commutative Banach algebras is not true.
Let $\mathcal {A}$ be a commutative complex semisimple Banach algebra. Denote by ${\rm kh}({\rm soc}(\mathcal {A}))$ the kernel of the hull of the socle of $\mathcal {A}$. In this work we give some new characterizations of this ideal in terms of minimal idempotents in $\mathcal {A}$. This allows us to show that a ``result'' from Riesz theory in commutative Banach algebras is not true.
DOI : 10.21136/MB.2019.0106-18
Classification : 46J05, 46J20, 47A10
Keywords: commutative Banach algebra; socle; kh-socle; inessential element 
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Hadder, Youness. The kh-socle of a commutative semisimple Banach algebra. Mathematica Bohemica, Tome 145 (2020) no. 4, pp. 387-399. doi: 10.21136/MB.2019.0106-18

[1] Aiena, P.: Fredholm and Local Spectral Theory, with Applications to Multipliers. Kluwer Academic Publishers, Dordrecht (2004). | DOI | MR | JFM

[2] Alexander, J. C.: Compact Banach algebras. Proc. Lond. Math. Soc., III. Ser. 18 (1968), 1-18. | DOI | MR | JFM

[3] Al-Moajil, A. H.: The compactum of a semi-simple commutative Banach algebra. Int. J. Math. Math. Sci. 7 (1984), 821-822. | DOI | MR | JFM

[4] Androulakis, G., Schlumprecht, T.: Strictly singular, non-compact operators exist on the space of Gowers and Maurey. J. Lond. Math. Soc., II. Ser. 64 (2001), 655-674. | DOI | MR | JFM

[5] Aupetit, B.: A Primer on Spectral Theory. Universitext. Springer, New York (1991). | DOI | MR | JFM

[6] Aupetit, B., Mouton, H. du T.: Spectrum preserving linear mappings in Banach algebras. Studia Math. 109 (1994), 91-100. | DOI | MR | JFM

[7] Barnes, B. A.: A generalized Fredholm theory for certain maps in the regular representations of an algebra. Can. J. Math. 20 (1968), 495-504. | DOI | MR | JFM

[8] Barnes, B. A.: The Fredholm elements of a ring. Can. J. Math. 21 (1969), 84-95. | DOI | MR | JFM

[9] Barnes, B. A., Murphy, G. J., Smyth, M. R. F., West, T. T.: Riesz and Fredholm Theory in Banach Algebras. Research Notes in Mathematics 67. Pitman Advanced Publishing Program, Boston (1982). | MR | JFM

[10] Boudi, N., Hadder, Y.: On linear maps preserving generalized invertibility and related properties. J. Math. Anal. Appl. 345 (2008), 20-25. | DOI | MR | JFM

[11] Puhl, J.: The trace of finite and nuclear elements in Banach algebras. Czech. Math. J. 28 (1978), 656-676. | DOI | MR | JFM

[12] Rickart, C. E.: General Theory of Banach Algebras. The University Series in Higher Mathematics. D. Van Nostrand, Princeton (1960). | MR | JFM

[13] Smyth, M. R. F.: Riesz theory in Banach algebras. Math. Z. 145 (1975), 145-155. | DOI | MR | JFM

[14] Wang, X., Cao, P.: Spectral characterization of the kh-socle in Banach Jordan algebras. J. Math. Anal. Appl. 466 (2018), 567-572. | DOI | MR | JFM

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