The \(\mathrm{kh}\)-socle of a commutative semisimple Banach algebra. (English) Zbl 1499.46108

Summary: Let \(\mathcal{A}\) be a commutative complex semisimple Banach algebra. Denote by \(\mathrm{kh}(\mathrm{soc}(\mathcal{A}))\) 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.


46J05 General theory of commutative topological algebras
46J20 Ideals, maximal ideals, boundaries
47A10 Spectrum, resolvent
Full Text: DOI


[1] Aiena, P., Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers, Dordrecht (2004)
[2] Alexander, J. C., Compact Banach algebras, Proc. Lond. Math. Soc., III. Ser. 18 (1968), 1-18
[3] Al-Moajil, A. H., The compactum of a semi-simple commutative Banach algebra, Int. J. Math. Math. Sci. 7 (1984), 821-822
[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
[5] Aupetit, B., A Primer on Spectral Theory, Universitext. Springer, New York (1991)
[6] Aupetit, B.; Mouton, H. du T., Spectrum preserving linear mappings in Banach algebras, Studia Math. 109 (1994), 91-100
[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
[8] Barnes, B. A., The Fredholm elements of a ring, Can. J. Math. 21 (1969), 84-95
[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)
[10] Boudi, N.; Hadder, Y., On linear maps preserving generalized invertibility and related properties, J. Math. Anal. Appl. 345 (2008), 20-25
[11] Puhl, J., The trace of finite and nuclear elements in Banach algebras, Czech. Math. J. 28 (1978), 656-676
[12] Rickart, C. E., General Theory of Banach Algebras, The University Series in Higher Mathematics. D. Van Nostrand, Princeton (1960)
[13] Smyth, M. R. F., Riesz theory in Banach algebras, Math. Z. 145 (1975), 145-155
[14] Wang, X.; Cao, P., Spectral characterization of the kh-socle in Banach Jordan algebras, J. Math. Anal. Appl. 466 (2018), 567-572
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.