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

Hadder, Youness

Math. Bohem. 145, No. 4, 387-399 (2020).

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.

MSC:

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

Keywords:

commutative Banach algebra; socle; \(\mathrm{kh}\)-socle; inessential element

References:

[1]	Aiena, P., Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers, Dordrecht (2004) · Zbl 1077.47001
[2]	Alexander, J. C., Compact Banach algebras, Proc. Lond. Math. Soc., III. Ser. 18 (1968), 1-18 · Zbl 0184.16502
[3]	Al-Moajil, A. H., The compactum of a semi-simple commutative Banach algebra, Int. J. Math. Math. Sci. 7 (1984), 821-822 · Zbl 0599.46072
[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 · Zbl 1015.46007
[5]	Aupetit, B., A Primer on Spectral Theory, Universitext. Springer, New York (1991) · Zbl 0715.46023
[6]	Aupetit, B.; Mouton, H. du T., Spectrum preserving linear mappings in Banach algebras, Studia Math. 109 (1994), 91-100 · Zbl 0829.46039
[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 · Zbl 0159.18502
[8]	Barnes, B. A., The Fredholm elements of a ring, Can. J. Math. 21 (1969), 84-95 · Zbl 0175.44101
[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) · Zbl 0534.46034
[10]	Boudi, N.; Hadder, Y., On linear maps preserving generalized invertibility and related properties, J. Math. Anal. Appl. 345 (2008), 20-25 · Zbl 1178.47023
[11]	Puhl, J., The trace of finite and nuclear elements in Banach algebras, Czech. Math. J. 28 (1978), 656-676 · Zbl 0394.46041
[12]	Rickart, C. E., General Theory of Banach Algebras, The University Series in Higher Mathematics. D. Van Nostrand, Princeton (1960) · Zbl 0095.09702
[13]	Smyth, M. R. F., Riesz theory in Banach algebras, Math. Z. 145 (1975), 145-155 · Zbl 0298.46049
[14]	Wang, X.; Cao, P., Spectral characterization of the kh-socle in Banach Jordan algebras, J. Math. Anal. Appl. 466 (2018), 567-572 · Zbl 1407.46041