Characterizations of a commutative semisimple modular annihilator Banach algebra through its socle




Commutative algebras, Socle, Inessential element, Completely continuous algebra, Modular annihilator algebra


Let A be a commutative complex semisimple Banach algebra. In this paper we continue the study of kh(soc(A)). Thus we will give, among other things, some new characterizations of this ideal in terms of the closure, in the spectral radius norm, of the socle of A.

Author Biographies

Youness Hadder, Sidi Mohamed Ben Abdellah University Atlas

Laboratoire de Sciences Math´ematiques et Applications (LaSMA).

Dhar El Mahraz Faculty of Sciences.

Abdelkhalek El Amrani, Sidi Mohamed Ben Abdellah University

Departement de Mathematiques, Faculte des Sciences Dhar-Mehraz


P. Aiena, Fredholm and local spectral theory with applications to multipliers. Dordrecht: Kluwer, 2004, doi: 10.1007/1-4020-2525-4

J. C. Alexander, “Compact Banach algebras”, Proceedings of the London Mathematical Society, vol. s3-18, no. 1, pp. 1–18, 1968, doi: 10.1112/plms/s3-18.1.1

J. C. Alexander, “Algebras of compact operators”, Thesis presented for the Degree of Doctor of Philosophy. University of Edinburgh, Faculty of Science, 1967. [On line]. Available:

B. Aupetit, A primer on spectral theory. New York, NY: Springer, 1991, doi: 10.1007/978-1-4612-3048-9

B. Aupetit and H. du T. Mouton, “Spectrum preserving linear mappings in Banach algebras,” Studia mathematica, vol. 109, no. 1, pp. 91–100, 1994, doi: 10.4064/sm-109-1-91-100

B. A. Barnes, “Modular annihilator algebras”, Canadian journal of mathematics, vol. 18, pp. 566-578, 1966, doi: 10.4153/cjm-1966-055-6

B. A. Barnes, “A generalised Fredholm theory for certain maps in the regular representation of an algebra”, Canadian journal of mathematics, vol. 20, pp. 495-504, 1968, doi: 10.4153/cjm-1968-048-2

D. Blecher and C. J. Read, “Operator algebras with contractive approximate identities, IV: a large operator algebra in c0”, Transactions of the American Mathematical Society, vol. 368, no. 5, pp. 3243-3270, 2016, doi: 10.1090/tran/6590

F. F. Bonsall and J. Duncan, Complete normed algebras. Berlin: Springer, 1973, doi: 10.1007/978-3-642-65669-9

N. Boudi and Y. Hadder, “On linear maps preserving generalized invertibility on commutative algebras”, The Rocky Mountain journal of mathematics, vol. 42, no. 4, pp. 1007-1014, 2012, doi: 10.1216/rmj-2012-42-4-1107

B. A. Barnes, G. J. Murphy, M. R. F. Smyth, and T. T. West, Riesz and Fredholm theory in Banach algebras. Boston: Pitman, 1982.

P. Enflo, “A counterexample to the approximation problem in Banach spaces”, Acta mathematica, vol. 130, pp. 309-317, 1973, doi: 10.1007/bf02392270

B. Forrest, “Arens regularity and discrete groups”, Pacific journal of mathematics, vol. 151, no. 2, pp. 217-227, 1991, doi 10.2140/pjm.1991.151.217

G. Gandroulakis and T. Schlumprecht, “Strictly singular, non-compact operators exist on the space of Gowers and Maurey”, Journal of the London Mathematical Society, vol. 64, no. 3, pp. 655-674, 2001, doi: 10.1112/S0024610701002769

Y. Hadder, “The Kh-socle of a commutative semisimple Banach algebra”, Mathematica bohemica, vol. 145, no. 4, pp. 387-399, 2020, doi: 10.21136/MB.2019.0106-18

C. S. Herz, “Harmonic synthesis for subgroups”, Annales de l'Institut Fourier, vol. 23, no. 3, pp. 91-123, 1973, doi: 10.5802/aif.473

L. Loomis, An introduction to abstract harmonic analysis. Toronto: Van Nostrand, 1953. [On line]. Available:

T. W. Palmer, Banach algebras and the general theory of *-algebras: algebras and Banach algebras, vol. 1. Cambridge: Cambridge University Press, 1994, doi: 10.1017/cbo9781107325777

A. E. Taylor and D. C. Lay, Introduction to functional analysis, 2nd ed. New York, NY: Wiley, 1980

C. E. Rickart, General theory of Banach algebras. Princeton, NJ: Van Nostrand, 1960.

M. R. F. Smyth, “Riesz theory in Banach algebras”, Mathematische zeitschrift, vol. 145, no. 2, pp. 145-155, 1975, doi: 10.1007/bf01214779

K. Vala, “Sur les éléments compacts d’une algèbre normée”, Annales Academiae. Scientiarum Fennicae. Series A, I. Mathematica, no. 407, pp. 1-8, 1967, doi: 10.5186/aasfm.1967.407



How to Cite

Y. Hadder and A. El Amrani, “Characterizations of a commutative semisimple modular annihilator Banach algebra through its socle”, Proyecciones (Antofagasta, On line), vol. 40, no. 3, pp. 697-709, May 2021.