Three spectral notions for representations of commutative Banach algebras
Annales de l'Institut Fourier, Volume 25 (1975) no. 2, p. 1-32

Let T be a bounded representation of a commutative Banach algebra B. The following spectral sets are studied. Λ 1 (T): the Gelfand space of the quotient algebra B/ Ker T. Λ 2 (T): the Gelfand space of the operator algebra Im T. Λ 3 (T): those characters ϕ of B for which the inequalities T b x-b ^(ϕ)x<εx, bF, have a common solution x0, for any ε>0 and any finite subset F of B. A theorem of Beurling on the spectrum of L -functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of our theory by taking for T the regular representation and its adjoint.

Soit T une représentation bornée d’une algèbre de Banach commutative B. Les ensembles spectraux suivants sont étudiés. Λ 1 (T) : le spectre de l’algèbre quotient B/ Ker T. Λ 2 (T) : le spectre de l’algèbre d’opérateurs Im T. Λ 3 (T) : les caractères ϕ de B, tels que les inégalités T b x-b ^(ϕ)x<εx, bF, admettent une solution commune x0, quels que soient ε>0 et F, sous-ensemble fini de B. Un théorème de Beurling sur le spectre des fonctions de L et des résultats de Slodkowski et Zelazko sur les diviseurs de zéro topologiques simultanés sont obtenus comme des cas particuliers de notre théorie en prenant pour T la représentation régulière et son adjoint.

@article{AIF_1975__25_2_1_0,
     author = {Domar, Yngve and Lindahl, Lars-Ake},
     title = {Three spectral notions for representations of commutative Banach algebras},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {25},
     number = {2},
     year = {1975},
     pages = {1-32},
     doi = {10.5802/aif.553},
     zbl = {0301.46045},
     mrnumber = {53 \#3714},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1975__25_2_1_0}
}
Domar, Yngve; Lindahl, Lars-Ake. Three spectral notions for representations of commutative Banach algebras. Annales de l'Institut Fourier, Volume 25 (1975) no. 2, pp. 1-32. doi : 10.5802/aif.553. http://www.numdam.org/item/AIF_1975__25_2_1_0/

[1] R. Arens, Extensions of Banach algebras, Pacific J. Math., 10 (1960), 1-16. | MR 23 #A4031 | Zbl 0095.09801

[2] A. Beurling, Un théorème sur les fonctions bornées et uniformément continues sur l'axe réel, Acta Math., 77 (1945), 127-136. | MR 7,61f | Zbl 0061.13311

[3] M.D. Choi and C. Davis, The spectral mapping theorem for joint approximate point spectrum, Bull. Amer. Math. Soc., 80 (1974), 317-321. | MR 48 #12104 | Zbl 0276.47001

[4] J. Dixmier, Sur un théorème de Banach, Duke Math. J., 15 (1948), 1057-1071. | MR 10,306g | Zbl 0031.36301

[5] Y. Domar, Harmonic analysis based on certain commutative Banach algebras, Acta Math., 96 (1956), 1-66. | MR 17,1228a | Zbl 0071.11302

[6] Y. Domar, On spectral analysis in the narrow topology, Math. Scand., 4 (1956), 328-332. | MR 19,413b | Zbl 0078.29401

[7] Y. Domar, Some results on narrow spectral analysis, Math. Scand., 20 (1967), 5-18. | MR 36 #690 | Zbl 0166.11201

[8] V.P. Gurarii, Spectral synthesis of bounded functions on the half-axis, Funkcional. Anal. i Prilozen, 3 n° 4 (1969), 34-48 (= Functional Anal. Appl., 3 (1969), 282-294). | MR 41 #743b | Zbl 0205.42304

[9] L-Å. Lindahl, On narrow spectral analysis, Math. Scand., 26 (1970), 149-164. | MR 41 #5893 | Zbl 0189.44601

[10] L-Å. Lindahl, On ideals of joint topological divisors of zero, to appear in Studia Math., 53. | MR 51 #13693 | Zbl 0268.46049

[11] Yu. I. Lyubich, On the spectrum of a representation of an abelian topological group, Dokl. Akad. Nauk. SSSR, 200 (1971), 777-780 (= Soviet Math. Dokl., 12 (1971), 1482-1486). | Zbl 0235.22008

[12] Yu. I. Lyubich, V.I. Matsaev and G.M. Fel'Dman, On representations with a separable spectrum, Funkcional. Anal. i Prilozen, 7 no. 2 (1973), 52-61 (= Functional Anal. Appl., 7 (1973), 129-136. | Zbl 0285.22004

[13] B. Nyman, On the one-dimensional translation group and semi-group in certain function spaces, Uppsala, (1950), 55 pp. (Thesis). | MR 12,108g | Zbl 0037.35401

[14] C. Rickart, General theory of Banach algebras, Van Nostrand, (1960). | MR 22 #5903 | Zbl 0095.09702

[15] W. Rudin, Boundary values of continuous analytic functions, Proc. Amer. Math. Soc., 7 (1956), 808-811. | MR 18,472c | Zbl 0073.29701

[16] Z. Slodkowski, On ideals consisting of joint topological divisors of zero, Studia Math., 48 (1973), 83-88. | MR 50 #1003 | Zbl 0271.46046

[17] N. Varopoulos, Groups of continuous functions in harmonic analysis, Acta Math., 125 (1970), 109-154. | MR 43 #7868 | Zbl 0214.38102

[18] C.R. Warner, Weak-* dense subspaces of L∞ (R), Math. Ann., 197 (1972), 180-181. | MR 47 #2336 | Zbl 0221.46027

[19] W. Żelazko, On a certain class of non-removable ideals in Banach algebras, Studia Math., 44 (1972), 87-92. | MR 47 #2376 | Zbl 0213.40603