Geodesic
Irregularity of the Rate of Decrease of Sequences of Powers in the Volterra Algebra
Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 320-324

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G. R. Allan and A. M. Sinclair proved in [1] that if a commutative radical Banach algebra possesses bounded approximate identities then for every sequence (αn ) of real numbers such that limn→∞ αn = 0 there exists such that In the other direction it is shown in [6] that if is separable and if the nilpotents are dense in then for every sequence (βn ) of positive reals there exists such that (This result was given in [2] for the Volterra algebra.) 
Esterle, J. Irregularity of the Rate of Decrease of Sequences of Powers in the Volterra Algebra. Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 320-324. doi: 10.4153/CJM-1981-026-6
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[1] 1. Allan, G. R. and Sinclair, A. M., Power factorization in Banach algebras with bounded approximate identity, Studia Math. 56 (1976), 31–38. Google Scholar

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[5] 5. Esterle, J., Theorems of Gelfand Mazur type and continuity of epimorphisms from , J. Functional Analysis, to appear. Google Scholar

[6] 6. Esterle, J., Rate of decrease of sequences of powers in commutative radical Banach algebras, Pacific J. Math., to appear. Google Scholar

[7] 7. Johnson, B. E., Continuity of centralizers on Banach algebras, J. London Math. Soc. J+l (1966), 639–640. Google Scholar

[8] 8. Varopoulos, N., Continuité des formes linéaires positives sur une algèbre de Banach avec involution, C.R. Acad., Sci. Paris Ser. A258 (1964), 1121. Google Scholar

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