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Irregularity of the Rate of Decrease of Sequences of Powers in the Volterra Algebra

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 lim n→∞ 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.)