Distance between the elements of a semi-group in a Banach algebra

Let (a(t))(t>0) be a semigroup in a Banach algebra, and let n be a positive integer. We show that if lim sup(n-->infinity) parallel toa(t) - a(t(n-1))parallel to < n/(n+1)(1+1/n) then either a(t) = 0 for t > 0, or the closed subalgebra A generated by the semigroup, (a(t))(t>0) is unital, and there exists u is an element of A such that a(t) = e(tu) for t > 0. A simple example shows that the constant n/(n+1)(1+1/n) is best possible. (C) 2002 Elsevier Science (USA).