Closed ideals in C∞ with the Duhamel product as multiplication

Let (C∞, ) denote the algebra of infinitely differentiable functions in [0, 1] with Duhamel product (f g) = d dx x 0 f (x − t)g(t)dt as multiplication. We describe all the closed ideals in (C∞, ). As a consequence we obtain that the integration operator I, (If )(x) = x 0 f (t)dt, is unicellular in the space C∞[0, 1], which is the solution of a long-standing problem.  2004 Elsevier Inc. All rights reserved