Analytic representation theory of (R,+)

We prove a theorem of Dixmier and Malliavin type for analytic vectors of bounded representations of (R,+): Let (π,E) be a bounded representation of (R,+) in a Banach space E and let A be the convolution algebra of analytic vectors for the left regular representation of R on L1(R), then A∗A=A and π(A)E=Eω.