Contractive weak star continuous representations of the Fourier binest algebra (of Katavolos and Power) are shown to be completely contractive. The proof depends on the approximation of by semicrossed product algebras and on the complete contractivity of contractive representations of such algebras. The latter result is obtained by two applications of the Sz.-Nagy-Foias lifting theorem. In the presence of an approximate identity of compact operators it is shown that an automorphism of a general weakly closed operator algebra is necessarily continuous for the weak star topology and leaves invariant the subalgebra of compact operators. This fact and the main result are used to show that isometric automorphisms of the Fourier binest algebra are unitarily implemented.
First published in Proceedings of the American Mathematical Society 126, (8), published by the American Mathematical Society.