Completely contractive representations for some doubly generated antisymmetric operator algebras

Abstract

Contractive weak star continuous representations of the Fourier binest algebra A \mathcal A (of Katavolos and Power) are shown to be completely contractive. The proof depends on the approximation of A \mathcal A by semicrossed product algebras A ( D ) × Z + A(\mathbb D)\times \mathbb Z_+ 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.