Multipliers and representations of noncommutative disc algebras

Abstract. The non-commutative disc algebra An, n ≥ 2, is the norm clo-sure of the non-selfadjoint algebra generated by the left regular represen-tation of F+n, the free semigroup on n generators. We present examples of contractive representation of An, n ≥ 2, which are not completely contrac-tive. This answers a question from [30]. We characterize the (completely) contractive Schur multipliers of An which are indexed by subsets of F n, and we show that for n ≥ 2, B(F+n) ⊂M0(F+n) and M0(F+n)\B(F+n) 6 = ∅. B(F+n) is the space of coefficients of contractive representations of F+n and M0(F n) is the space of completely bounded Schur multipliers of An. 1