Erratum to: The algebraic structure of non-commutative analytic Toeplitz algebras

Abstract

The algebraic structure of the non-commutative analytic Toeplitz algebra Ln is developed in the original article. Some of the results fail for the case n = ∞, and this implies that certain other results are not established in this case. In Theorem 3.2 of the original article, we showed there is continuous surjection πn,k from Repk(Ln), the space of completely contractive representations of Ln into the k × k matrices Mk , onto the closed unit ball Bn,k ofRn(Mk) by evaluation at the generators. It is further claimed that if T = [T1, . . . , Tn] ∈ Rn(Mk) with ‖T ‖ < 1, then there is a unique representation in π−1 n,k (T ). Further information is obtained for k = 1 in Theorem 3.3 of the original article. Our proof of these results is valid for n < ∞, however, for n = ∞ the uniqueness claim is incorrect. An example due to Michael Hartz (see [2, Example 2.4]) shows that π−1 ∞,1(0) is very large—it contains a copy of the βN\N. The difficulty in the proof of Theorems 3.2 and 3.3 of the original article stems from the use of the factorization A = W X used in Lemma 3.1 of the original article. In the case n = ∞, this factorization comes from Corollary 2.9. The problem is that the infinite sum in Corollary 2.9 converges in the strong topology, not the norm topology, so that when the representation is not strongly continuous (or equivalently,