1 Erratum to: Math. Ann. 311, 275–303 (1998) DOI 10.1007/s002080050188

The algebraic structure of the non-commutative analytic Toeplitz algebra \({\mathfrak {L}}_n\) is developed in the original article. Some of the results fail for the case \(n=\infty \), 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 \(\pi _{n,k}\) from \( {\text {Rep}}_k({\mathfrak {L}}_n)\), the space of completely contractive representations of \({\mathfrak {L}}_n\) into the \(k\times k\) matrices \({\mathfrak {M}}_k\), onto the closed unit ball \(\overline{{\mathbb {B}}_{n,k}}\) of \({\mathcal {R}}_n({\mathfrak {M}}_k)\) by evaluation at the generators. It is further claimed that if \(T = [T_1,\dots ,T_n] \in {\mathcal {R}}_n({\mathfrak {M}}_k)\) with \(\Vert T\Vert <1\), then there is a unique representation in \(\pi _{n,k}^{-1}(T)\). Further information is obtained for \(k=1\) in Theorem 3.3 of the original article. Our proof of these results is valid for \(n<\infty \), however, for \(n=\infty \) the uniqueness claim is incorrect. An example due to Michael Hartz (see [2, Example 2.4]) shows that \(\pi _{\infty ,1}^{-1}(0)\) is very large—it contains a copy of the \(\beta {\mathbb {N}}{\setminus }{\mathbb {N}}\).

The difficulty in the proof of Theorems 3.2 and 3.3 of the original article stems from the use of the factorization \(A=WX\) used in Lemma 3.1 of the original article. In the case \(n=\infty \), 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 \(\Phi \) is not strongly continuous (or equivalently, wot-continuous), the calculation of the norm in the last lines of the proof of Lemma 3.1 is invalid.

Theorem 3.3 is used throughout Section 4 of the original article. The results of Section 4 are valid when \(n<\infty \). However, we are no longer certain of the validity of the following results when \(n=\infty \): Theorem 4.1, Proposition 4.3, Theorem 4.6, Theorem 4.7, and Corollary 4.12. Proposition 4.3 is used in Theorem 4.6 to establish that all automorphisms are wot-continuous. When attention is restricted to the class of wot-continuous automorphisms, our proofs of Theorem 4.1 and Theorem 4.7 remain valid when \(n=\infty \). We do not know whether wot-continuity is automatic.

It is worth noting that the results of Section 4 are valid for the class of isometric isomorphisms even when \(n=\infty \). This is because any isometric automorphism is wot-continuous. Indeed, by [3, Theorem 3.3], \({\mathfrak {L}}_n\) has a unique predual, \({\mathfrak {L}}_n{}_*\), which sits naturally inside the dual space \({\mathfrak {L}}_n^*\). The uniqueness of the predual implies that \(\Theta ^*\) fixes the image of \({\mathfrak {L}}_n{}_*\), so there is a surjective isometry \(\Theta _*:{\mathfrak {L}}_n{}_*\rightarrow {\mathfrak {L}}_n{}_*\) such that \(\Theta =(\Theta _*)^*\). Hence \(\Theta \) is weak-\(*\) continuous. By [1, Corollary 2.12], the weak-\(*\) and wot topologies on \({\mathfrak {L}}_n\) coincide, so \(\Theta \) is wot-continuous.