## Mathematics, Department of

#### Date of this Version

2015

#### Citation

Math. Ann. (2015) 361:1123–1124

#### Abstract

The algebraic structure of the non-commutative analytic Toeplitz algebra L*n *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 *π**n**,**k *from Rep*k **(*L*n**)*, the space of completely contractive representations of L*n *into the *k *× *k *matrices M*k *, onto the closed unit ball B_{n}*,**k *of *R*_{n}*(*M*k **) *by evaluation at the generators. It is further claimed that if *T *= [*T*1*, . . . , **Tn*] ∈ *R**n**(*M*k **) *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 *β*N\N.

## Comments

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