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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

The non-commutative analytic Toeplitz algebra is the WOT-closed algebra generated by the left regular representation of the free semigroup onn generators. We obtain a distance formula to an arbitrary WOT-closed right ideal and thereby show that the quotient is completely isometrically isomorphic to the compression of the algebra to the orthogonal complement of the range of the ideal. This is used to obtain Nevanlinna-Pick type interpolation theorems

The non-commutative analytic Toeplitz algebra is the wot– closed algebra generated by the left regular representation of the free semigroup on n generators. We develop a detailed picture of the algebraic structure of this algebra. In particular, we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n-ball. The k-dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k × kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of H∞ over the unit disk. In [6, 17, 18, 20], a good case is made that the appropriate analogue for the analytic Toeplitz algebra in n non-commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators. The papers cited obtain a compelling analogue of Beurling’s theorem and inner–outer factorization. In this paper, we add further evidence. The main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of Cn. The kernel is the subgroup of quasi-inner automorphisms, which are trivial modulo the wot-closed commutator ideal. Additional evidence of analytic properties comes from the structure of k-dimensional (completely contractive) representations, which have a structure very similar to the fibration of the maximal ideal space of H∞ over the unit disk. An important tool in our analysis is a detailed structure theory for wot-closed right ideals. Curiously, left ideals remain more obscure. The non-commutative analytic Toeplitz algebra Ln is determined by the left regular representation of the free semigroup Fn on n generators z1, . . . , zn which acts on `2(Fn) by λ(w)ξv = ξwv for v, w in Fn. In particular, the algebra Ln is the unital, wot-closed algebra generated by the isometries Li = λ(zi) for 1 ≤ i ≤ n. This algebra and its norm-closed version (the noncommutative disk algebra) were introduced by Popescu [19] in an abstract sense in connection with a non-commutative von Neumann inequality and 1991 Mathematics Subject Classification. 47D25. March 9, 1997; October 9, 1997 final draft. First author partially supported by an NSERC grant and a Killam Research Fellowship. Second author partially supported by an NSF grant.

In this note, we classify completely distributive CSLs up to approximate unitary equivalence. Our proof uses a new characterization of complete distributivity and leads to a generalization of a result of Arveson on ordered group lattices. As consequences of our results, we obtain a similarity theorem for hyperreflexive, completely distributive CSLs and some new perturbation results.

In 1972 Kadison and Kastler [12] asked whether two close von Neumann subalgebras of ~ ( J f ) are unitarily equivalent by a unitary close to the identity. This question was answered affirmatively for certain classes of von Neumann algebras in [5, 4, 6, 15]. The nonselfadjoint case was first considered by Lance who showed in [13] that close nest algebras are similar via a similarity which is close to the identity. For nest algebras, there is a close connection between the question of whether closeness of algebras implies similarity and the classification of nest algebras up to similarity: this connection may be seen in the papers [7] and [14] and we shall discuss it further in Remark 8 below. The desire to extend the classification of nest algebras to the class of reflexive algebras with commutative subspace lattice was one of the motivations for the work on perturbations of matrix algebras begun in [3] and continuing with perturbations of suboolean operator algebras in [8]. In [16], we proved that if ~ ' is a hyperreflexive CSL algebra whose lattice is atomic and satisfies a certain technical condition, then any other CSL algebra whose unit ball is sufficiently close to the unit ball of d has the property that ~ is similar to d . This result was extended in [9] to the class of all hyperreflexive CSL algebras whose lattice is completely distributive. The purpose of the present paper is to remove the hypothesis of hyperreflexivity and complete distributivity from the results described above. We show that the class of all CSL algebras is stable under small perturbations. In particular, we prove Theorem 6 below, which gives a complete answer to Problem 5 of the section entitled "Open Problems" of [11].