A free semigroup algebra is the weak operator topology closed algebra generated by a set of isometries with pairwise orthogonal ranges. The most important example is the left regular free semigroup algebra generated by the left regular representation of the free semigroup on n generators. This algebra is the appropriate non‐commutative n‐dimensional analogue of the analytic Toeplitz algebra. We develop a detailed picture of the invariant subspace structure analogous to Beurling's theorem and show that this algebra is hyper‐reflexive with distance constant at most 51. The free semigroup algebras, known as atomic, for which the range projections of words in the generators lie in an atomic masa are completely classified. This provides a complete classification for a large class of representations of the Cuntz C*‐algebras On. This allows us to describe completely the invariant subspace structure of these algebras, and thereby show that these algebras are all hyper‐reflexive. 1991 Mathematics Subject Classification: 47D25.

Abstract. The non-commutative analytic Toeplitz algebra is the wotclosed 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 C n . 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 L n is determined by the left regular representation of the free semigroup F n on n generators z 1 , . . . , z n which acts on 2 (F n ) by λ(w)ξ v = ξ wv for v, w in F n . In particular, the algebra L n is the unital, wot-closed algebra generated by the isometries L i = λ(z i ) 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

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 obtain a distance formula to an arbitrary wotclosed 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.In [9,10], we studied the non-commutative analytic Toeplitz algebras L n associated to the free semi-group on n generators. This is the wot-closed algebra generated by the left regular representation. This algebra and its norm-closed version (the non-commutative disk algebra) were introduced by Popescu [19] in an abstract sense in connection with a non-commutative von Neumann inequality and further studied in several papers [17,19,20,21,23]. We established a strong connection with the function theory on the ball B n in C n through our characterization [10] of the automorphism group. In particular, there is a natural homomorphism of L n into H ∞ (B n ). This leads to the natural question of which analytic functions are in the range of this map, and in particular, interpolation questions of the Nevanlinna-Pick type. We let F n denote the free semigroup on n generators z 1 , . . . , z n . (We allow n = ∞, but for convenience of notation, shall act as if n is finite.) Form the Hilbert space H n = 2 (F n ) with orthonormal basis ξ w for w ∈ F n . Then define the left regular representation by isometries L v ξ w = ξ vw for v, w ∈ F n . For simplification of notation, we write L i instead ofThe algebra L n is the wot-closed algebra generated by {L 1 , . . . , L n }. It is sometimes convenient to identify 2 (F n ) with the Fock space of C n = span{e 1 , . . . , e n } by sendingIn [3], Arias and Popescu use a Beurling type theorem from [17] to obtain the reflexivity of L n . We discovered this Beurling theorem independently and used it to establish hyper-reflexivity [9, Theorem 2.9]. The Beurling theorem shows that cyclic invariant subspaces of L n correspond to the ranges of isometries in the commutant R n , the right regular representation algebra. Moreover, every invariant subspace is the direct sum of cyclic invariant subspaces. Indeed an invariant subspace M is determined by a Wold decomposition byIn addition to this description of invariant subspaces, we needed to know all the invariant subspaces of codimension one. These correspond to eigenvectors of the adjoint algebra L * n . They are classified [9, Theorem 2.6] by the points of the complex n-ballThese vectors are given by the formulaThey yield the set of all wot-continuous multiplicative linear functionals on L n by the formula ϕ λ (A) = Aν λ , ν λ .Moreover the map taking A to the function Theorem 3.3]. Finally, a calculation shows that for NON-COMMUTATIVE TOEPLITZ ALGEBRAS 3Further evidence of the strong connection with analytic functions is provided by the automorphism group. There is a natural map of Aut(L n ) ...

A free semigroup algebra is the wot-closed algebra generated by an n-tuple of isometries with pairwise orthogonal ranges. The interest in these algebras arises primarily from two of their interesting features. The ®rst is that they provide useful information about unitary invariants of representations of the Cuntz-Toeplitz algebras. The second is that they form a class of nonself-adjoint operator algebras which are of interest in their own right. This class contains a distinguished representative, the``non-commutative Toeplitz algebra'', which is generated by the left regular representation of the free semigroup on n letters and denoted L n . This paper provides a general structure theorem for all free semigroup algebras, Theorem 2.6, which extends results for important special cases in the literature. The structure theorem highlights the importance of the type L representations, which are the representations which provide a free semigroup algebra isomorphic to L n . Indeed, every free semigroup algebra has a 2 Â 2 lower triangular form where the ®rst column is a slice of a von Neumann algebra and the 22 entry is a type L algebra. We develop the structure of type L algebras in more detail. In particular, we show in Corollary 1.9 that every type L representation has a ®nite ampliation with a spanning set of wandering vectors. As an application of our structure theorem, we are immediately able to characterize the radical in Corollary 2.9. With additional work, we obtain Theorem 4.5 of Russo-Dye type showing that the convex hull of the isometries in any free semigroup algebra contains the whole open unit ball. Finally we obtain some information about invariant subspaces and hyper-re¯exivity.Background. The study of the non-commutative analytic Toeplitz algebra was initiated by Popescu [19], [20], [21] in the context of dilation theory. In particular, he obtains an analogue of Beurling's theorem for the structure of its invariant subspaces. A detailed analysis of this algebra is contained in the authors' papers [10], [11], [12] and Kribs [16] and Arias-Popescu [1], [2] which develop the analytic structure of these algebras. In particular, there is a natural map from the automorphism group onto the group of conformal automorphisms of the complex n-ball. The connection with dilation theory comes from a theo-

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra A n . Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a * -extendible representation σ. A * -extendible representation of A n is regular if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given * -extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of σ(A n ) is weak- * dense in the unit ball of the associated free semigroup algebra if and only if σ is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts. 3