Factoring in Non-commutative Analytic Toeplitz Algebras

The non-commutative analytic Toeplitz algebra is the wot-closed algebra generated by the left regular representation of the free semigroup on n generators. The structure theory of contractions in these algebras is examined. Each is shown to have an H∞ functional calculus. The isometries defined by words are shown to factor only as the words do over the unit ball of the algebra. This turns out to be false over the full algebra. The natural identification of wot-closed left ideals with invariant subspaces of the algebra is shown to hold only for a proper subcollection of the subspaces. In [2] and [3], the algebraic and invariant subspace structures of the non-commutative analytic Toeplitz algebras were developed extensively. Several analogues of the analytic Toeplitz algebra were obtained. Many of these results came from a lucid characterization of the wotclosed right ideals of these algebras. Although technical difficulties were encountered, a similar characterization of the left ideals was expected. In this paper, it is shown that, although it holds for a subcollection, the analogous characterization of the wot-closed left ideals fails. The reason for this failure is a deep factorization problem in these algebras. Typically, when norm conditions are placed on possible factors of operators in these algebras, reasonable factorization results can be obtained. Indeed, positive results regarding isometries in the unit ball are included. However, in the general setting it turns out that even seemingly obvious unique factorizations do not hold. The examples provided go toward understanding the fabric of these algebras as well as the pathologies of factorization involved. Many of these examples rely on an understanding of the structure theory of contractions in these algebras. The minimal isometric dilation of these contractions is determined. Further, each contraction is shown to have an H∞ functional calculus. The author would like to thank his supervisor, Ken Davidson, for all of his assistance. 1991 Mathematics Subject Classification. 47D25. Partially supported by an NSERC graduate scholarship. 1