# Journal of Operator Theory

Volume 45, Issue 1, Winter 2001 pp. 175-193.

Non-commutative analytic Toeplitz algebras**Authors**: David W. Kribs

**Author institution:**Pure Mathematics Department, University of Waterloo, Waterloo, Ontario N2L--3G1, Canada

**Summary:**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 $\hinf$ functional calculus. The isometries defined by words are shown to factor only as the words do over th e 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 subspace s.

**Keywords:**Analytic Toeplitz algebra, left regular representation, isometric dilation

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