Factoring in Non-commutative Analytic Toeplitz Algebras

TL;DR

This paper studies the structure of non-commutative analytic Toeplitz algebras, revealing their functional calculus, factorization properties, and the relationship between ideals and invariant subspaces.

Contribution

It provides new insights into the structure and factorization of contractions and ideals within non-commutative analytic Toeplitz algebras.

Findings

Contractions have an $H^$ functional calculus.

Word-defined isometries factor only as words do over the unit ball.

Identification of weakly closed left ideals with invariant subspaces holds only for a proper subset.

Abstract

The non-commutative analytic Toeplitz algebra is the weak operator topology 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 $H^\infty$ 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 weakly closed left ideals with invariant subspaces of the algebra is shown to hold only for a proper subcollection of the subspaces.