The algebraic structure of non-commutative analytic Toeplitz algebras

Abstract

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.