The algebraic structure of non-commutative analytic Toeplitz algebras

Authors

Kenneth R. Davidson, Pure Math. Dept. U. WaterlooFollow
David R. Pitts, University of Nebraska-LincolnFollow

Document Type

Article

Date of this Version

1998

Citation

Math. Ann. 311, 275–303 (1998)

Comments

Copyright Springer-Verlag 1998

Abstract

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 wot-closed 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.