## Mathematics, Department of

## Date of this Version

1998

## Citation

Math. Ann. 311, 275–303 (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 C^{n}* *. 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.

## Comments

Copyright Springer-Verlag 1998