Toeplitz algebras on the disk ✩

Let B be a Douglas algebra and let B be the algebra on the disk generated by the harmonic extensions of the functions in B. In this paper we show that B is generated by H∞(D) and the complex conjugates of the harmonic extensions of the interpolating Blaschke products invertible in B. Every element S in the Toeplitz algebra TB generated by Toeplitz operators (on the Bergman space) with symbols in B has a canonical decomposition S = TS˜ + R for some R in the commutator ideal CTB ; and S is in CTB iff the Berezin transform S˜ vanishes identically on the union of the maximal ideal space of the Douglas algebra B and the setM1 of trivial Gleason parts. © 2006 Elsevier Inc. All rights reserved.