 
Summary: Toeplitz Algebras on the Disk
Sheldon Axler Dechao Zheng
28 September 2006
Abstract. 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 = T~S + 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 set M1 of trivial Gleason parts.
1 Introduction
Let dA denote Lebesgue area measure on the open unit disk D, normalized
so that the measure of D equals 1. The Bergman space L2
a is the Hilbert space
consisting of the analytic functions on D that are also in L2
(D, dA). For z D,
the Bergman reproducing kernel is the function Kz L2
