Ideals in Toeplitz Algebras

We determine the ideal structure of the Toeplitz C∗-algebra on the bidisk 0 Introduction A large part of doing research in mathematics is asking the right question. Posing a timely question can trigger thought and provoke insights, even when the matter has no good resolution. That is what happened in the case at hand. After devoting much time to the study of Hilbert modules and quotient Hilbert modules in particular, I had occasion to wonder if a C∗-algebra perspective on this topic might not be useful. More precisely, I asked myself whether there was any relationship between the C∗-algebras for a Hilbert module and its quotient module generated by the operators obtained from module multiplication. The answer turned out to be negative but the techniques and approach used to reach that conclusion, namely an analysis of the ideal structure of the Toeplitz C∗-algebras, seemed worth reporting on. Although this general topic has been considered earlier (cf. [6]), these studies focused almost exclusively on index theory which will not concern us here. Let Ω be a bounded domain in C and A(Ω) be the function algebra obtained from the closure in the supremum norm on Ω of all functions holomorphic on some neighborhood of the closure of Ω. In this paper we will focus on the case of the unit ball B and the polydisk D for m = 2. 2000 Mathematical Subject Classification. 46L06, 47L05, 47L15, 47L20, 47L80.