Irreducibility of Toeplitz \(C^ *\)-algebras (Q1093872)

The irreducibility of the Toeplitz \(C^*\)-algebra generated by Toeplitz operators on the Hardy space \({\mathcal H}^ 2(S^ 1)\) was discovered in 1967 by \textit{L. A. Coburn} [Bull. Am. Math. Soc. 73, 722-726 (1967; Zbl 0153.166)]. There is a similar result for the Toeplitz \(C^*\)-algebra on the Bergman space \({\mathcal H}^ 2(D^ 1)\) on the unit disc D. The Cayley transform of \(D^ 1\) onto the upper half plane D leads to the same results on \({\mathcal H}^ 2(D)\) and \({\mathcal H}^ 2(H)\). Here these results are generalized to a type of generalized upper half planes, called Siegel domains, in several complex variables. This extends earlier results of Dynin where a stratification of the Siegel domains was required.