Toeplitz algebras on the disk
Let \(B\) be a Douglas algebra and let \(\mathcal B\) be the algebra on the unit disk generated by the harmonic extensions of functions in \(B\). Denote by \(L^2_a\) the Hilbert space of analytic functions on the unit disk \(D\) that belong to \(L^2(D, dA)\) for \(dA\) denoting the Lebesgue area measure on \(D\). For an operator \(S\in {\mathcal L}(L^2_a)\) the Berezin transform of \(S\) is the function \(\widetilde S\) defined on \(D\) by \[ \widetilde S (z)= \langle Sk_z, k_z\rangle, \] where \(k_z\) is the normalized Bergman reproducing kernel. Further, let \(\mathcal T_\mathcal B\) be the closed subalgebra of \({\mathcal L}(L^2_a)\) generated by Toeplitz operators with symbol in \(\mathcal B\). The main goal of this paper is to characterize the algebra \(\mathcal T_B\) using the properties of the Berezin transform of the operators in \(\mathcal T_\mathcal B\). It is shown that every element \(S\in\mathcal T_\mathcal B\) has a canonical decomposition \[ S=T_{\widetilde S} + R, \] where \(T_{\widetilde S} \) is the Toeplitz operator with symbol \(\widetilde S\), the Berezin transform of \(S\), and \(R\) belongs to the commutator ideal \({\mathcal C}_{{\mathcal T}_{\mathcal B}}\), the smallest closed two-sided ideal of \(\mathcal T_\mathcal B\) containing all operators of the form \(R\, U-UR\) with \(R,U\in \mathcal T_\mathcal B\). Moreover, \(S\in {\mathcal C}_{{\mathcal T}_{\mathcal B}}\) if and only if the Berezin transform \(\widetilde S\) vanishes on the union of the maximal ideal space of the Douglas algebra \(B\) and the set \(\mathcal M_1\) of trivial Gleason parts. The set \(\mathcal M_1\) is constituted by those \(m\in \mathcal M\) (\(\mathcal M\) stands for the maximal ideal space of \(H^\infty (D)\)) such that the Hoffman map \(\varphi_m:D\rightarrow \mathcal M\) defined by \(\varphi_m(w)=\lim_{z\rightarrow w} \varphi_z(w)\), \(w\in D\), is constant on \(\mathcal M\).
scientific article; zbMATH DE number 5129769
scientific article