Maximal Abelian von Neumann algebras and Toeplitz operators with separately radial symbols (Q1035223)

The author studies the algebras generated by Toeplitz operators \(T_a\) with symbol \(a\) acting on the standard weighted Bergman spaces \(A^2_{\alpha}(\mathbb{D})\), \(\alpha > -1\), over the unit disk \(\mathbb(D)\) on the complex plane. Given a pencil \(\mathcal{P}\) of hyperbolic geodesics, denote by \(\mathcal{A}(\mathbb{D})\) the set of \(L^{\infty}\)-symbols which are constant on the corresponding cycles. It is well-known that the \(C^*\)-algebra \(\mathcal{T}_\alpha(\mathcal{A}(\mathbb{D}))\) generated by all Toeplitz operators with symbols \(a \in \mathcal{A}(\mathbb{D})\) is abelian on each weighted Bergman space \(A^2_{\alpha}(\mathbb{D})\). The author shows that the \(w^*\)-closure of \(\mathcal{T}_\alpha(\mathcal{A}(\mathbb{D}))\) is maximal abelian. That is, any operator in the commutator of \(\mathcal{T}_\alpha(\mathcal{A}(\mathbb{D}))\) lies in the \(w^*\)-closure of \(\mathcal{T}_\alpha(\mathcal{A}(\mathbb{D}))\).