Commutative \(C^{\ast}\)-algebras of Toeplitz operators and quantization on the unit disk (Q2491611)

Suppose that \(d\mu(z)=(1/2\pi i)[d\bar z\Lambda dz/(1-|z|^2)^2]\) denotes the Möbius invariant normalized measure on the unit disk \(D\) in the complex plane and, for \(-1<\lambda<\infty\), set \(d\mu_{\lambda}(z)=(\lambda+1)(1-|z|^2)^{\lambda+2}d\mu(z)\). Let \(L^2(D, d\mu_{\lambda})\) denote the Hilbert space of all square integrable functions with respect to the measure \(d\mu_{\lambda}(z)\) on \(D\) and let \(A^2_{\lambda}(D)\) denote a subspace of \(L^2(D, d\mu_\lambda)\) which consists of functions analytic in \(D\). \(A^2_{\lambda}(D)\) is said to be the weighted Bergman space on \(D\), in particular, if \(\lambda=0\) it is the classical Bergman space \(A^2(D)\). For a given essential bounded measurable function \(\phi\) on \(D\), the Toeplitz operator \(T_\phi\) on \(A^2_{\lambda}(D)\) with symbol \(\phi\) is defined as \[ T_\phi f=P(\phi f),\quad\text{for }f\in A^2_\lambda(D), \] where \(P\) denotes the orthogonal projection from \(L^2(D, d\mu_\lambda)\) onto \(A^2_\lambda(D)\). It is well-known that all commutative C*-algebras of Toeplitz operators can be classified by pencils of geodesics on the unit disk, considered as the hyperbolic plane. More precisely, given a pencil of geodesics, consider the set of symbols constant on the corresponding cycles, the orthogonal trajectories to geodesics forming the pencil. The C*-algebra generated by Toeplitz operators with such symbols turns out to be commutative. However, the principal question -- whether the above classes are the only possible sets of symbols which might generate the commutative C*-algebras of Toeplitz operators on each weighted Bergman space? -- is unsolved. In the present paper, an affirmative answer to this question is given.