Bergman space structure, commutative algebras of Toeplitz operators, and hyperbolic geometry (Q1402343)

Recall that the Bergman space \(A^2(\mathbb{D})\) is the subspace in \(L^2(\mathbb{D})\) of all holomorphic functions on the unit disc \(\mathbb{D}\); and for \(\phi \in L^\infty(\mathbb{D})\), the Toeplitz operator \(T_\phi\) on \(A^2(\mathbb{D})\) is defined by \(T_\phi f=P(\phi f)\), where \(P:L^2(\mathbb{D})\to A^2(\mathbb{D})\) is the orthogonal projection. The author exhibits three types of commutative \(C^*\)-algebras generated by Toeplitz operators on \(A^2(\mathbb{D})\): the one generated by \(T_\phi\) with \(\phi\) radial (i.e., \(\phi(z)\) depending only on \(| z| \)); the one generated by \(T_\phi\) with \(\phi\) constant on each of the horocycles corresponding to a fixed boundary point (upon passing from the disc to the upper half-plane, this corresponds to \(\phi(w)\) depending only on \text {Im\(\,w\)}); and the one generated by \(T_\phi\) with \(\phi\) constant on each of the circles passing through a given pair of boundary points (passing again to the upper half-plane, this corresponds to \(\phi(w)\) satisfying \(\phi(w)=\phi(aw)\) for all \(a>0\)). Further, in each case, an explicit unitary equivalence \(U\) is exhibited such that \(U^* T_\phi U\), for any \(\phi\) in the corresponding class, is the operator of multiplication by a certain Fourier-like transform \(\widetilde\phi\) of \(\phi\) on a certain \(L^2\)-space. It is conjectured that any commutative \(C^*\)-algebra generated by Toeplitz operators on \(A^2(\mathbb{D})\) must be contained in a \(C^*\)-algebra of one of the above three types.