C*-Algebras Generated by Radial Toeplitz Operators on Polyanalytic Weighted Bergman Spaces

Abstract

In a previous paper (Barrera-Castelán et al. in Bol Soc Mat Mex 27:43, 2021. https://doi.org/10.1007/s40590-021-00348-w), using disk polynomials as an orthonormal basis in the n-analytic weighted Bergman space, we showed that for every bounded radial generating symbol a, the associated Toeplitz operator, acting in this space, can be identified with a matrix sequence \(\gamma (a)\), where the entries of the matrices are certain integrals involving a and Jacobi polynomials. In this paper, we suppose that the generating symbols a have finite limits on the boundary and prove that the C*-algebra generated by the corresponding matrix sequences \(\gamma (a)\) is the C*-algebra of all matrix sequences having scalar limits at infinity. We use Kaplansky’s noncommutative analog of the Stone–Weierstrass theorem and some ideas from several papers by Loaiza, Lozano, Ramírez-Ortega, Ramírez-Mora, and Sánchez-Nungaray. We also prove that for \(n\ge 2\), the closure of the set of matrix sequences \(\gamma (a)\) is not equal to the generated C*-algebra.