C*-algebras generated by radial Toeplitz operators on polyanalytic weighted Bergman spaces

In a previous paper (Radial operators on polyanalytic weighted Bergman spaces, Bol. Soc. Mat. Mex. 27, 43), 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.