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

TL;DR

The paper analyzes C*-algebras generated by radial Toeplitz operators on polyanalytic weighted Bergman spaces. By representing these operators as matrix sequences $\\gamma_{n,\alpha}(a)$ and employing Jacobi polynomial structures, the authors identify the generated C*-algebra with the algebra of matrix sequences having scalar limits at infinity, denoted $\\mathcal{L}_n$. They prove $\\mathcal{X}_{n,\alpha}=\\mathcal{L}_n$ and show that for $n\ge 2$ the closure of the generating matrix sequences is a proper subspace of $\\mathcal{L}_n$, while in the analytic case $n=1$ the generating set is dense in $c(\\mathbb{N}_0)$. The main technique combines pure-state separation with Kaplansky’s noncommutative Stone–Weierstrass theorem, using specially crafted Jacobi-based generators and a finite-generating-set mechanism to realize full matrix algebras within frequency blocks. The results extend the structure theory of Toeplitz-type algebras to polyanalytic Bergman spaces and provide a robust framework for separating limit and pure-state behaviors in noncommutative harmonic analysis.

Abstract

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 $γ(a)$, 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 $γ(a)$ 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 $γ(a)$ is not equal to the generated C*-algebra.

Theorems & Definitions (68)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Remark 3.1
• Example 3.2
• Remark 3.3
• Remark 3.4
• Lemma 4.1
• proof
• Proposition 4.2