On the density of Toeplitz operators in the Toeplitz algebra over the Bergman space of the unit ball

TL;DR

The paper provides a streamlined, QHA–based proof that Toeplitz operators are norm dense in the Toeplitz algebra on the Bergman space $\mathcal{A}^2(\mathbb{B}^n)$, focusing on the unit ball and the noncommutative group $\mathrm{SU}(n,1)$. It develops a framework of $\beta$-weakly localized operators, derives an integral representation for such operators, and shows their $S_{\boldsymbol{B_r}}$ pieces approximate the operator via Toeplitz operators, culminating in Xia's density result. A key contribution is the integral decomposition $S=\iint_G \langle S\pi(h)1,\pi(g)1\rangle \pi(g)1\otimes\pi(h)1\, d\mu_G(g)d\mu_G(h)$ and the identification $\mathfrak{T}(L^\infty)=\overline{L^\infty(G)\ast \mathcal{S}^1(\mathcal{A}^2)}$, which bypasses separated-set arguments. The approach clarifies how QHA methods apply to Bergman spaces on the unit ball and confirms Xia's density result within this noncommutative, group–representation framework.

Abstract

We use quantum harmonic analysis and representation theory to provide a new proof of Xia's theorem: "Toeplitz operators are norm dense in the Toeplitz algebra over the Bergman space of the unit ball."

Theorems & Definitions (25)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Proposition 3.3