Toeplitz algebra of bounded symmetric domains: A quantum harmonic analysis approach via localization

TL;DR

This paper demonstrates that Toeplitz operators are dense in the Toeplitz algebra on weighted Bergman spaces over bounded symmetric domains, using quantum harmonic analysis and representation theory.

Contribution

It introduces a novel approach combining quantum harmonic analysis and localization techniques to analyze Toeplitz algebras on symmetric domains.

Findings

Toeplitz operators are norm dense in the Toeplitz algebra.

Weakly-localized operators form a self-adjoint algebra containing Toeplitz operators.

The norm closure of weakly-localized operators equals the Toeplitz algebra.

Abstract

We prove that Toeplitz operators are norm dense in the Toeplitz algebra $\mathfrak{T}(L^\infty)$ over the weighted Bergman space $\mathcal{A}^2_\nu(\Omega)$ of a bounded symmetric domain $\Omega\subset\mathbb{C}^n$. Our methods use representation theory, quantum harmonic analysis, and weakly-localized operators. Additionally, we note that the set of all $\alpha$-weakly-localized operators form a self-adjoint algebra, containing the set of all Toeplitz operators, whose norm closure coincides with the Toeplitz algebra.