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

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.