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

We prove that Toeplitz operators are norm dense in the Toeplitz algebra $\mathfrak{T}(L^{\mathrm{\infty }})$ over the weighted Bergman space ${\mathcal{A}}_{\nu }^2(\mathrm{\Omega })$ of a bounded symmetric domain $\mathrm{\Omega }\subset {\mathbb{C}}^n$. 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.