Uniform Continuity and Quantization on Bounded Symmetric Domains

We consider Toeplitz operators $T_f^{\lambda }$ with symbol $f$ acting on the standard weighted Bergman spaces over a bounded symmetric domain $\Omega\subset $\mathbb{C}$^n$\mathrm{.}Here$\lambda > genus-1$ is the weight parameter. The classical asymptotic semi-commutator relation $\lim_{\lambda \rightarrow \infty} \big{|}T_f^{\lambda} T_g^{\lambda} -T_{fg}^{\lambda} \big{|}=0$with$f,g \in C(\overline{$\mathbb{B}$^n})$,where$\Omega=$\mathbb{B}$^n$ denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol-functions and to more general domains. We deal with operator symbols that generically are neither continuous inside $\mathrm{\Omega }$ (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let $\beta$ denote the Bergman metric distance function on $\mathrm{\Omega }$. We prove that the semi-commutator relation remains true for $f$ and $g$ in the space $\mathrm{U}\mathrm{C}(\mathrm{\Omega })$ of all $\beta$-uniformly continuous functions on $\mathrm{\Omega }$. Note that this space contains also unbounded functions. In case of the complex unit ball $\mathrm{\Omega }={\mathbb{B}}^n\subset {\mathbb{C}}^n$ we show that the semi-commutator relation holds true for bounded symbols in $\mathrm{V}\mathrm{M}\mathrm{O}({\mathbb{B}}^n)$, where the vanishing oscillation inside ${\mathbb{B}}^n$ is measured with respect to $\beta$. At the same time the semi-commutator relation fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in $\mathrm{\Omega }$.