On the Berezin-Toeplitz calculus

We consider the problem of composing Berezin-Toeplitz operators on the Hilbert space of Gaussian square-integrable entire functions on complex $n$-space, $\mathbf{C}^n$. For several interesting algebras of functions on $\mathbf{C}^n$, we have $T_\varphi T_\psi =T_{\varphi \diamond \psi }$for all $\varphi ,\psi$ in the algebra, where $T_\varphi$ is the Berezin-Toeplitz operator associated with $\varphi$ and $\varphi \diamond \psi$ is a ``twisted'' associative product on the algebra of functions. On the other hand, there is a $C^\infty$ function $\varphi$ for which $T_\varphi$ is bounded but $T_\varphi T_\varphi \neq T_\psi$ for any $\psi$.