A geometric construction of representations of the Berezin-Toeplitz quantization

For a K"ahler manifold $X$ equipped with a prequantum line bundle $L$, we give a geometric construction of a family of representations of the Berezin-Toeplitz deformation quantization algebra $(C^{\mathrm{\infty }}(X)[[\mathrm{\hslash }]],{\star }_{BT})$ parametrized by points $z_0\in X$. The key idea is to use peak sections to suitably localize the Hilbert spaces $H^0(X,L^{\otimes m})$ around $z_0$ in the large volume limit.