Toeplitz quantization on Fock space

For Toeplitz operators T_f ^(t) acting on the weighted Fock space H_t ², we consider the semi-commutator T_f ^(t)T_g ^(t)−T_fg ^(t), where t>0 is a certain weight parameter that may be interpreted as Planck's constant ħ in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit (⁎)limt→0⁡‖T_f ^(t)T_g ^(t)−T_fg ^(t)‖_t. It is well-known that ‖T_f ^(t)T_g ^(t)−T_fg ^(t)‖_t tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to f,g∈BUC(Cⁿ) by Bauer and Coburn. We now further generalize (⁎) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra VMO∩L^∞ of bounded functions having vanishing mean oscillation on Cⁿ. Our approach is based on the algebraic identity T_f ^(t)T_g ^(t)−T_fg ^(t)=−(H_f¯ ^(t))^⁎H_g ^(t), where H_g ^(t) denotes the Hankel operator corresponding to the symbol g, and norm estimates in terms of the (weighted) heat transform. As a consequence, only f (or likewise only g) has to be contained in one of the above classes for (⁎) to vanish. For g we only have to impose limsup_t→0‖H_g ^(t)‖_t<∞ e.g. g∈L^∞(Cⁿ). We prove that the set of all symbols f∈L^∞(Cⁿ) with the property that lim_t→0⁡‖T_f ^(t)T_g ^(t)−T_fg ^(t)‖_t=lim_t→0⁡‖T_g ^(t)T_f ^(t)−T_gf ^(t)‖_t=0 for all g∈L^∞(Cⁿ) coincides with VMO∩L^∞. Additionally, we show that limt→0⁡‖T_f ^(t)‖_t=‖f‖_∞ holds for all f∈L^∞(Cⁿ). Finally, we present new examples, including bounded smooth functions, where (⁎) does not vanish.