Uniformly continuous functions and quantization on the Fock space

With a family (μ_t)_t>0 of Gaussian probability measures we consider the scale (H²_t)_>0 of μ_t -square integrable entire functions on ℂⁿ. Here t plays the role of Planck's constant. For f and g in the space BUC(ℂⁿ) of all bounded and uniformly continuous complex valued functions on Cn we show the asymptotic composition formula (Equation Presented) where ||·||_t denotes the norm in Ⅎ(H²_t) and T^(t)_f is the Toeplitz operator with symbol f. Different from previously knownresults (e.g. Borthwick, Perspectives on quantization. Contemporary mathematics, vol 214. AMS, Providence, pp 23-37, 1998; Coburn, Commun Math Phys 149:415-424, 1992) neither differentiability nor compact support of the operator symbols is assumed. We provide an example which indicates that (1) in general fails for rapidly oscillating bounded symbols.