Localization and Berezin transform on the Fock space

We introduce the notion of sufficiently localized operators on the Fock space. We show that if A is in the C⁎-algebra generated by the class of sufficiently localized operators, then A is compact if and only if its Berezin transform vanishes at infinity. Moreover, we show that this class contains many familiar operators, including all the Toeplitz operators with bounded symbols.