In this Note we give a proof for the particular case of the group $SU(2)$ of a local characterization of the central Fourier algebra $A_{C}(G)$ of a compact Lie group $G$ in terms of the Fourier algebra of a maximal torus of $G$. The proof for the general case will appear elsewhere [5]. We derive a result on local symbolic calculus for $A_{C}(G)$ which partially extends a theorem of M. P. and P. Malliavin.