BEST BOUNDS FOR THE APPROXIMATE UNITS FOR CERTAIN IDEALS OF L(1)(G) AND OF A(P)(G)

We compute the best bound for the approximate units of the augmentatio n ideal of the group algebra L(1)(G) of a locally compact amenable gro up G. More generally such a calculation is performed for the kernel of the canonical map from L(1)(G) onto L(1)(G/H), H being a closed amena ble sub-group of G. Analogous results involving certain ideals of the Fourier algebra of an amenable group are also discussed.