Harmonic synthesis for subgroups

Let G be a locally compact group and H a closed subgroup. Then H is always a set of local spectral synthesis with respect to the algebra A p (G), where A 2 (G) is the Fourier algebra in the sense of Eymard. Global synthesis holds if and only if a certain condition (C) is satisfied; it is whenever the subgroup H is amenable or normal. Global synthesis implies that each convolution operator on L p (G) with support in H which is the ultraweak limit of measures carried by H. The problem of passing from local to global synthesis is examined in an abstract context.