Harmonic synthesis for subgroups

Abstract

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.