Invariant projections and convolution operators

We prove the existence of invariant projections $\mathcal{P}$ from the Banach space $PM_{p}(G)$ of $p$-pseudomeasures onto $PM_{p}(H)$ with $\operatorname{supp} {\mathcal{P}}(T)\subset \operatorname{supp}T$ for $H$ closed neutral subgroup of a locally compact group $G$. As a main application we obtain that every closed neutral subgroup is a set of $p$-synthesis in $G$ and in fact locally $p$-Ditkin in $G$. We also obtain an extension theorem concerning the Fourier algebra.