A property of \(B_p(G)\). Applications to convolution operators (Q1000539)

One of the main results of this paper is Theorem 7. Let \(G\) be a locally compact group, \(H\) a dense subgroup, \(1<p<\infty,\) and \(u\in C(G,{\mathbb C}).\) Suppose that \(\text{Res}_H u\in B_p(H_d),\) where \(B_p(G)\) is Herz's algebra. Then \(u\in B_p(G)\) and \(\|u\|_{B_p(G)}=\|\text{Res}_H u\|_{B_p(H_d)}.\) This result is new even for \(G\) compact. In the paper, it is applied to spectral synthesis and to obtaining an extended version of the Lohoué's monomorphism theorem.