A property of \(B_p(G)\). Applications to convolution operators. (English) Zbl 1161.43001
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.
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.
Reviewer: Elijah Liflyand (Ramat-Gan)
MSC:
| 43A05 | Measures on groups and semigroups, etc. |
| 43A22 | Homomorphisms and multipliers of function spaces on groups, semigroups, etc. |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
| 43A45 | Spectral synthesis on groups, semigroups, etc. |
Keywords:
Measures on group; convolution operators; multipliers; Figà-Talamanca-Herz algebra; Amenable groupsReferences:
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