Atomization process for convolution operators on locally compact groups. (English) Zbl 1108.43001
Let \(G\) be a locally compact amenable group, and let \(CV_p(G)\) be the Banach space of all \(p\)-convolution operators on \(L_p(G)\). Let \(T\) denote such an operator and \(U\) an open subset containing the support of \(T\). The problem addressed in the present paper is whether \(T\) lies in the weak closure of the set of all \(p\)-convolution operators that are defined by measures finitely supported in \(U\) and whose \(p\)-norms are dominated by the \(p\)-norm of \(T\). A similar question was investigated by N. Lohoue for the case of abelian \(G\). The main result of the present paper is that if \(G\) is amenable and has a dense net of closed subgroups then the above question has an affirmative answer.
Reviewer: Benjamin B. Wells jun. (Charlottesville)
MSC:
| 43A05 | Measures on groups and semigroups, etc. |
| 43A22 | Homomorphisms and multipliers of function spaces on groups, semigroups, etc. |
| 47B38 | Linear operators on function spaces (general) |
| 47B48 | Linear operators on Banach algebras |
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