Uniqueness of the topology on \(L^1(G)\) (Q2773435)

The uniqueness of the topology on \(L^1(G)\) is discussed, where \(G\) is a locally compact Abelian group, and \(\widehat G\) is the dual of \(G\), \(X\subset L^1(G)\) is a translation invariant linear subspace. The main results are as follows.NEWLINENEWLINENEWLINE(1) For \(G\), \(\widehat G,\) \(X\subset L^1(G)\), let \(t\in G\setminus \{0\}\), such that \(T_t(X) \subset X\). Then every Banach space topology on \(X\), making the translation operator \(T_t\) on \(X\) continuous is not weaker than the topology of convergence in mean. There is at most one Banach space topology on \(X\) with the translation operator \(T_t\) on \(X\) continuous.NEWLINENEWLINENEWLINE(2) Let \(G\) be a noncompact locally compact Abelian group and \(X\) a translation invariant linear subspace of \(L^1(G)\). Then every Banach space topology on \(X\) with translations on \(X\) continuous is not weaker than the topology of convergence in mean on \(X\). Moreover, there is at most one Banach space topology on \(X\) with the translation operator \(T_t\) on \(X\) continuous. NEWLINENEWLINENEWLINE(3) Let \(G\) be a compact Abelian group and \(X\) a translation invariant linear subspace of \(L^1(G)\) with \(1\in X\). Moreover, suppose that \(X\) is endowed with a Banach space topology making translations on \(X\) continuous and that \(X\) has a discontinuous translation invariant linear functional. Then \(X\) does not carry a unique Banach space topology with translations on \(X\) continuous.NEWLINENEWLINENEWLINE(4) Let \(X\) and \(Y\) be group spaces on a locally compact Abelian group \(G\) and \(\Phi\) be a translation invariant linear operator from \(X\) into \(Y\). Then the following assertions hold:NEWLINENEWLINENEWLINE(a) If \(G\) is noncompact, then \(\Phi\) is continuous;NEWLINENEWLINENEWLINE(b) if \(G\) is compact, then \(\dim D(\Phi)< \infty\);NEWLINENEWLINENEWLINE(c) if \(G\) is connected and compact and \(X=L^p(G)\) for some \(p\), \(1<p< \infty\), then \(\Phi\) is continuous.