Compactness properties of a locally compact group and analytic semigroups in the group algebra (Q1174945)

Let \(L^ 1(G)\) be the \(L^ 1\) group algebra of a locally compact group \(G\). In investigating the structure of \(G\) in terms of the analytic semigroups in \(L^ 1(G)\), it was asked ``If \(L^ 1(G)\) has a nonzero analytic semigroup \(\{a^ z: \hbox{Re }z>0\}\) with the property that \(\{a^{1+iy}:\;y\in\mathbb{R}\}\) is bounded, does it follow that \(G\) is compact?'' In this connection, the author considers a locally compact group \(G\) for which \(G/Z(G)\) is compact where \(Z(G)\) is the centre of \(G\). For such a group \(G\) he proves that ``\(L^ 1(G)\) has a nonzero analytic semigroup \(\{a^ z:\;\hbox{Re }z>0\}\) with the property that \(\{a^{1+iy}:\;y\in\mathbb{R}\}\) is relatively weakly compact, if and only if \(G\) contains a compact open subgroup. If, moreover, \(G\) is connected then \(G\) must be compact.''.