On some special measures on \(\beta G\) (Q1318959)

Let \(S\) be an amenable cancellative discrete semigroup. Let \(\nu\) be a positive left-invariant normalised element of \(\ell^ \infty(S)^*\). The author begins with the interesting elementary observation that if \(X\) is any infinite subset of \(S\) and \(h \in \ell^ \infty(S)\) is such that \(\nu(h_ s \overline{h}_ t) = 0\) for all distinct \(s, t \in X\), then \(\nu(\chi h) = 0\) for every semicharacter of \(S\). This result permits the construction, for some particular abelian groups \(G\), of functions \(f \in \ell^ \infty(G)\) with \(| f| = 1\) which are \(\widehat{G}\)-almost convergent to zero (that is, \(\nu(f) = 0\) for every \(\nu \in \ell^ \infty(G)^*\) for which \(\chi \nu = \nu\) for all \(\chi \in \widehat{G}\)). These methods are then applied to find, for many groups \(G\), elements \(\mu \in C_ 0(G)^ \perp\) for which \(\gamma \mu = 0\) for all \(\gamma \in C_ 0(G)^ \perp\) (the product here is the first Arens product in the second dual of the Banach algebra \(\ell^ 1(G)\)). This conclusion means that there are left ideals in \(\ell^ \infty(G)^*\) which contain no minimal left ideal. Analogous results are also given for the dual \(UC(\mathbb{R}^ n)^*\) of bounded uniformly continuous functions on \(\mathbb{R}^ n\).