The \(L^{\infty{}}\)-representation algebra of a foundation topological semigroup (Q1210032)

For \(S\) a commutative topological semigroup, \(R(S)\) denotes the representation algebra generated by weak continuous representations of \(S\) in \(L^ \infty\) spaces, and \(S^ \wedge\) denotes the set of continuous semicharacters of \(S\). It is shown that for a large class of such semigroups \(S\), including all products of discrete semigroups and locally compact groups, the algebra \(R(S)\) is identical to the set of Gelfand transforms of bounded measures on \(S^ \wedge\).