The \(L^{\infty{}}\)-representation algebra of an idempotent topological semigroup (Q1206777)

This paper considers the representation algebra of an idempotent commutative topological semigroup \(S\). The authors obtain a description of this algebra in terms of an algebra of measures on the semigroup \(\Gamma\) of all semicharacters on the semigroup \(S\). Using a result of S. E. Newman, they realize the algebra as the set of functions of bounded variation on \(S\). When \(S\) is a finite product of commutative semigroups, each being compact in the order topology and having maximum as multiplication, then the representation algebra is the linear span of the space of the Fourier transforms of the set of all positive regular Borel measures \(\mu\) on the product of the linear ordered semigroups.