The primitivity of semigroup algebras of free products
Let \(S\) be the free product of semigroups \(A\), \(B\). It is well known that the semigroup algebra \(F[S]\) over any field \(F\) is left and right primitive provided that one of \(A\), \(B\) is nontrivial. In this paper a faithful irreducible representation of \(F[S]\) is constructed. The same is done in the more difficult case of the algebra \(\ell^1(S)\) of infinite sums \(\sum^\infty_{i=1}\alpha_is_i\), \(s_i\in S\), \(\alpha_i\in K\), satisfying \(\sum_i|\alpha_i|<\infty\), over the field \(K\) of reals or complexes. The method is similar to that used by the third author [in Bull. Lond. Math. Soc. 8, 294-298 (1976; Zbl 0344.20002)].
scientific article; zbMATH DE number 994597