Ring isomorphisms of \(H^*\)-algebras (Q1913954)

The authors prove that ring homomorphisms with dense image between complex infinite-dimensional topologically simple \(H^*\)-algebras are continuous. Moreover they are either linear or conjugate-linear isomorphisms. This result and another previous one by \textit{A. Rodríguez-Palacios} and \textit{J. A. Cuenca Mira} [J. Algebra 106, 1-14 (1987; Zbl 0616.46047)] allow a deep study of the ring isomorphisms between \(H^*\)-algebras with zero annihilator. More questions are treated, so the following assertions are proved: i) there are discontinuous ring isomorphisms between finite-dimensional topologically simple \(H^*\)-algebras; ii) ring isomorphisms between complex topologically simple \(H^*\)-algebras are \(\tau\)-linear for a suitable field automorphism \(\tau\) of \(\mathbb{C};\) iii) ring isomorphic complex associative \(H^*\)-algebras with zero annihilator are \(*\)-isomorphic.