A Kaplansky theorem for \(JB^*\)-algebras (Q1293508)

In 1949, Irving Kaplansky proved that any algebra norm on \(C(X)\) dominates the standard uniform norm. This result was extended to noncommutative \(C^*\)-algebras (modulo some constant) by S. B. Cleveland. This result was extended, in 1994, to \(JB^*\)-algebras by J. Perez, L. Rico and A. Rodriguez Palacios. The two authors of this paper give a slightly different proof of this last result, essentially based on Cleveland's ideas. I only mention that the proof of Theorem 4 is not complete, because every element of the separating space could be quasinilpotent. But then, having noticed that it is easy to finish.