A proof of the Russo-Dye theorem for \(JB^*\)-algebras (Q983419)

The so called Russo-Dye theorem for (unital) \(C^*\)-algebras \(A\) states that the closed unit ball of \(A\) is the closed convex hull of its unitary elements. A similar result for \(JB^*\)-algebras was obtained by Wright and Youngson by modifying the proof of the Russo-Dye theorem for \(J^*\)-algebras given by Harris. In the present paper the author gives a new proof of the Russo-Dye theorem for \(JB^*\)-algebras (inspired by results of Gardner, Kadison and Pedersen for \(C^*\)-algebras), which is independent of the classification and gives more information about the number of unitaries needed in the approximation. As a main tool he uses the fact that the unitary isotope of a unital \(JB^*\)-algebra is itself a \(JB^*\)-algebra.