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A proof of the Russo-Dye theorem for \(JB^*\)-algebras. (English) Zbl 1231.46015

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.

MSC:

46K70 Nonassociative topological algebras with an involution
17C65 Jordan structures on Banach spaces and algebras
46L70 Nonassociative selfadjoint operator algebras
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