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Self-adjointness in unitary isotopes of \(JB^*\)-algebras. (English) Zbl 1142.46020

Let \(u\) be a unitary element of a \(JB^*\)-algebra \(J\). Then the unitary isotope \(J^{[u]}\) of \(J\) is a \(JB^*\)-algebra having \(u\) as its unit, the original norm of \(J\), and the involution \(*_u\) given by \(x^{*_u} = \{ u x^* u\}\). In the paper under review, the author investigates certain links between the elements of the unit ball of a \(JB^*\)-algebra \(J\) which are self-adjoint in some unitary isotope \(J^{[u]}\), and those elements which are the average of two unitaries. (Both \(J\) and \(J^{[u]}\) have the same unitary elements.)

MSC:

46H70 Nonassociative topological algebras
17C65 Jordan structures on Banach spaces and algebras
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