On unitaries in \(JB^*\)-algebras. (English) Zbl 1115.46058

Summary: We look at relationships between the distances from a point to the set of invertible elements and to the set of unitaries in a \(JB^*\)-algebra \({\mathcal J}\); various formulae are deduced for cases when the point is of norm \(\leq 1\), when \({\mathcal J}\) is of tsr 1 and when \({\mathcal J}\) is finite-dimensional. In the sequel, some results are obtained with particular focus on the notion of unitary rank of an element in connections with its distances to the unitaries and to the invertibles.


46L70 Nonassociative selfadjoint operator algebras
17C99 Jordan algebras (algebras, triples and pairs)