Surjective isometries between unitary sets of unital JB⁎-algebras

This paper is, in a first stage, devoted to establishing topological–algebraic characterization of the principal component, U0(M), set unitary elements, U(M), unital JB⁎-algebra M. We arrive conclusion that, as case C⁎-algebras,U0(M)=M1−1∩U(M)={Ueihn⋯Ueih1(1):n∈N,hj∈Msa∀1≤j≤n}={u∈U(M): there exists w∈U0(M) with ‖u−w‖<2} is analytically arcwise connected. Actually, U0(M) smallest quadratic subset U(M) containing eiMsa. Our second goal provide complete description surjective isometries between components two JB⁎-algebras M and N. Contrary C⁎-algebras, we shall deduce existence connected which are not isometric metric spaces. also establish necessary sufficient conditions guarantee that isometry Δ:U(M)→U(N) admits an extension linear N, always true. Among consequences it proved N Jordan ⁎-isomorphic if, only their spaces mapping unit element U0(N). These results setting obtained by O. Hatori for C⁎-algebras.