Isometries of JB-algebras
The main theorem of this paper proves that for a bijective linear map \(F: A\to B\) between JB-algebras the nice equivalence of each of the following assertions is satisfied: i) \(F\) is an isometry; ii) \(F= b\circ G\) where \(G\) is an algebra isomorphism between both JB-algebras and \(b^2= \text{Id}\) with \(b\) belonging to the algebra of multipliers of \(B\); iii) \(F(x^3)= F(x)^3\) for all \(x\) in \(A\); iv) \(F( \{xyz\})= \{F(x) F(y) F(z)\}\) for any \(x, y,z\in A\). In the particular case of JBW-factors the bijective linear isometries are of the form \(\varepsilon G\) for \(\varepsilon= \pm1\) and \(G\) an algebra isomorphism. Section 2 is then devoted to the determination of the algebra isomorphisms between JBW- factors of type I different from \(H_3 (\mathbb{O})\).
scientific article; zbMATH DE number 784033