Complete metric Jordan algebras (Q2644746)

The following theorem is proved: Let F be a complete non-discrete topological field, in which the set \((F\setminus V)^{-1}\) is bounded for every neighbourhood V of zero and A be a complete metric Jordan algebra. If A is algebraic over the field F then A is algebraic over F of bounded degree. From this result it follows in particular: An algebraic Jordan-Banach algebra is algebraic over \({\mathbb{R}}\) and of bounded index.