Topological division algebras without corresponding topological affine plane
Let \(D\) be a topological (non-associative) division algebra. If \(D\) is alternative, then \(D\) gives rise to a topological projective plane, see \textit{H. Salzmann} [Math. Z. 67, 436-466 (1957; Zbl 0078.341)]. The author shows that this is not true for arbitrary (non-alternative) division algebras. More precisely, he proves the following result. Let \(K\) be a field of infinite transcendency degree over its prime field, and let \(D\) be a division algebra of finite dimension over its center \(K\). If \(D\) is not alternative, then \(D\) can be topologized in such a way that \(D\) becomes a topological division algebra which does not belong to any topological projective (or affine) plane. If \(K\) is real closed, then one can take the product topology on \(D\) derived from any nondiscrete valuation topology of \(K\) which is distinct from the order topology.
scientific article; zbMATH DE number 404313
scientific article