Absolute valued algebras containing a central idempotent
A nonassociative algebra over the reals is called an absolute valued algebra if it is a normed space satisfying \(\| xy\| =\| x\| \| y\|\). It is proved here that any absolute valued algebra containing a central idempotent is actually an inner product space, so that the norm comes from the inner product. This allows the author to classify the third-power associative finite dimensional absolute valued algebras. These are, up to isomorphism, the Hurwitz and para-Hurwitz division algebras and the pseudo-octonion algebra defined by \textit{S. Okubo} [Hadronic J. 5, 1564-1612 (1982; Zbl 0547.17001)].
scientific article; zbMATH DE number 4127459