Involutions in the Cayley-Dickson construction

Abstract

We determine all involutions in the Cayley-Dickson construction that extend the involution of the original $\ast$-algebra. We also find all algebra isomorphisms between the resulting Cayley doubles that extend the identity automorphism of the original $\ast$-algebra, and consequently classify the resulting $\ast$-algebras up to $\ast$-algebra isomorphism. As applications, we show that Cayley doubles without zero divisors admit exactly one additional involution, prove that the classical Cayley-Dickson involution is the unique scalar involution, and obtain a classification of the $\ast$-algebras arising from $\mathbb{R}$ up to dimension 4.

18 pages; minor update