Commutative nearly absolute valued algebras with square roots (Q721492)

A real or complex algebra $A\neq 0$ is a nearly absolute valued algebra if it is a normed vector space and there exists a real number $k>0$ such that $k\|x\|\|y\|\leq \|xy\|\leq\|x\|\|y\|$. The authors prove that all the elements of a commutative nearly absolute valued algebra have square roots if and only if the algebra is a two-dimensional real division algebra. As a corollary they show that for a commutative real algebra $A$, the following are equivalent: \begin{itemize} \item[(1)] $A$ is a division nearly absolute valued algebra \item[(2)] either $A$ is isomorphic to the real numbers or $A$ is a nearly absolute valued algebra with all elements having some square root \item[(3)] $A$ is a division algebra with $\dim A\leq2$ \item[(4)] $A$ is a finite dimensional division algebra \item[(5)] $A$ is a nearly absolute valued algebra with some element for which left translation has a dense image. \end{itemize} The implication $(4)\Rightarrow (3)$ is Hopf's Theorem: [\textit{H. Hopf}, Comment. Math. Helv. 13, 219--239 (1941; Zbl 0024.36002; JFM 67.0737.03)].