Calculus in O-algebras with positive squares

Abstract

Let $A$ be an $\mathcal{O}$-algebra with positive squares and $F\left( X_{1},...,X_{n}\right) \in\linebreak\mathbb{R}^{+}\left[ X_{1},...,X_{n}\right] $ be a homogeneous polynomial of degree $p$ $\left( p\in\mathbb{N}^{\ast },\text{ }p\neq2\right) $. It is shown that for all $0\leq a_{1},...,a_{n}\in A$ there exists $0\leq a$ $\in A$ such that $F\left( a_{1},...,a_{n}\right) =a^{p}$ . As an application we show that every algebra homomorphism $T$ from an $\mathcal{O}$-algebra $A$\textit{ }with positive squares into an Archimedean semiprime \textit{f-}algebra $B$ is positive. This improves a result of Render [14, Theorem 4.1], who proved it for the case of order bounded multiplicative functional $T$ from an $\mathcal{O}$-algebra $A$ with positive squares into $\mathbb{R}$.