On a problem by Arens, Goldberg, and Luxemburg

Abstract

We construct a normed algebra ${\calA}$ with norm $N(\cdot )$ over the reals, which is {\em quadrative} in the sense that $N(A^2)\le N(A{)}^2$ for all $A\in \mathcal{A}$, but is not 3-{\em bounded} in the sense that $N(A^3)\le N(A{)}^3$. This answers a question of Arens, Goldberg, and Luxemburg.

7 pages, 2 pictures, submitted, Pacific J. Math