On a problem by Arens, Goldberg, and Luxemburg

We construct a normed algebra $\text{calA}$ with norm $N(\cdot )$ over the reals, which is {\em quadrative} in the sense that $N(A^2)\le N(A{)}^2$ for all $A \in {\cal A}$,butisnot3-\text{em}boundedinthesensethat$N(A^3) \le N(A)^3$. This answers a question of Arens, Goldberg, and Luxemburg.