On a problem by Arens, Goldberg, and Luxemburg
  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 $A \in {\cal A}$, 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. 
Raymond Redheffer, Terence Tao