Grothendieck's inequalities for JB${}^{\ast }$-triples: Proof of the Barton-Friedman conjecture

Abstract

We prove that, given a constant $K>2$ and a bounded linear operator $T$ from a JB${}^{\ast }$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $\psi \in E^{\ast }$ satisfying $$\Vert T(x)\Vert \le K\Vert T\Vert \Vert x{\Vert }_{\psi },$$ for all $x\in E$. Applying this result we show that, given $G>8(1+2\sqrt{3})$ and a bounded bilinear form $V$ on the Cartesian product of two JB${}^{\ast }$-triples $E$ and $B$, there exist norm-one functionals $\phi \in E^{\ast }$ and $\psi \in B^{\ast }$ satisfying $$|V(x,y)|\le G\Vert V\Vert \Vert x{\Vert }_{\phi }\Vert y{\Vert }_{\psi }$$ for all $(x,y)\in E\times B$. These results prove a conjecture pursued during almost twenty years.

23 pages. We corrected a misprint, added some comments and precised one reference