Grothendieck’s inequalities for JB*-triples: Proof of the Barton–Friedman conjecture

Hamhalter, Jan; Kalenda, Ondřej; Peralta, Antonio; Pfitzner, Hermann

doi:10.1090/tran/8227

Grothendieck's inequalities for JB-triples: Proof of the Barton-Friedman conjecture

Authors: Jan Hamhalter, Ondřej F.K. Kalenda, Antonio M. Peralta and Hermann Pfitzner
Journal: Trans. Amer. Math. Soc. 374 (2021), 1327-1350
MSC (2010): Primary 46L70, 17C65
DOI: https://doi.org/10.1090/tran/8227
Published electronically: November 12, 2020
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Abstract: We prove that, given a constant and a bounded linear operator from a JB-triple into a complex Hilbert space , there exists a norm-one functional $\psi \in E^*$ satisfying

$\displaystyle \Vert T(x)\Vert \leq K \, \Vert T\Vert \, \Vert x\Vert _{\psi }$

for all $x\in E$ . Applying this result we show that, given $G > 8 (1+2\sqrt {3})$ and a bounded bilinear form

on the Cartesian product of two JB

-triples

and

, there exist norm-one functionals $\varphi \in E^{*}$ and $\psi \in B^{*}$ satisfying

$\displaystyle \vert V(x,y)\vert \leq G \ \Vert V\Vert \, \Vert x\Vert _{\varphi } \, \Vert y\Vert _{\psi }$

for all $(x,y)\in E \times B$ . These results prove a conjecture pursued during almost twenty years.

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