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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: November 12, 2020
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Abstract: We prove that, given a constant $ K> 2$ and a bounded linear operator $ T$ from a JB$ ^*$-triple $ E$ into a complex Hilbert space $ H$, 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 $ V$ on the Cartesian product of two JB$ ^*$-triples $ E$ and $ B$, 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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Additional Information

Jan Hamhalter
Affiliation: Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technicka 2, 166 27, Prague 6, Czech Republic

Ondřej F.K. Kalenda
Affiliation: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 86, 186 75 Praha 8, Czech Republic

Antonio M. Peralta
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain.

Hermann Pfitzner
Affiliation: Université d’Orléans, BP 6759, F-45067 Orléans Cedex 2, France

Keywords: Grothendieck's inequality, little Grothendieck inequality, JB$^*$-triple, JBW$^*$-triple
Received by editor(s): May 8, 2020
Received by editor(s) in revised form: June 19, 2020
Published electronically: November 12, 2020
Additional Notes: The first two authors were supported in part by the Research Grant GAČR 17-00941S. The first author was partly supported further by the project OP VVV Center for Advanced Applied Science CZ.02.1.01/0.0/0.0/16_019/000077. The third author was partially supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European Regional Development Fund project no. PGC2018-093332-B-I00, Programa Operativo FEDER 2014-2020 and Consejería de Economía y Conocimiento de la Junta de Andalucía grant number A-FQM-242-UGR18 and by Junta de Andalucía grant FQM375.
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