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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Grothendieck’s inequalities for JB$^*$-triples: Proof of the Barton–Friedman conjecture
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by Jan Hamhalter, Ondřej F.K. Kalenda, Antonio M. Peralta and Hermann Pfitzner PDF
Trans. Amer. Math. Soc. 374 (2021), 1327-1350 Request permission

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 \begin{equation*} \|T(x)\| \leq K \|T\| \|x\|_{\psi } \end{equation*} 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 \begin{equation*} |V(x,y)| \leq G \ \|V\| \|x\|_{\varphi } \|y\|_{\psi } \end{equation*} 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
  • MR Author ID: 80430
  • Email: hamhalte@fel.cvut.cz
  • Ondřej F.K. Kalenda
  • Affiliation: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 86, 186 75 Praha 8, Czech Republic
  • ORCID: 0000-0003-4312-2166
  • Email: kalenda@karlin.mff.cuni.cz
  • Antonio M. Peralta
  • Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain.
  • MR Author ID: 666723
  • ORCID: 0000-0003-2528-8357
  • Email: aperalta@ugr.es
  • Hermann Pfitzner
  • Affiliation: Université d’Orléans, BP 6759, F-45067 Orléans Cedex 2, France
  • MR Author ID: 333993
  • Email: hermann.pfitzner@univ-orleans.fr
  • 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.
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 1327-1350
  • MSC (2010): Primary 46L70, 17C65
  • DOI: https://doi.org/10.1090/tran/8227
  • MathSciNet review: 4196395