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
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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.

References

Charalambos D. Aliprantis and Kim C. Border, Infinite dimensional analysis, 3rd ed., Springer, Berlin, 2006. A hitchhiker’s guide. MR 2378491
T. Barton and Y. Friedman, Grothendieck’s inequality for $JB^*$-triples and applications, J. London Math. Soc. (2) 36 (1987), no. 3, 513–523. MR 918642, DOI 10.1112/jlms/s2-36.3.513
T. Barton and Richard M. Timoney, Weak$^\ast$-continuity of Jordan triple products and its applications, Math. Scand. 59 (1986), no. 2, 177–191. MR 884654, DOI 10.7146/math.scand.a-12160
Robert Braun, Wilhelm Kaup, and Harald Upmeier, A holomorphic characterization of Jordan $C^*$-algebras, Math. Z. 161 (1978), no. 3, 277–290. MR 493373, DOI 10.1007/BF01214510
Leslie J. Bunce, Francisco J. Fernández-Polo, Juan Martínez Moreno, and Antonio M. Peralta, A Saitô-Tomita-Lusin theorem for JB*-triples and applications, Q. J. Math. 57 (2006), no. 1, 37–48. MR 2204259, DOI 10.1093/qmath/hah059
Miguel Cabrera García and Ángel Rodríguez Palacios, Non-associative normed algebras. Vol. 1, Encyclopedia of Mathematics and its Applications, vol. 154, Cambridge University Press, Cambridge, 2014. The Vidav-Palmer and Gelfand-Naimark theorems. MR 3242640, DOI 10.1017/CBO9781107337763
Miguel Cabrera García and Ángel Rodríguez Palacios, Non-associative normed algebras. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 167, Cambridge University Press, Cambridge, 2018. Representation theory and the Zel’manov approach. MR 3791798, DOI 10.1017/9781107337817
Cho-Ho Chu, Jordan structures in geometry and analysis, Cambridge Tracts in Mathematics, vol. 190, Cambridge University Press, Cambridge, 2012. MR 2885059
Cho-Ho Chu, Bruno Iochum, and Guy Loupias, Grothendieck’s theorem and factorization of operators in Jordan triples, Math. Ann. 284 (1989), no. 1, 41–53. MR 995380, DOI 10.1007/BF01443503
Seán Dineen, Complete holomorphic vector fields on the second dual of a Banach space, Math. Scand. 59 (1986), no. 1, 131–142. MR 873493, DOI 10.7146/math.scand.a-12158
C. Martin Edwards, Francisco J. Fernández-Polo, Christopher S. Hoskin, and Antonio M. Peralta, On the facial structure of the unit ball in a $\rm JB^*$-triple, J. Reine Angew. Math. 641 (2010), 123–144. MR 2643927, DOI 10.1515/CRELLE.2010.030
Marián J. Fabian, Gâteaux differentiability of convex functions and topology, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. Weak Asplund spaces; A Wiley-Interscience Publication. MR 1461271
Yaakov Friedman and Bernard Russo, Structure of the predual of a $JBW^\ast$-triple, J. Reine Angew. Math. 356 (1985), 67–89. MR 779376, DOI 10.1515/crll.1985.356.67
A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79 (French). MR 94682
Uffe Haagerup, The Grothendieck inequality for bilinear forms on $C^\ast$-algebras, Adv. in Math. 56 (1985), no. 2, 93–116. MR 788936, DOI 10.1016/0001-8708(85)90026-X
Jan Hamhalter, Ondřej F. K. Kalenda, Antonio M. Peralta, and Hermann Pfitzner, Measures of weak non-compactness in preduals of von Neumann algebras and $\rm JBW^\ast$-triples, J. Funct. Anal. 278 (2020), no. 1, 108300, 69. MR 4027746, DOI 10.1016/j.jfa.2019.108300
Harald Hanche-Olsen and Erling Størmer, Jordan operator algebras, Monographs and Studies in Mathematics, vol. 21, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 755003
Lawrence A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces, Proceedings on Infinite Dimensional Holomorphy (Internat. Conf., Univ. Kentucky, Lexington, Ky., 1973) Lecture Notes in Math., Vol. 364, Springer, Berlin, 1974, pp. 13–40. MR 0407330
Günther Horn, Classification of JBW$^*$-triples of type $\textrm {I}$, Math. Z. 196 (1987), no. 2, 271–291. MR 910832, DOI 10.1007/BF01163661
G. Horn and E. Neher, Classification of continuous $JBW^*$-triples, Trans. Amer. Math. Soc. 306 (1988), no. 2, 553–578. MR 933306, DOI 10.1090/S0002-9947-1988-0933306-7
Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Graduate Studies in Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1997. Advanced theory; Corrected reprint of the 1986 original. MR 1468230, DOI 10.1090/gsm/016/01
O. F. K. Kalenda, A. M. Peralta, and H. Pfitzner On optimality of constants in the Little Grothendieck theorem, arXiv:2002.12273.
Wilhelm Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), no. 4, 503–529. MR 710768, DOI 10.1007/BF01173928
Wilhelm Kaup and Harald Upmeier, Jordan algebras and symmetric Siegel domains in Banach spaces, Math. Z. 157 (1977), no. 2, 179–200. MR 492414, DOI 10.1007/BF01215150
Antonio M. Peralta, Little Grothendieck’s theorem for real $\textrm {JB}^*$-triples, Math. Z. 237 (2001), no. 3, 531–545. MR 1845336, DOI 10.1007/PL00004878
Antonio M. Peralta, New advances on the Grothendieck’s inequality problem for bilinear forms on JB*-triples, Math. Inequal. Appl. 8 (2005), no. 1, 7–21. MR 2137902, DOI 10.7153/mia-08-02
Antonio M. Peralta and Angel Rodríguez Palacios, Grothendieck’s inequalities for real and complex $\textrm {JBW}^\ast$-triples, Proc. London Math. Soc. (3) 83 (2001), no. 3, 605–625. MR 1851084, DOI 10.1112/plms/83.3.605
Antonio M. Peralta and Angel Rodríguez Palacios, Grothendieck’s inequalities revisited, Recent progress in functional analysis (Valencia, 2000) North-Holland Math. Stud., vol. 189, North-Holland, Amsterdam, 2001, pp. 409–423. MR 1861775, DOI 10.1016/S0304-0208(01)80064-5
Gilles Pisier, Grothendieck’s theorem for noncommutative $C^{\ast }$-algebras, with an appendix on Grothendieck’s constants, J. Functional Analysis 29 (1978), no. 3, 397–415. MR 512252, DOI 10.1016/0022-1236(78)90038-1
Gilles Pisier, Grothendieck’s theorem, past and present, Bull. Amer. Math. Soc. (N.S.) 49 (2012), no. 2, 237–323. MR 2888168, DOI 10.1090/S0273-0979-2011-01348-9
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
J. D. Maitland Wright, Jordan $C^*$-algebras, Michigan Math. J. 24 (1977), no. 3, 291–302. MR 487478