Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-25T01:09:01.379Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE EXTENSION OF ISOMETRIES BETWEEN THE UNIT SPHERES OF A $\text{JBW}^{\ast }$-TRIPLE AND A BANACH SPACE

Part of: Normed linear spaces and Banach spaces; Banach lattices Linear function spaces and their duals Topological linear spaces and related structures Special classes of linear operators

Published online by Cambridge University Press:  15 April 2019

Julio Becerra-Guerrero Open the ORCID record for Julio Becerra-Guerrero [Opens in a new window] ,
María Cueto-Avellaneda Open the ORCID record for María Cueto-Avellaneda [Opens in a new window] ,
Francisco J. Fernández-Polo Open the ORCID record for Francisco J. Fernández-Polo [Opens in a new window]  and
Antonio M. Peralta Open the ORCID record for Antonio M. Peralta [Opens in a new window]
Julio Becerra-Guerrero
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071Granada, Spain (juliobg@ugr.es; mcueto@ugr.es; pacopolo@ugr.es; aperalta@ugr.es)
María Cueto-Avellaneda
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071Granada, Spain (juliobg@ugr.es; mcueto@ugr.es; pacopolo@ugr.es; aperalta@ugr.es)
Francisco J. Fernández-Polo
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071Granada, Spain (juliobg@ugr.es; mcueto@ugr.es; pacopolo@ugr.es; aperalta@ugr.es)
Antonio M. Peralta
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071Granada, Spain (juliobg@ugr.es; mcueto@ugr.es; pacopolo@ugr.es; aperalta@ugr.es)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that if $M$ is a $\text{JBW}^{\ast }$-triple and not a Cartan factor of rank two, then $M$ satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another real Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$.

Keywords

Tingley’s problemMazur–Ulam propertyextension of isometriesJBW∗ -triples

MSC classification

Primary: 47B49: Transformers, preservers (operators on spaces of operators)
Secondary: 46A22: Theorems of Hahn-Banach type; extension and lifting of functionals and operators 46B20: Geometry and structure of normed linear spaces 46B04: Isometric theory of Banach spaces 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) 46E40: Spaces of vector- and operator-valued functions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 1 , January 2021 , pp. 277 - 303
Check for updates
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akemann, C. A. and Pedersen, G. K., Facial structure in operator algebra theory, Proc. Lond. Math. Soc. 64 (1992), 418–448.10.1112/plms/s3-64.2.418CrossRefGoogle Scholar
Bader, U., Furman, A., Gelander, T. and Monod, N., Property (T) and rigidity for actions on Banach spaces, Acta Math. 198 (2007), 57–105.10.1007/s11511-007-0013-0CrossRefGoogle Scholar
Barton, T. and Timoney, R. M., Weak∗ -continuity of Jordan triple products and applications, Math. Scand. 59 (1986), 177–191.10.7146/math.scand.a-12160CrossRefGoogle Scholar
Barton, T. J. and Friedman, Y., Bounded derivations of JB∗ -triples, Q. J. Math. 41(2) (1990), 255–268.10.1093/qmath/41.3.255CrossRefGoogle Scholar
Braun, R. B., Kaup, W. and Upmeier, H., A holomorphic characterization of Jordan-C ∗ -algebras, Math. Z. 161 (1978), 277–290.CrossRefGoogle Scholar
Bunce, L. J. and Chu, C.-H., Compact operations, multipliers and Radon-Nikodym property in JB ∗ -triples, Pacific J. Math. 153 (1992), 249–265.10.2140/pjm.1992.153.249CrossRefGoogle Scholar
Bunce, L. J., Fernández-Polo, F. J., Martínez Moreno, J. and Peralta, A. M., A Saitô-Tomita-Lusin theorem for JB∗ -triples and applications, Q. J. Math. 57 (2006), 37–48.10.1093/qmath/hah059CrossRefGoogle Scholar
Burgos, M., Fernández-Polo, F. J., Garcés, J., Martínez, J. and Peralta, A. M., Orthogonality preservers in C∗ -algebras, JB∗ -algebras and JB∗ -triples, J. Math. Anal. Appl. 348 (2008), 220–233.10.1016/j.jmaa.2008.07.020CrossRefGoogle Scholar
Cabrera García, M. and Rodríguez Palacios, A., Non-associative normed algebras, in Vol. 2. Representation Theory and the Zel’manov Approach, Encyclopedia of Mathematics and its Applications, Volume 167 (Cambridge University Press, Cambridge, 2018).Google Scholar
Cheng, L. and Dong, Y., On a generalized Mazur–Ulam question: extension of isometries between unit spheres of Banach spaces, J. Math. Anal. Appl. 377 (2011), 464–470.10.1016/j.jmaa.2010.11.025CrossRefGoogle Scholar
Chu, C.-H. and Iochum, B., Complementation of JBW∗ -triples in von Neumann algebras, Proc. Amer. Math. Soc. 108 (1990), 19–24.10.1090/S0002-9939-1990-0990418-4CrossRefGoogle Scholar
Cueto-Avellaneda, M. and Peralta, A. M., The Mazur–Ulam property for commutative von Neumann algebras, Linear Multilinear Algebra, to appear, https://doi.org/10.1080/03081087.2018.1505823.Google Scholar
Dang, T. and Friedman, Y., Classification of JBW∗ -triple factors and applications, Math. Scand. 61(2) (1987), 292–330.10.7146/math.scand.a-12206CrossRefGoogle Scholar
Day, M. M., Some characterizations of inner-product spaces, Trans. Amer. Math. Soc. 62 (1947), 320–337.10.1090/S0002-9947-1947-0022312-9CrossRefGoogle Scholar
Dineen, S., The second dual of a JB∗ -triple system, in Complex Analysis, Functional Analysis and Approximation Theory (ed. Múgica, J.), North-Holland Mathematics Studies, Volume 125, pp. 67–69 (North-Holland, Amsterdam-New York, 1986).Google Scholar
Ding, G. G., The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space, Sci. China Ser. A 45(4) (2002), 479–483.CrossRefGoogle Scholar
Ding, G. G., On the extension of isometries between unit spheres of E and C (𝛺), Acta Math. Sin. (Engl. Ser.) 19 (2003), 793–800.10.1007/s10114-003-0240-zCrossRefGoogle Scholar
Ding, G. G., The isometric extension of the into mapping from a 𝓛 ∞ (𝛤)-type space to some Banach space, Illinois J. Math. 51(2) (2007), 445–453.CrossRefGoogle Scholar
Ding, G. G., On isometric extension problem between two unit spheres, Sci. China Ser. A 52 (2009), 2069–2083.CrossRefGoogle Scholar
Edwards, C. M., Fernández-Polo, F. J., Hoskin, C. S. and Peralta, A. M., On the facial structure of the unit ball in a JB∗ -triple, J. Reine Angew. Math. 641 (2010), 123–144.Google Scholar
Edwards, C. M. and Rüttimann, G. T., On the facial structure of the unit balls in a JBW∗ -triple and its predual, J. Lond. Math. Soc. 38 (1988), 317–332.CrossRefGoogle Scholar
Edwards, C. M. and Rüttimann, G. T., Compact tripotents in bi-dual JB∗ -triples, Math. Proc. Cambridge Philos. Soc. 120 (1996), 155–173.10.1017/S0305004100074740CrossRefGoogle Scholar
Fernández-Polo, F. J., Garcés, J. J., Peralta, A. M. and Villanueva, I., Tingley’s problem for spaces of trace class operators, Linear Algebra Appl. 529 (2017), 294–323.10.1016/j.laa.2017.04.024CrossRefGoogle Scholar
Fernández-Polo, F. J., Jordá, E. and Peralta, A. M., Tingley’s problem for p-Schatten von Neumann classes, J. Spectr. Theory, to appear, arXiv:1803:00763.Google Scholar
Fernández-Polo, F. J. and Peralta, A. M., Closed tripotents and weak compactness in the dual space of a JB∗ -triple, J. Lond. Math. Soc. 74 (2006), 75–92.CrossRefGoogle Scholar
Fernández-Polo, F. J. and Peralta, A. M., Non-commutative generalisations of Urysohn’s lemma and hereditary inner ideals, J. Funct. Anal. 259 (2010), 343–358.10.1016/j.jfa.2010.04.003CrossRefGoogle Scholar
Fernández-Polo, F. J. and Peralta, A. M., Tingley’s problem through the facial structure of an atomic JBW∗ -triple, J. Math. Anal. Appl. 455 (2017), 750–760.CrossRefGoogle Scholar
Fernández-Polo, F. J. and Peralta, A. M., On the extension of isometries between the unit spheres of von Neumann algebras, J. Math. Anal. Appl. 466 (2018), 127–143.10.1016/j.jmaa.2018.05.062CrossRefGoogle Scholar
Fernández-Polo, F. J. and Peralta, A. M., Low rank compact operators and Tingley’s problem, Adv. Math. 338 (2018), 1–40.CrossRefGoogle Scholar
Friedman, Y. and Russo, B., Structure of the predual of a JBW∗ -triple, J. Reine Angew. Math. 356 (1985), 67–89.Google Scholar
Friedman, Y. and Russo, B., The Gelfand–Naimark Theorem for JB∗ -triples, Duke Math. J. 53 (1986), 139–148.10.1215/S0012-7094-86-05308-1CrossRefGoogle Scholar
Harris, L. A., Bounded symmetric homogeneous domains in infinite dimensional spaces, in Proceedings on Infinite Dimensional Holomorphy (Kentucky 1973), Lecture Notes in Mathematics, Volume 364, pp. 13–40 (Springer, Berlin-Heidelberg-New York, 1974).10.1007/BFb0069002CrossRefGoogle Scholar
Horn, G., Characterization of the predual and ideal structure of a JBW∗ -triple, Math. Scand. 61(1) (1987), 117–133.CrossRefGoogle Scholar
Horn, G., Classification of JBW∗ -triples of type I, Math. Z. 196 (1987), 271–291.10.1007/BF01163661CrossRefGoogle Scholar
Jiménez-Vargas, A., Morales-Campoy, A., Peralta, A. M. and Ramírez, M. I., The Mazur–Ulam property for the space of complex null sequences, Linear and Multilinear Algebra 67(4) (2019), 799–816.10.1080/03081087.2018.1433625CrossRefGoogle Scholar
Kadison, R. V. and Ringrose, J. R., Fundamentals of the Theory of Operator Algebras. Vol. II. Advanced Theory, Pure and Applied Mathematics, Volume 100 (Academic Press, Inc., Orlando, FL, 1986).Google Scholar
Kaup, W., A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), 503–529.10.1007/BF01173928CrossRefGoogle Scholar
Kaup, W., On real Cartan factors, Manuscripta Math. 92 (1997), 191–222.10.1007/BF02678189CrossRefGoogle Scholar
Kaup, W. and Upmeier, H., Jordan algebras and symmetric Siegel domains in Banach spaces, Math. Z. 157 (1977), 179–200.10.1007/BF01215150CrossRefGoogle Scholar
Li, B., Real Operator Algebras (World Scientific Publishing Co., Inc., River Edge, NJ, 2003).CrossRefGoogle Scholar
Liu, R., On extension of isometries between unit spheres of 𝓛 ∞ (𝛤)-type space and a Banach space E , J. Math. Anal. Appl. 333 (2007), 959–970.10.1016/j.jmaa.2006.11.044CrossRefGoogle Scholar
Mankiewicz, P., On extension of isometries in normed linear spaces, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 20 (1972), 367–371.Google Scholar
Megginson, R. E., An Introduction to Banach Space Theory (Springer, New York, 1998).10.1007/978-1-4612-0603-3CrossRefGoogle Scholar
Mori, M., Tingley’s problem through the facial structure of operator algebras, J. Math. Anal. Appl. 466(2) (2018), 1281–1298.10.1016/j.jmaa.2018.06.050CrossRefGoogle Scholar
Mori, M. and Ozawa, N., Mankiewicz’s theorem and the Mazur–Ulam property for C∗ -algebras, Studia Math. accepted for publication, arXiv:1804.10674.Google Scholar
Navarro-Pascual, J. C. and Navarro, M. A., Unitary operators in real von Neumann algebras, J. Math. Anal. Appl. 386(2) (2012), 933–938.10.1016/j.jmaa.2011.08.049CrossRefGoogle Scholar
Peralta, A. M., Extending surjective isometries defined on the unit sphere of ℓ ∞(𝛤), Rev. Mat. Complut. 32(1) (2019), 99–114.CrossRefGoogle Scholar
Peralta, A. M., A survey on Tingley’s problem for operator algebras, Acta Sci. Math. (Szeged) 84 (2018), 81–123.CrossRefGoogle Scholar
Peralta, A. M. and Rodríguez, A., Grothendieck’s inequalities for real and complex JBW∗ -triples, Proc. Lond. Math. Soc. 83 (2001), 605–625.10.1112/plms/83.3.605CrossRefGoogle Scholar
Peralta, A. M. and Tanaka, R., A solution to Tingley’s problem for isometries between the unit spheres of compact C∗ -algebras and JB∗ -triples, Sci. China Math. 62(3) (2019), 553–568.CrossRefGoogle Scholar
Russo, B. and Dye, H. A., A note on unitary operators in C∗ -algebras, Duke Math. J. 33 (1966), 413–416.CrossRefGoogle Scholar
Siddiqui, A. A., Average of two extreme points in JBW∗ -triples, Proc. Japan Acad. Ser. A 83(9–10) (2007), 176–178.CrossRefGoogle Scholar
Siddiqui, A. A., A proof of the Russo–Dye theorem for JB∗ -algebras, New York J. Math. 16 (2010), 53–60.Google Scholar
Tan, D., Huang, X. and Liu, R., Generalized-lush spaces and the Mazur–Ulam property, Studia Math. 219 (2013), 139–153.10.4064/sm219-2-4CrossRefGoogle Scholar
Tan, D., Extension of isometries on unit sphere of L ∞ , Taiwanese J. Math. 15 (2011), 819–827.10.11650/twjm/1500406236CrossRefGoogle Scholar
Tan, D., On extension of isometries on the unit spheres of L p -spaces for 0 < p⩽1, Nonlinear Anal. 74 (2011), 6981–6987.CrossRefGoogle Scholar
Tan, D., Extension of isometries on the unit sphere of L p -spaces, Acta Math. Sin. (Engl. Ser.) 28 (2012), 1197–1208.10.1007/s10114-011-0302-6CrossRefGoogle Scholar
Tan, D. and Liu, R., A note on the Mazur–Ulam property of almost-CL-spaces, J. Math. Anal. Appl. 405 (2013), 336–341.CrossRefGoogle Scholar
Tanaka, R., A further property of spherical isometries, Bull. Aust. Math. Soc. 90 (2014), 304–310.CrossRefGoogle Scholar
Tanaka, R., The solution of Tingley’s problem for the operator norm unit sphere of complex n × n matrices, Linear Algebra Appl. 494 (2016), 274–285.10.1016/j.laa.2016.01.020CrossRefGoogle Scholar
Tingley, D., Isometries of the unit sphere, Geom. Dedicata 22 (1987), 371–378.10.1007/BF00147942CrossRefGoogle Scholar
Wright, J. D. M., Jordan C∗ -algebras, Michigan Math. J. 24 (1977), 291–302.Google Scholar
Wright, J. D. M. and Youngson, M. A., A Russo–Dye theorem for Jordan C∗ -algebras, in Functional Analysis: Surveys and Recent Results (Proc. Conf., Paderborn, 1976), North-Holland Mathematics Studies, Volume 27; Notas de Mat., No. 63, pp. 279–282 (North-Holland, Amsterdam, 1977).10.1016/S0304-0208(08)70537-1CrossRefGoogle Scholar
Yang, X. and Zhao, X., On the extension problems of isometric and nonexpansive mappings, in Mathematics Without Boundaries (ed. Rassias, T. M. and Pardalos, P. M.), pp. 725–748 (Springer, New York, 2014).Google Scholar