×

Extending surjective isometries defined on the unit sphere of \(\ell_\infty(\Gamma)\). (English) Zbl 07037768

Summary: Let \(\Gamma\) be an infinite set equipped with the discrete topology. We prove that the space \(\ell_\infty(\Gamma ,\mathbb{C})\), of all complex-valued bounded functions on \(\Gamma\), satisfies the Mazur-Ulam property, that is, every surjective isometry from the unit sphere of \(\ell_\infty(\Gamma ,\mathbb{C})\) onto the unit sphere of an arbitrary complex Banach space \(X\) admits a unique extension to a surjective real linear isometry from \(\ell_\infty(\Gamma ,\mathbb{C})\) to \(X\).

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
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
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Diestel, J.: Sequences and Series in Banach Spaces, Graduate Texts in Mathematics, vol. 92. Springer, New York (1984) · doi:10.1007/978-1-4612-5200-9
[2] 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), 479-483 (2002) · Zbl 1107.46302 · doi:10.1007/BF02872336
[3] Ding, G.G.: The isometric extension problem in the spheres of \[l^p (\Gamma )\] lp(Γ)\[(p>1)\](p>1) type spaces. Sci. China Ser. A 46, 333-338 (2003) · Zbl 1217.46010
[4] Ding, G.G.: The representation theorem of onto isometric mappings between two unit spheres of \[l^\infty\] l∞-type spaces and the application on isometric extension problem. Sci. China Ser. A 47, 722-729 (2004) · Zbl 1093.46007 · doi:10.1360/03ys0049
[5] Ding, G.G.: The representation theorem of onto isometric mappings between two unit spheres of \[l^1 (\Gamma )\] l1(Γ) type spaces and the application to the isometric extension problem. Acta. Math. Sin. (Engl. Ser.) 20, 1089-1094 (2004) · Zbl 1093.46008 · doi:10.1007/s10114-004-0447-7
[6] Ding, G.G.: The isometric extension of the into mapping from a \[{\cal{L}}^\infty (\Gamma )\] L∞(Γ)-type space to some Banach space. Ill. J. Math. 51(2), 445-453 (2007) · Zbl 1136.46007
[7] Fernández-Polo, F.J., Garcés, J.J., Peralta, A.M., Villanueva, I.: Tingley’s problem for spaces of trace class operators. Linear Algebra Appl. 529, 294-323 (2017) · Zbl 1388.46013 · doi:10.1016/j.laa.2017.04.024
[8] Fernández-Polo, F.J., Peralta, A.M.: Low rank compact operators and Tingley’s problem (2016). arXiv:1611.10218v1 · Zbl 1491.46004
[9] Fernández-Polo, F.J., Peralta, A.M.: On the extension of isometries between the unit spheres of a \[\text{ C }^*C\]∗-algebra and \[B(H)B(H)\] Trans. Amer. Math. Soc. 5, 63-80 (2018) · Zbl 06843537
[10] Fernández-Polo, F.J., Peralta, A.M.: Tingley’s problem through the facial structure of an atomic \[\text{ JBW }^*\] JBW∗-triple. J. Math. Anal. Appl. 455, 750-760 (2017) · Zbl 1387.46015 · doi:10.1016/j.jmaa.2017.06.002
[11] Fernández-Polo, F.J., Peralta, A.M.: On the extension of isometries between the unit spheres of von Neumann algebras. J. Math. Anal. Appl. 466, 127-143 (2018) · Zbl 1403.46009 · doi:10.1016/j.jmaa.2018.05.062
[12] Jiménez-Vargas, A., Morales-Campoy, A., Peralta, A.M., Ramírez, M.I.: The Mazur-Ulam property for the space of complex null sequences. Linear Multilinear Algebra (2017). https://doi.org/10.1080/03081087.2018.1433625 · Zbl 1421.46012
[13] Kadets, V., Martín, M.: Extension of isometries between unit spheres of infite-dimensional polyhedral Banach spaces. J. Math. Anal. Appl. 396, 441-447 (2012) · Zbl 1258.46004 · doi:10.1016/j.jmaa.2012.06.031
[14] Liu, R.: On extension of isometries between unit spheres of \[{\cal{L}}^{\infty }(\Gamma )\] L∞(Γ)-type space and a Banach space \[E\] E. J. Math. Anal. Appl. 333, 959-970 (2007) · Zbl 1124.46005 · doi:10.1016/j.jmaa.2006.11.044
[15] Peralta, A.M., Tanaka, R.: A solution to Tingley’s problem for isometries between the unit spheres of compact \[\text{ C }^*C\]∗-algebras and \[\text{ JB }^*\] JB∗-triples. Sci. China Math. (2018). https://doi.org/10.1007/s11425-017-9188-6. (to appear in)
[16] Tan, D.: Extension of isometries on unit sphere of \[L^\infty\] L∞. Taiwanese J. Math. 15, 819-827 (2011) · Zbl 1244.46003 · doi:10.11650/twjm/1500406236
[17] Tan, D.: On extension of isometries on the unit spheres of \[L^p\] Lp-spaces for \[0<p \le 10\]<p≤1. Nonlinear Anal. 74, 6981-6987 (2011) · Zbl 1235.46005 · doi:10.1016/j.na.2011.07.035
[18] Tan, D.: Extension of isometries on the unit sphere of \[L^p\] Lp-spaces. Acta. Math. Sin. (Engl. Ser.) 28, 1197-1208 (2012) · Zbl 1271.46011 · doi:10.1007/s10114-011-0302-6
[19] Tanaka, R.: The solution of Tingley’s problem for the operator norm unit sphere of complex \[n \times n\] n×n matrices. Linear Algebra Appl. 494, 274-285 (2016) · Zbl 1382.15034 · doi:10.1016/j.laa.2016.01.020
[20] Tanaka, R.: Spherical isometries of finite dimensional \[C^*C\]∗-algebras. J. Math. Anal. Appl. 445(1), 337-341 (2017) · Zbl 1371.46008 · doi:10.1016/j.jmaa.2016.07.073
[21] Tanaka, R.: Tingley’s problem on finite von Neumann algebras. J. Math. Anal. Appl. 451, 319-326 (2017) · Zbl 1371.46009 · doi:10.1016/j.jmaa.2017.02.013
[22] Tingley, D.: Isometries of the unit sphere. Geom. Dedicata 22, 371-378 (1987) · Zbl 0615.51005 · doi:10.1007/BF00147942
[23] Wang, R.S.: Isometries between the unit spheres of \[C_0(\Omega )\] C0(Ω) type spaces. Acta Math. Sci. (Engl. Ed.) 14(1), 82-89 (1994) · doi:10.1016/S0252-9602(18)30093-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.