Let f : S(E) → S(B) be a surjective isometry between the unit spheres of two weakly compact JB*-triples not containing direct summands of rank less than or equal to 3. Suppose E has rank greater than or equal to 5. Applying techniques developed in JB*-triple theory, we prove that f admits an extension to a surjective real linear isometry T : E → B. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C*-algebras A and B, without assuming any restriction on the rank of their direct summands(and in particular when A = K(H) and B = K(H′)), extends to a surjective real linear isometry from A into B. These results provide new examples of infinite-dimensional Banach spaces where Tingley's problem admits a positive answer.
Let f : S(E) → S(B) be a surjective isometry between the unit spheres of two weakly compact JB*-triples not containing direct summands of rank less than or equal to 3. Suppose E has rank greater than or equal to 5. Applying techniques developed in JB*-triple theory, we prove that f admits an extension to a surjective real linear isometry T : E → B. Among the consequences, we show that every surjective isometry between the unit spheres of two compact C*-algebras A and B, without assuming any restriction on the rank of their direct summands(and in particular when A = K(H) and B = K(H′)), extends to a surjective real linear isometry from A into B. These results provide new examples of infinite-dimensional Banach spaces where Tingley's problem admits a positive answer.
1 Akemann C A,Pedersen G K.Facial structure in operator algebra theory.Proc Lond Math Soc(3),1992,64:418-448
2 Alexander J C.Compact Banach algebras.Proc Lond Math Soc(3),1968,18:1-18
3 Barton J T,Timoney R M.Weak*-continuity of Jordan triple products and applications.Math Scand,1986,59:177-191
4 Bunce L J.The theory and structure of dual JB-algebras.Math Z,1982,180:525-534
5 Bunce L J,Chu C-H.Compact operations,multipliers and Radon-Nikodym property in JB*-triples.Pacific J Math,1992,153:249-265
6 Bunce L J,Fern′andez-Polo F J,Mart′?nez Moreno J,et al.A Sait?o-Tomita-Lusin theorem for JB*-triples and applications.Q J Math,2006,57:37-48
7 Burgos M,Fern′andez-Polo F J,Garc′es J,et al.Orthogonality preservers in C*-algebras,JB*-algebras and JB*-triples.J Math Anal Appl,2008,348:220-233
8 Chu C-H.Jordan Structures in Geometry and Analysis.Cambridge:Cambridge University Press,2012
9 Chu C-H,Dang T,Russo B,et al.Surjective isometries of real C*-algebras.J Lond Math Soc(2),1993,47:97-118
10 Dang T.Real isometries between JB*-triples.Proc Amer Math Soc,1992,114:971-980
11 Dineen S.The second dual of a JB*-triple system,In:Complex Analysis,Functional Analysis and Approximation Theory.North-Holland Mathematics Studies,vol.125.Amsterdam-New York:North-Holland,1986,67-69
12 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,2002,45:479-483
13 Ding G G.The isometric extension problem in the spheres of lp(Γ)(p>1)type spaces.Sci China Ser A,2003,46:333-338
14 Ding G G.On the extension of isometries between unit spheres of E and C(?).Acta Math Sin Engl Ser,2003,19,793-800
15 Ding G G.The representation theorem of onto isometric mappings between two unit spheres of l∞-type spaces and the application on isometric extension problem.Sci China Ser A,2004,47:722-729
16 Ding G G.The representation theorem of onto isometric mappings between two unit spheres of l1(Γ)type spaces and the application to the isometric extension problem.Acta Math Sin Engl Ser,2004,20:1089-1094
17 Ding G G.On isometric extension problem between two unit spheres.Sci China Ser A,2009,52:2069-2083
18 Ding G G.The isometric extension problem between unit spheres of two separable Banach spaces.Acta Math Sin Engl Ser,2015,31:1872-1878
19 Edwards C M,Fern′andez-Polo F J,Hoskin C S,et al.On the facial structure of the unit ball in a JB*-triple.J Reine Angew Math,2010,641:123-144
20 Edwards C M,H¨ugli R V.M-orthogonality and holomorphic rigidity in complex Banach spaces.Acta Sci Math(Szeged),2004,70:237-264
21 Edwards C M,R¨uttimann G T.On the facial structure of the unit balls in a JBW*-triple and its predual.J Lond Math Soc(2),1988,38:317-322
22 Edwards C M,R¨uttimann G T.Compact tripotents in bi-dual JB*-triples.Math Proc Cambridge Philos Soc,1996,120:155-173
23 Fern′andez-Polo F J,Mart′?nez J,Peralta A M.Surjective isometries between real JB*-triples.Math Proc Cambridge Philos Soc,2004,137:709-723
24 Fern′andez-Polo F J,Mart′?nez J,Peralta A M.Geometric characterization of tripotents in real and complex JB*-triples.J Math Anal Appl,2004,295:435-443
25 Fern′andez-Polo F J,Mart′?nez J,Peralta A M.Contractive perturbations in JB*-triples.J Lond Math Soc(2),2012,85:349-364
26 Fern′andez-Polo F J,Peralta A M.Closed tripotents and weak compactness in the dual space of a JB*-triple.J Lond Math Soc,2006,74:75-92
27 Fern′andez-Polo F J,Peralta A M.Non-commutative generalisations of Urysohn’s lemma and hereditary inner ideals.J Funct Anal,2010,259,343-358
28 Fern′andez-Polo F J,Peralta A M.Low rank compact operators and
Tingley’s problem.ArXiv:1611.10218v1,2016
29 Friedman Y,Russo B.Structure of the predual of a JBW*-triple.J Reine Angew Math,1985,356:67-89
30 Harris L A.Bounded symmetric homogeneous domains in infinite dimensional spaces.In:Proceedings on Infinite Dimensional Holomorphy.Lecture Notes in Mathematics,vol.364.Berlin-Heidelberg-New York:Springer,1974,13-40
31 Hatori O,Moln′ar L.Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in C*-algebras.J Math Anal Appl,2014,409:158-167
32 Horn G.Characterization of the predual and ideal structure of a JBW*-triple.Math Scand,1987,61:117-133
33 Isidro J M,Kaup W,Rodr′?guez A.On real forms of JB*-triples.Manuscripta Math,1995,86:311-335
34 Kadets V,Mart′?n M.Extension of isometries between unit spheres of infite-dimensional polyhedral Banach spaces.JMath Anal Appl,2012,396,441-447
35 Kadison R V,Pedersen G K.Means and convex combinations of unitary operators.Math Scand,1985,57:249-266
36 Kaup W.A Riemann mapping theorem for bounded symmentric domains in complex Banach spaces.Math Z,1983,183:503-529
37 Kaup W.On real Cartan factors.Manuscripta Math,1997,92:191-222
38 Mankiewicz P.On extension of isometries in normed linear spaces.Bull Pol Acad Sci Math,1972,20:367-371
39 Megginson R E.An Introduction to Banach Space Theory.New York:Springer-Verlag,1998
40 Russo B,Dye H A.A note on unitary operators in C*-algebras.Duke Math J,1966,33,413-416
41 Tan D.Extension of isometries on unit sphere of L∞.Taiwanese J Math,2011,15:819-827
42 Tan D.On extension of isometries on the unit spheres of Lp-spaces for 0
43 Tan D.Extension of isometries on the unit sphere of Lp-spaces.Acta Math Sin Engl Ser,2012,28:1197-1208
44 Tan D,Huang X,Liu R.Generalized-lush spaces and the Mazur-Ulam property.Studia Math,2013,219:139-153
45 Tan D,Liu R.A note on the Mazur-Ulam property of almost-CL-spaces.J Math Anal Appl,2013,405:336-341
46 Tanaka R.A further property of spherical isometries.Bull Aust Math Soc,2014,90:304-310
47 Tanaka R.The solution of Tingley’s problem for the operator norm unit sphere of complex n×n matrices.Linear Algebra Appl,2016,494:274-285
48 Tanaka R.Spherical isometries of finite dimensional C*-algebras.J Math Anal Appl,2017,445:337-341
49 Tanaka R.Tingley’s problem on finite von Neumann algebras.J Math Anal Appl,2017,451:319-326
50 Tingley D.Isometries of the unit sphere.Geom Dedicata,1987,22:371-378
51 Ylinen K.Compact and Finite-Dimensional Elements of Normed Algebras.Annales Academi?Scientiarium Fennic?Series A I,vol.428.Helsink:Acad Sci Fennica,1968