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A survey on Tingley’s problem for operator algebras

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Abstract

We survey the most recent results on the extension of isometries between special subsets of the unit spheres of C*-algebras, von Neumann algebras, trace class operators, preduals of von Neumann algebras,and p-Schatten–von Neumann spaces, with special interest on Tingley’s problem.

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References

  1. C. A. Akemann and G. K. Pedersen, Facial structure in operator algebra theory, Proc. Lond. Math. Soc., 64 (1992), 418–448.

    MathSciNet  MATH  Google Scholar 

  2. J. C. Alexander, Compact Banach algebras, Proc. London Math. Soc., 18 (1968), 1–18.

    MathSciNet  MATH  Google Scholar 

  3. L. J. Bunce and J. D. M. Wright, The Mackey–Gleason problem, Bull. Amer. Math. Soc., 26 (1992), 288–293.

    MathSciNet  MATH  Google Scholar 

  4. L. J. Bunce and J. D. M. Wright, The Mackey–Gleason problem for vector measures on projections in von Neumann algebras, J. London Math. Soc., 49 (1994), 133–149.

    MathSciNet  MATH  Google Scholar 

  5. L. Cheng and Y. Dong, On a generalized Mazur-Ulam question: extension of isometries between unit spheres of Banach spaces, J. Math. Anal. Appl., 377 (2011), 464–470.

    MathSciNet  MATH  Google Scholar 

  6. Ch. -H. Chu, Jordan structures in geometry and analysis, Cambridge, Cambridge University Press, 2012.

    MATH  Google Scholar 

  7. Ch. H. Chu, T. Dang, B. Russo and B. Ventura, Surjective isometries of real C*-algebras, J. London Math. Soc., 47 (1993), 97–118.

    MathSciNet  MATH  Google Scholar 

  8. T. Dang, Real isometries between JB*-triples, Proc. Amer. Math. Soc., 114 (1992), 971–980.

    MathSciNet  MATH  Google Scholar 

  9. G. G. Ding, The representation theorem of onto isometric mappings between two unit spheres of l1-type spaces and the application on isometric extension problem, Sci. China Ser. A, 47 (2004), 722–729.

    MathSciNet  MATH  Google Scholar 

  10. G. G. Ding, 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.), 20 (2004), 1089–1094.

    MathSciNet  MATH  Google Scholar 

  11. G. G. Ding, The isometric extension of the into mapping from a L1(??)-type space to some Banach space, Illinois J. Math., 51 (2007), 445–453.

    MathSciNet  MATH  Google Scholar 

  12. G. G. Ding, On isometric extension problem between two unit spheres, Sci. China Ser. A, 52 (2009), 2069–2083.

    MathSciNet  MATH  Google Scholar 

  13. H. A. Dye, On the geometry of projections in certain operator algebras, Ann. of Math., 61 (1955), 73–89.

    MathSciNet  MATH  Google Scholar 

  14. C. M. Edwards, F. J. Fernández-Polo, C. S. Hoskin and A. M. Peralta, On the facial structure of the unit ball in a JB*-triple, J. Reine Angew. Math., 641 (2010), 123–144.

    MathSciNet  MATH  Google Scholar 

  15. C. M. Edwards and G. T. Rüttimann, On the facial structure of the unit balls in a GL-space and its dual, Math. Proc. Cambridge Philos. Soc., 98 (1985), 305–322.

    MathSciNet  MATH  Google Scholar 

  16. C. M. Edwards and G. T. Rüttimann, On the facial structure of the unit balls in a GM-space and its dual, Math. Z., 193 (1986), 597–611.

    MathSciNet  MATH  Google Scholar 

  17. C. M. Edwards and G. T. Rüttimann, On the facial structure of the unit balls in a JBW*-triple and its predual, J. Lond. Math. Soc., 38 (1988), 317–332.

    MathSciNet  MATH  Google Scholar 

  18. C. M. Edwards and G. T. Rüttimann, Compact tripotents in bi-dual JB*-triples, Math. Proc. Camb. Phil. Soc., 120 (1996), 155–173.

    MathSciNet  MATH  Google Scholar 

  19. A. J. Ellis, Real characterizations of function algebras amongst function spaces, Bull. London Math. Soc., 22 (1990), 381–385.

    MathSciNet  MATH  Google Scholar 

  20. X. N. Fang and J. H. Wang, Extension of isometries between the unit spheres of normed space E and C(Ω), Acta Math. Sinica (Engl. Ser. ), 22 (2006), 1819–1824.

    MathSciNet  MATH  Google Scholar 

  21. F. J. Fernández-Polo, J. J. Garcés, A. M. Peralta and I. Villanueva, Tingley’s problem for spaces of trace class operators, Linear Algebra Appl., 529 (2017), 294–323.

    MathSciNet  MATH  Google Scholar 

  22. F. J. Fernández-Polo, E. Jordá and A. M. Peralta, Tingley’s problem for p-Schatten von Neumann classes for 2 < p < ∞, arXiv (2018), arXiv: 1803. 00763v1.

    Google Scholar 

  23. F. J. Fernández-Polo, J. Martínez and A. M. Peralta, Surjective isometries between real JB*-triples, Math. Proc. Cambridge Phil. Soc., 137 (2004), 709–723.

    MathSciNet  MATH  Google Scholar 

  24. F. J. Fernández-Polo and A. M. Peralta, On the facial structure of the unit ball in the dual space of a JB*-triple, Math. Ann., 348 (2010), 1019–1032.

    MathSciNet  MATH  Google Scholar 

  25. F. J. Fernández-Polo and A. M. Peralta, Low rank compact operators and Tingley’s problem, arXiv (2016), arXiv: 1611. 10218v1.

    Google Scholar 

  26. F. J. Fernández-Polo and A. M. Peralta, On the extension of isometries between the unit spheres of a C*-algebra and B(H), Trans. Amer. Math. Soc., 5 (2018), 63–80.

    MathSciNet  MATH  Google Scholar 

  27. F. J. Fernández-Polo and A. M. Peralta, Tingley’s problem through the facial structure of an atomic JBW*-triple, J. Math. Anal. Appl., 455 (2017), 750–760.

    MathSciNet  MATH  Google Scholar 

  28. F. J. Fernández-Polo and A. M. Peralta, On the extension of isometries between the unit spheres of von Neumann algebras, arXiv (2017), arXiv: 1709. 08529v1.

    Google Scholar 

  29. O. Hatori and L. Molnár, Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in C*-algebras, J. Math. Anal. Appl., 409 (2014), 158–167.

    MathSciNet  MATH  Google Scholar 

  30. R. V. Kadison and G. K. Pedersen, Means and convex combinations of unitary operators, Math. Scand., 57 (1985), 249–266.

    MathSciNet  MATH  Google Scholar 

  31. W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z., 183 (1983), 503–529.

    MathSciNet  MATH  Google Scholar 

  32. P. Mankiewicz, On extension of isometries in normed linear spaces, Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys., 20 (1972), 367–371.

    MathSciNet  MATH  Google Scholar 

  33. T. Miura, Real-linear isometries between function algebras, Cent. Eur. J. Math., 9 (2011), 778–788.

    MathSciNet  MATH  Google Scholar 

  34. M. Mori, Tingley’s problem through the facial structure of operator algebras, arXiv (2017), arXiv: 1712. 09192v1.

    Google Scholar 

  35. L. Molnár and G. Nagy, Isometries and relative entropy preserving maps on density operators, Linear Multilinear Algebra, 60 (2012), 93–108.

    MathSciNet  MATH  Google Scholar 

  36. L. Molnár and W. Timmermann, Isometries of quantum states, J. Phys. A: Math. Gen., 36 (2003), 267–273.

    MathSciNet  MATH  Google Scholar 

  37. G. Nagy, Isometries on positive operators of unit norm, Publ. Math. Debrecen, 82 (2013), 183–192.

    MathSciNet  MATH  Google Scholar 

  38. G. Nagy, Isometries of spaces of normalized positive operators under the operator norm, Publ. Math. Debrecen, 92 (2018), 243–254.

    MathSciNet  MATH  Google Scholar 

  39. C. L. Olsen, Unitary approximation, J. Funct. Anal., 85 (1989), 392–419.

    MathSciNet  MATH  Google Scholar 

  40. C. L. Olsen and G. K. Pedersen, Convex combinations of unitary operators in von Neumann algebras, J. Funct. Anal., 66 (1986), 365–380.

    MathSciNet  MATH  Google Scholar 

  41. G. K. Pedersen, Measure theory for C* algebras. II, Math. Scand., 22 (1968), 63–74.

    MathSciNet  Google Scholar 

  42. G. K. Pedersen, C*-algebras and their automorphism groups, London Mathematical Society Monographs 14, Academic Press, London, 1979.

    MATH  Google Scholar 

  43. A. M. Peralta, On the unit sphere of positive operators, arXiv (2017), arXiv: 1711. 05652v1.

    Google Scholar 

  44. A. M. Peralta and R. Tanaka, A solution to Tingley’s problem for isometries between the unit spheres of compact C*-algebras and JB*-triples, Sci. China Math., to appear; arXiv: 1608. 06327v1.

  45. I. Raeburn and A. M. Sinclair, The C*-algebra generated by two projections, Math. Scand., 65 (1989), 278–290.

    MathSciNet  MATH  Google Scholar 

  46. S. Sakai, C*-algebras and W*-algebras, Springer Verlag, Berlin, 1971.

    MATH  Google Scholar 

  47. M. Takesaki, Theory of operator algebras I, Springer, New York, 2003.

    MATH  Google Scholar 

  48. D. Tan, Extension of isometries on unit sphere of L1∞, Taiwanese J. Math., 15 (2011), 819–827.

    MathSciNet  MATH  Google Scholar 

  49. D. Tan, On extension of isometries on the unit spheres of Lp-spaces for 0 < p ≤ 1, Nonlinear Anal., 74 (2011), 6981–6987.

    MathSciNet  MATH  Google Scholar 

  50. R. Tanaka, A further property of spherical isometries, Bull. Aust. Math. Soc., 90 (2014), 304–310.

    MathSciNet  MATH  Google Scholar 

  51. R. Tanaka, The solution of Tingley’s problem for the operator norm unit sphere of complex n × n matrices, Linear Algebra Appl., 494 (2016), 274–285.

    MathSciNet  MATH  Google Scholar 

  52. R. Tanaka, Spherical isometries of finite dimensional C*-algebras, J. Math. Anal. Appl., 445 (2017), 337–341.

    MathSciNet  MATH  Google Scholar 

  53. R. Tanaka, Tingley’s problem on finite von Neumann algebras, J. Math. Anal. Appl., 451 (2017), 319–326.

    MathSciNet  MATH  Google Scholar 

  54. D. Tingley, Isometries of the unit sphere, Geom. Dedicata, 22 (1987), 371–378.

    MathSciNet  MATH  Google Scholar 

  55. R. S. Wang, Isometries between the unit spheres of C0(Ω) type spaces, Acta Math. Sci. (English Ed.), 14 (1994), 82–89.

    MathSciNet  Google Scholar 

  56. X. Yang and X. Zhao, On the extension problems of isometric and nonexpansive mappings, Mathematics without boundaries (Eds.: Themistocles M. Rcassias and Panos M. Pardalos), Springer, New York, 2014, 725–748.

    Google Scholar 

  57. K. Ylinen, Compact and finite-dimensional elements of normed algebras, Ann. Acad. Sci. Fenn. Ser. A I, 428 (1968).

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  1. Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

    Antonio M. Peralta

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  1. Antonio M. Peralta
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Correspondence to Antonio M. Peralta.

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Communicated by L. Molnár

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Peralta, A.M. A survey on Tingley’s problem for operator algebras. ActaSci.Math. 84, 81–123 (2018). https://doi.org/10.14232/actasm-018-255-0

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