Tingley’s problem through the facial structure of operator algebras. (English) Zbl 1411.46007
Tingley’s problem asks whether every onto isometry between unit spheres of Banach spaces can be extended to an isometry between the whole spaces. In this paper, positive answers are given for the cases when both spaces are preduals of von Neumann algebras, the spaces of self-adjoint operators in von Neumann algebras, or the spaces of self-adjoint normal functionals on von Neumann algebras. The techniques to get the results include the study of the facial structure of operator algebras.
More information on Tingley’s problem can be found in [D. Tingley, Geom. Dedicata 22, 371–378 (1987; Zbl 0615.51005)], the surveys [G.-G. Ding, Sci. China, Ser. A 52, No. 10, 2069–2083 (2009; Zbl 1190.46013)] and [X.-Z. Yang and X.-P. Zhao, in: Mathematics without boundaries. Surveys in pure mathematics. New York, NY: Springer. 725–748 (2014; Zbl 1328.46012)]. For more results on Tingley’s problem for operator algebras, we refer to the recent survey paper [A. M. Peralta, Acta Sci. Math. 84, No. 1–2, 81–123 (2018; Zbl 1413.47065)] and references therein.
More information on Tingley’s problem can be found in [D. Tingley, Geom. Dedicata 22, 371–378 (1987; Zbl 0615.51005)], the surveys [G.-G. Ding, Sci. China, Ser. A 52, No. 10, 2069–2083 (2009; Zbl 1190.46013)] and [X.-Z. Yang and X.-P. Zhao, in: Mathematics without boundaries. Surveys in pure mathematics. New York, NY: Springer. 725–748 (2014; Zbl 1328.46012)]. For more results on Tingley’s problem for operator algebras, we refer to the recent survey paper [A. M. Peralta, Acta Sci. Math. 84, No. 1–2, 81–123 (2018; Zbl 1413.47065)] and references therein.
Reviewer: Miguel Martín (Granada)
MSC:
| 46B04 | Isometric theory of Banach spaces |
| 46B20 | Geometry and structure of normed linear spaces |
| 46H35 | Topological algebras of operators |
| 46L10 | General theory of von Neumann algebras |
| 47C15 | Linear operators in \(C^*\)- or von Neumann algebras |
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