The isometric extension of “into” mappings on unit spheres of AL-spaces

Abstract

In this paper, we show that if V0 is an isometric mapping from the unit sphere of an AL-space onto the unit sphere of a Banach space E, then V0 can be extended to a linear isometry defined on the whole space.

The isometrical extensions of 1-Lipschitz mappings on Gâteaux differentiability spaces

“…They obtained a positive answer of the above problem. And using the idea in [11], a similar result as the one in [10] for the case of E = L ∞ (µ) is obtained in [20].…”

“…Moreover, we studied in [11] the Tingley's problem for an "into" isometric mappings from S 1 (L 1 (µ)) to S 1 (F ) (where F is a Banach space) and got some results. In particular, we obtained a positive answer to the Tingley's problem for the "onto" mappings, i.e., the above linear extension problem of isometries 713 is solved between the space L 1 (µ) (also AL-space, see [21]) and a general Banach space F .…”

“…In particular, we obtained a positive answer to the Tingley's problem for the "onto" mappings, i.e., the above linear extension problem of isometries 713 is solved between the space L 1 (µ) (also AL-space, see [21]) and a general Banach space F . We must say that the proof in [11] is not easy, where many techniques are used.…”

Extension of isometries on the unit sphere of L p spaces

“…In 2003, Ding [9] first studied Tingley's problem in this case and proved that surjective isometries between the unit spheres of a Banach space E and C(Ω), where Ω is a compact Hausdorff space under some conditions can be linearly extended to the whole space E. Fang and Wang [19] proved that these conditions can be removed where Ω is a compact metric space. Ding [10] showed the same result for L 1 (μ) where the additional conditions are obviously satisfied if the isometries are assumed to be surjective. Liu [12] established a more general result yielding an affirmative answer to Tingley's problem for L ∞ (μ) space.…”

“…In recent years, Ding and his students have been working on this topic and have obtained many important results (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). Until now, the isometric extension problem for the surjective isometries between unit spheres of the same type classical Banach spaces has almost been solved.…”