Tingley's Problem on Symmetric Absolute Normalized Norms on R^2

In this paper,we study Tingley＇s problem on symmetric absolute normalized norms on R^2.We construct new methods for Tingley＇s problem on two-dimensional spaces by using isosceles orthogonality,which does not make use of the notion of natural extension.Furthermore,using our methods,several sufficient conditions for Tingley＇s problem on symmetric absolute normalized norms on R2 are given.As applications,we present various new examples including the two-dimensional Lorentz sequence space d^（2）（ω,q） and its dual d^（2）（ω,q）＊by simple arguments.

A solution to Tingley's problem for isometries between the unit spheres of compact C~*-algebras and JB~*-triples

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.

Extension of Isometries Between the Unit Spheres of Normed Space E and C(Ω)

In this paper, we study the extension of isometries between the unit spheres of normed space E and C（Ω）. We obtain that any surjective isometry between the unit spheres of normed space E and C（Ω） can be extended to be a linear isometry on the whole space E and give an affirmative answer to the corresponding Tingley＇s problem （where Ω be a compact metric space）.

Extension of Isometries on the Unit Sphere of L^p Spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of L^p（μ） （1 〈 p 〈∞, p ≠ 2） and a Banach space E can be extended to a linear isometry from L^p（μ） onto E, which means that if the unit sphere of E is （metrically） isometric to the unit sphere of L^P（μ）, then E is linearly isometric to L^p（μ）. We also prove that every surjective 1-Lipschitz or anti-l-Lipschitz map between the unit spheres of L^p（μ1, H1） and L^p（μ2, H2） must be an isometry and can be extended to a linear isometry from L^p（μ2, H2） to L^p（μ2, H2）, where H1 and H2 are Hilbert spaces.

A Remark on Extension of Into Isometries

In this paper, we prove that an into isometry form S（l（n）^∞） to S（E）, which under some conditions, can be extended to be a linear isometry defined on the whole space. Therefore we improve the results of [Ding, G. G.： The isometric extension of an into mapping from the unit sphere S（l（2）^∞） to S（Lμ^1）. Acta Mathematica Sinica, English Series, 22（6）, 1721-1724 （2006）].

复Banach空间lp(Γ)(1≤p<∞)的Mazur-Ulam性质

1978年,Tingley提出著名的Tingley问题(等距延拓问题),受到许多学者的重视.遗憾的是到目前为止,即使对于二维Banach空间,这个问题仍是一个开问题.目前的研究主要集中在同类型或不同类型的经典Banach空间之间,并得到了肯定的回答.本文对复Banach空间lp(Γ) (1≤p<∞)与复Banach空间E之间的Tingley问题给出了肯定的回答,即复Banach空间lp(r) (1≤p<∞)满足Mazur-Ulam性质.