On extension of isometries and approximate isometries between unit spheres

In this paper, we will study the isometric extension problem for L1-spaces and prove that every surjective isometry from the unit sphere of L1(μ) onto that of a Banach space E can be extended to a linear surjective isometry from L1(μ) onto E. Moreover, we introduce the approximate isometric extension problem and show that, if E and F are Banach spaces and E satisfies the property (m) (special cases are L∞(Γ), C0(Ω) and L∞(μ)), then every bijective ϵ-isometry between the unit spheres of E and F can be extended to a bijective 5ϵ-isometry between their closed unit balls. At last, we will give an example to show that the surjectivity assumption cannot be omitted. Using this, we solve the non-surjective isometric extension problem in the negative.