ON THE FIGIEL TYPE PROBLEM AND EXTENSION OFε-ISOMETRY BETWEEN UNIT SPHERES

This paper studies two isometric problems between unit spheres of Banach spaces.In the first part,we introduce and study the Figiel type problem of isometric embeddings between unit spheres.However,the classical Figiel theorem on the whole space cannot be trivially generalized to this case,and this is pointed out by a counterexample.After establishing this,we find a natural necessary condition required by the existence of the Figiel operator.Furthermore,we prove that when X is a space with the T-property,this condition is also sufficient for an isometric embedding T:S_(X)→S_(Y) to admit the Figiel operator.This answers the Figiel type problem on unit spheres for a large class of spaces.In the second part,we consider the extension of bijectiveε-isometries between unit spheres of two Banach spaces.It is shown that every bijectiveε-isometry between unit spheres of a local GL-space and another Banach space can be extended to be a bijective 5ε-isometry between the corresponding unit balls.In particular,whenε=0,this recovers the MUP for local GL-spaces obtained in[40].

STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES

A pair of Banach spaces (X, Y) is said to be stable if for every ￡-isometry f : X→Y,there exist γ> 0 and a bounded linear operator T : L(f)→X with ||T||≤α such that ||Tf(x)- x||≤γε for all x ∈ X, where L(f) is the closed linear span of f(X). In this article, we study the stability of a pair of Banach spaces (X, Y) when X is a C(K) space. This gives a new positive answer to Qian's problem. Finally, we also obtain a nonlinear version for Qian's problem.

STABILITY OF ε-ISOMETRIES ON L_∞-SPACES

In this article,we discuss the stability ofε-isometries for L∞,λ-spaces.Indeed,we first study the relationship among separably injectivity,injectivity,cardinality injectivity and universally left stability of L∞,λ-spaces as well as we show that the second duals of universally left-stable spaces are injective,which gives a partial answer to a question of Bao-Cheng-Cheng-Dai,and then we prove a sharpen quantitative and generalized Sobczyk theorem which gives examples of nonseparable L∞-spaces X(but not injective)such that the couple(X,Y)is stable for every separable space Y.This gives a new positive answer to Qian's problem.

ε-isometric Approximation Problem

In this paper, some problems for isometric approximation are discussed. It is shown that an almost isometry from the unit ball of C(X) into the unit ball of C(Y) is near to an isometry. We also give a counterexample to this problem on the unit ball of l1.

Surjective Lp-isometries on Grassmann spaces

Letϕ:Pc(M1)→Pc(M2)be a surjective Lp-isometry between Grassmann spaces of projections with the trace value c in semifinite factors M1 and M2.Based on the characterization of surjective Lp-isometries of unitary groups in finite factors,we show thatϕor I−ϕcan be extended to a∗-isomorphism or a∗-antiisomorphism.In particular,ϕis given by a∗-(anti-)isomorphism unless M1 and M2 are finite and c=12.

On Universally Left-stability of ε-Isometry

Let X, Y be two real Banach spaces and ε≥0. A map f ： X → Y is said to be a standard ε-isometry if│││f/（x） - f（y）││ - ]ix - Y││x-y││ ε for all x,y C X and with f（O） = O. We say that a pair of Banach spaces （X, Y） is stable if there exists γ〉 0 such that, for every such ε and every standard v-isometry f ： X → Y, there is a bounded linear operator T ： L（f） → f（X） → X so that ││Tf（x） - x││ ≤γε for all x E X. X（Y） is said to be universally left-stable if （X, Y） is always stable for every Y（X）. In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w＊-closed subspace of ∞ （1） for some set F, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant.

Stability Characterizations of ε-isometries on Certain Banach Spaces

Suppose that X, Y are two real Banach Spaces. We know that for a standard ε-isometry f : X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of *. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ε-isometry to be stable in assuming that N is w*-closed in Y*.Making use of this result, we improve several known results including Figiel’s theorem in reflexive spaces.We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f)≡span[f(X)] contains a complemented linear isometric copy of X;Moreover, if X =Y, then for every e-isometry f: X → X, there exists a surjective linear isometry S:X → X such that f-S is uniformly bounded by 2ε on X.

Surjective L^(2)-isometries on the Projection Lattice

Recently,Gehér and Semrl have characterized the general form of surjective isometries of the set of all projections on an infinite-dimensional separable Hilbert space using unitaries and antiunitaries.In this paper,we study the surjective L^(2)-isometries of the projection lattice of an infinite dimensional Hilbert space and show that every such isometry can also be described by unitaries and antiunitaries.

On Almost Isometric Embedding from C(Ω) into C_0(Ω_0)

In this paper we give the sufficient and necessary condition for the existence of any almost isometric operator from C（Ω） into C0（Ω0）. As a corollary, we show that there is no e-isometry from 1 any abstract M space with a strong unit into C0（Г） if 0 〈 e 〈 1/9.

THE ISOMETRIC APPROXIMATION PROBLEM IN B(l^1(Γ), C(Ω) )

In this note we use the scalar field K(=R or C)and always suppose that Γ is an infinite index set and Ω is a compact Hausdorff space. All the keywords (except the last one) are from Ref. [1].