On weak stability of ε-isometries on wedges and its applications

In this paper, we study weak stability properties of an ε-isometry defined on a wedge W of a Banach space X, instead of the whole space X  . As a result, we show that if f:W→Yf:W→Y is an ε  -isometry with f(0)=0f(0)=0 for some Banach space Y  , then there exists a w⁎w⁎-compact absolutely convex set B⊂BX⁎B⊂BX⁎ satisfying that (a) p(x)≡supx⁎∈B⁡〈x⁎,x〉=‖x‖p(x)≡supx⁎∈B⁡〈x⁎,x〉=‖x‖ for all x∈W∪−Wx∈W∪−W; and (b) for every x⁎∈Bx⁎∈B, there is ϕ∈BY⁎ϕ∈BY⁎ so that|〈ϕ,f(x)〉−〈x⁎,x〉|≤2ε,for all x∈W. This is a generalization of a recent result so called a universal theorem for stability of ε-isometries (but the proof is more technical). As its application, we prove that if the ε-isometry f is defined on the positive cone W   of a C(K)C(K)-space, or, an abstract M  -space with a strong unit (in particular, ℓ∞(Γ)ℓ∞(Γ), and L∞(μ)L∞(μ) for a finite measure μ), then we can choose the set B   to be BX⁎BX⁎; the closed unit ball of the dual X⁎X⁎; and further show that X⁎⁎X⁎⁎ is w⁎w⁎-to-w⁎w⁎ continuously isometric to a subspace of Y⁎⁎Y⁎⁎.