On non-linear ε-isometries between the positive cones of certain continuous function spaces

Let X, Y be two w*-almost smooth Banach spaces, C(B(X*), w*) be the Banach space of all continuous real-valued functions on B(X*) endowed with the supremum norm and C+(B(X*), w*) be the positive cone of C(B(X*), w*). In this paper, we show that if F : C+(B(X*), w*) -> C+(B(Y*), w*) is a standard almost surjective epsilon-isometry, then there exists a homeomorphism tau : B(X*) -> B(Y*) in the w*-topology such that for any x* is an element of B(X*), we have vertical bar <delta(chi*), f > - <delta(tau(chi*)), F(f)>vertical bar <= 2 epsilon, for all f is an element of C+(B(X*), w*). As its application, we show that if U : C(B(X*), w*) -> C(B(Y*), w*) is the canonical linear surjective isometry induced by the homeomorphism gamma = tau(-1) : B(Y*) -> B(X*) in the w*-topology, then vertical bar vertical bar F(f) - U(f)vertical bar vertical bar <= 2 epsilon, for all f is an element of C+(B(X*), w*).