On universally left-stability of .-isometry

Let X, Y be two real Banach spaces and . . 0. A map f: X . Y is said to be a standard .-isometry if |.f(x) . f(y). . .x . y.| . . for all x, y . X and with f(0) = 0. We say that a pair of Banach spaces (X, Y) is stable if there exists . > 0 such that, for every such . and every standard .-isometry f: X . Y, there is a bounded linear operator T:L(f).span..f(X).X so that .Tf(x)t-x. . .. for all x . 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 . .(.) for some set ., 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.