Banach-Stone theorem

Stone–Banach theorem

For a compact Hausdorff space $ X $, let $ C(X) $ denote the Banach space of all continuous scalar-valued functions on $ X $, equipped with the usual sup-norm: $$ \forall f \in C(X): \qquad \| f \| \stackrel{\text{df}}{=} \sup(\{ |f(x)| \mid x \in X \}). $$ If $ \varphi $ is a surjective homeomorphism from a compact space $ Y $ onto a compact space $ X $, and if $ \chi $ is a continuous and uni-modular scalar-valued function on $ Y $, then $$ \forall f \in {C_{0}}(X): \qquad T(f) \stackrel{\text{df}}{=} \chi \cdot (f \circ \varphi), $$ defines a linear isometry from $ {C_{0}}(X) $ onto $ {C_{0}}(Y) $. The Banach–Stone theorem asserts that any linear surjective isometry $ T: C(X) \to C(Y) $ is of the form above. Here, if $ X $ is not necessarily compact, then $ {C_{0}}(X) $ is the space of continuous scalar-valued functions that vanish at infinity (i.e., the continuous scalar-valued functions $ f $ such that for all $ \epsilon > 0 $, there is a compact subset $ K $ of $ X $ such that $ |f(x)| < \epsilon $ for $ x \in X \setminus K $). A uni-modular function $ \chi $ is one for which $ |\chi(y)| = 1 $ for all $ y \in Y $.

The theorem was proved in the real case for compact metric spaces by S. Banach ([a2]); M.H. Stone ([a10]) proved that the assumption of metrizability was superfluous. Subsequently, the theorem was extended to spaces of both real- or complex-valued functions defined on a locally compact Hausdorff space.

The theorem has been extended further into several directions:

1. A Banach space $ E $ has the Banach–Stone Property if and only if, the Banach space $ {C_{0}}(X,E) $ of continuous $ E $-valued functions is isometric with $ {C_{0}}(Y,E) $ if and only if $ X $ and $ Y $ are homeomorphic; the class of Banach spaces with this property includes the strictly convex Banach spaces and the Banach spaces with strictly convex dual (see e.g. [a3], [a8]).
2. If there is an isomorphism $ T: C(X) \to C(Y) $ such that $ \| T \| \| T^{-1} \| = 1 + \epsilon < 2 $, then $ X $ and $ Y $ are homeomorphic and consequently, $ C(X) $ and $ C(Y) $ are isometric ([a1], [a4]). Here, $ 2 $ is the best bound ([a5]).
3. The theorem holds for several classes of sub-spaces of the spaces $ C(X) $, the most important one being the class of uniform algebras.

A joint approach to the last two extensions gave rise to the perturbation theory of uniform algebras. A Banach algebra $ B $ is an $ \epsilon $-perturbation (or $ \epsilon $-metric perturbation) of a Banach algebra $ A $ if and only if there is an isomorphism $ T: A \to B $ such that $ \| T \| \| T^{-1} \| < 1 + \epsilon $. In such a situation, the algebras $ A $ and $ B $ must share several important properties ([a7], [a9]); if the algebra $ B $ is forced to be isometrically isomorphic with $ A $, then the algebra $ A $ is called stable. Examples of stable uniform algebras include $ C(X) $, $ A(D) $ and $ {H^{\infty}}(D) $, with $ D $ the unit disc.

Isometries of several other classes of Banach spaces have been shown to follow the same general pattern, similar to (a1); one often refers to such results also as Banach–Stone theorems (for a given class of spaces; see e.g. [a6]).

References

[a1]	D. Amir, “On isomorphisms of continuous function spaces”, Israel J. Math., 3 (1965), pp. 205–210.
[a2]	S. Banach, “Théorie des opérations linéaires”, PWN (1932).
[a3]	E. Behrends, “$ M $-structure and the Banach–Stone theorem”, Lecture Notes in Mathematics, 736, Springer (1979).
[a4]	M. Cambern, “On isomorphisms with small bound”, Proc. Amer. Math. Soc., 18 (1967), pp. 1062–1066.
[a5]	H.B. Cohen, “A bound-two isomorphism between $ C(X) $ Banach spaces”, Proc. Amer. Math. Soc., 50, (1975), pp. 215–217.
[a6]	R.J. Fleming, J.E. Jamison, “Isometries on Banach spaces: a survey”, Analysis, Geometry and Groups: a Riemann Legacy Volume, Hadronic Press (1993), pp. 52–123.
[a7]	K. Jarosz, “Perturbations of Banach algebras”, Lecture Notes in Mathematics, 1120, Springer (1985).
[a8]	K. Jarosz, “Small isomorphisms of $ C(X,E) $ spaces”, Pacific J. Math., 138: 2 (1989), pp. 295–315.
[a9]	R. Rochberg, “Deformation of uniform algebras on Riemann surfaces”, Pacific J. Math., 121: 1 (1986), pp. 135–181.
[a10]	M.H. Stone, “Applications of the theory of Boolean rings to general topology”, Trans. Amer. Math. Soc., 41 (1937), pp. 375–481.