# Banach-Stone theorem

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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]).

How to Cite This Entry:
Banach–Stone theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach%E2%80%93Stone_theorem&oldid=22064