View source for Banach-Stone theorem

''Stone–Banach theorem''

For a compact [[Hausdorff space|Hausdorff space]] $ X $, let $ C(X) $ denote the [[Banach space|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|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 ([[#References|[a2]]]); M.H. Stone ([[#References|[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:

# 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. [[#References|[a3]]], [[#References|[a8]]]).
# 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 ([[#References|[a1]]], [[#References|[a4]]]). Here, $ 2 $ is the best bound ([[#References|[a5]]]).
# The theorem holds for several classes of sub-spaces of the spaces $ C(X) $, the most important one being the class of [[Uniform algebra|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|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 ([[#References|[a7]]], [[#References|[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. [[#References|[a6]]]).

====References====

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