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''Stone–Banach theorem''
 
''Stone–Banach theorem''
  
For a compact [[Hausdorff space|Hausdorff space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b1300501.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b1300502.png" /> be the [[Banach space|Banach space]] of all continuous scalar-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b1300503.png" />, with the usual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b1300504.png" />-norm:
+
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 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b1300505.png" /></td> </tr></table>
+
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.
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b1300506.png" /> is a surjective [[Homeomorphism|homeomorphism]] from a compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b1300507.png" /> onto a compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b1300508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b1300509.png" /> is a continuous and unimodular scalar-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005010.png" />, then
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
 
 
 
defines a linear isometry from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005012.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005013.png" />. The Banach–Stone theorem asserts that any linear surjective isometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005014.png" /> is of the above form. Here, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005015.png" /> is not necessarily compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005016.png" /> is the space of continuous functions that vanish at infinity (i.e. the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005017.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005018.png" /> there is a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005020.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005021.png" />). A unimodular function is one for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005022.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005023.png" />.
 
 
 
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:
 
The theorem has been extended further into several directions:
  
1) a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005024.png" /> has the Banach–Stone property if the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005026.png" />-valued continuous functions is isometric with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005027.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005029.png" /> 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]]]);
+
# 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]].
  
2) if there is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005033.png" /> are homeomorphic (and consequently <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005035.png" /> are isometric) [[#References|[a1]]], [[#References|[a4]]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005036.png" /> is the best bound [[#References|[a5]]];
+
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.
 
 
3) the theorem holds for several classes of subspaces of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005037.png" />, the most important one being the class of uniform algebras (cf. also [[Uniform algebra|Uniform algebra]]).
 
 
 
A joint approach to the last two extensions gave rise to the perturbation theory of uniform algebras. A Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005038.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005040.png" />-perturbation (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005042.png" />-metric perturbation) of a [[Banach algebra|Banach algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005043.png" /> if there is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005044.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005045.png" />. In such a situation the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005047.png" /> must share several important properties [[#References|[a7]]], [[#References|[a9]]]; if the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005048.png" /> is forced to be isometrically isomorphic with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005049.png" />, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005050.png" /> is called stable. Examples of stable uniform algebras include <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005053.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005054.png" /> 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]]]).
 
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====
 
====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,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005055.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005056.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130050/b13005057.png" /> 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>
+
 
 +
<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>

Latest revision as of 17:08, 6 January 2017

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.
How to Cite This Entry:
Banach-Stone theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach-Stone_theorem&oldid=22063
This article was adapted from an original article by K. Jarosz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article