View source for Banach-Stone 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:

<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>

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:

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]]]);

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]]];

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]]]).

====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>