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Stone–Banach theorem
For a compact Hausdorff space , let
be the Banach space of all continuous scalar-valued functions on
, with the usual
-norm:
![]() |
If is a surjective homeomorphism from a compact space
onto a compact space
and
is a continuous and unimodular scalar-valued function on
, then
![]() | (a1) |
defines a linear isometry from onto
. The Banach–Stone theorem asserts that any linear surjective isometry
is of the above form. Here, if
is not necessarily compact, then
is the space of continuous functions that vanish at infinity (i.e. the functions
such that for all
there is a compact set
such that
for
). A unimodular function is one for which
for all
.
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 has the Banach–Stone property if the Banach space
of
-valued continuous functions is isometric with
if and only if
and
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 such that
, then
and
are homeomorphic (and consequently
and
are isometric) [a1], [a4], where
is the best bound [a5];
3) the theorem holds for several classes of subspaces of the spaces , the most important one being the class of uniform algebras (cf. also Uniform algebra).
A joint approach to the last two extensions gave rise to the perturbation theory of uniform algebras. A Banach algebra is an
-perturbation (or
-metric perturbation) of a Banach algebra
if there is an isomorphism
such that
. In such a situation the algebras
,
must share several important properties [a7], [a9]; if the algebra
is forced to be isometrically isomorphic with
, the algebra
is called stable. Examples of stable uniform algebras include
,
and
, with
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, "![]() |
[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 ![]() |
[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 ![]() |
[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 |
Banach-Stone theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach-Stone_theorem&oldid=15162