Banach function algebra
A Banach algebra of continuous functions on a compact Hausdorff space separating the points of
and containing the constant functions (cf. also Algebra of functions). One speaks of a real or complex Banach function algebra according to whether the functions are real- or complex-valued. If the Banach algebra norm is the supremum norm, one speaks of a function algebra (or a uniform algebra). An example of a Banach function algebra which is not a uniform algebra is the algebra of all real- (or complex-) valued continuously differentiable functions on the unit interval
with norm given by
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where for a continuous function
on the interval
. See, e.g., [a1], [a2].
Let be a commutative semi-simple Banach algebra with unit and let
be the maximal ideal space (cf. also Commutative Banach algebra) for
. One can identify the elements of
with continuous functions on the compact Hausdorff space
via the Gel'fand transform and then
is a Banach function algebra on
. More generally, if
is a compact subset of
such that the
-element of
is the only function in
vanishing on
, then the restriction to
of the functions in
is a Banach function algebra on
. Conversely, if
is a Banach function algebra on
, then
is a semi-simple commutative Banach algebra, and
can be imbedded in the maximal ideal space for
via the mapping that sends a point of
to the evaluation functional at that point.
There is the following characterization of , the algebra of all real-valued continuous functions on a compact Hausdorff space
, among the real Banach function algebras: If
is a Banach function algebra on
and if there is a number
between 0 and 1 such that
is in
whenever
is in
and
is non-negative on
, then
, see [a2], [a3].
References
[a1] | R.B. Burckel, "Characterizations of ![]() |
[a2] | O. Hatori, "Symbolic calculus on a Banach algebra of continuous functions" J. Funct. Anal. , 115 (1993) pp. 247–280 |
[a3] | Y. Katznelson, "Sur les algèbres dont les éléments non-négatifs admettent des racines carrés" Ann. Sci. École Norm. Sup. , 77 (1960) pp. 167–174 |
Banach function algebra. E. Briem (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach_function_algebra&oldid=15556