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

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 among its subalgebras" , M. Dekker (1972)
[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