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[edit]

[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