View source for Banach function algebra

A [[Banach algebra|Banach algebra]] of continuous functions on a compact [[Hausdorff space|Hausdorff space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b1100801.png" /> separating the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b1100802.png" /> and containing the constant functions (cf. also [[Algebra of functions|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b1100803.png" /> with norm given by

<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/b110/b110080/b1100804.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b1100805.png" /> for a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b1100806.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b1100807.png" />. See, e.g., [[#References|[a1]]], [[#References|[a2]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b1100808.png" /> be a commutative semi-simple Banach algebra with unit and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b1100809.png" /> be the [[Maximal ideal|maximal ideal]] space (cf. also [[Commutative Banach algebra|Commutative Banach algebra]]) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008010.png" />. One can identify the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008011.png" /> with continuous functions on the compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008012.png" /> via the Gel'fand transform and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008013.png" /> is a Banach function algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008014.png" />. More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008015.png" /> is a compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008016.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008017.png" />-element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008018.png" /> is the only function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008019.png" /> vanishing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008020.png" />, then the restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008021.png" /> of the functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008022.png" /> is a Banach function algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008023.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008024.png" /> is a Banach function algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008025.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008026.png" /> is a semi-simple commutative Banach algebra, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008027.png" /> can be imbedded in the maximal ideal space for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008028.png" /> via the mapping that sends a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008029.png" /> to the evaluation functional at that point.

There is the following characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008030.png" />, the algebra of all real-valued continuous functions on a compact Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008031.png" />, among the real Banach function algebras: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008032.png" /> is a Banach function algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008033.png" /> and if there is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008034.png" /> between 0 and 1 such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008035.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008036.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008037.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008039.png" /> is non-negative on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008041.png" />, see [[#References|[a2]]], [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.B. Burckel,   "Characterizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110080/b11008042.png" /> among its subalgebras" , M. Dekker  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Hatori,   "Symbolic calculus on a Banach algebra of continuous functions"  ''J. Funct. Anal.'' , '''115'''  (1993)  pp. 247–280</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  Y. Katznelson,   "Sur les algèbres dont les éléments non-négatifs admettent des racines carrés"  ''Ann. Sci. École Norm. Sup.'' , '''77'''  (1960)  pp. 167–174</TD></TR></table>