maximal ideals of the algebra of continuous functions on a compact set
In this entry we are interested in identifying the maximal ideals and the character space of . Since is a Banach algebra with an identity element, there is a bijective correspondence between the character space of and the set of maximal ideals of this algebra, given by
Hence, by identifying the character space of we are able to identify its maximal ideals.
Theorem 1 - Let be the character space of . For each let be the point-evaluation at , i.e.
Thus, the character space of is homeomorphic to via point-evaluations.
Now, the maximal ideals of correspond to the kernels of the point-evaluation functions. The kernel of , the point-evaluation at , is just
i.e., the functions that vanish at .
Thus, each maximal ideal of is just the set of functions that vanish in a given point.
0.1 Generalization to locally compact Hausdorff spaces
There is a generalization of Theorem 1 above that allows one to identify the character space of , but since this algebra is not unital unless is compact, we cannot identify its maximal ideals by the above method.
Theorem 2- Let be the character space of . For each let be the point-evaluation at , i.e.
Then the mapping is an homeomorphism between and .
Thus, the character space of is also homeomorphic to via point-evaluations.
|Title||maximal ideals of the algebra of continuous functions on a compact set|
|Date of creation||2013-03-22 17:44:57|
|Last modified on||2013-03-22 17:44:57|
|Last modified by||asteroid (17536)|
|Synonym||character space of the algebra of continuous functions on a compact set|