characterization of maximal ideals of the algebra of continuous functions on a compact set
Let be a compact topological space and be an ideal of . Then either or there exists a point such that for all .
Assume that, for every point , there exists a continuous function such that . Then, by continuity, there must exist an open set containing so that for all . Thus, we may assign to each point a continuous function and an open set of such that for all . Since this collection of open sets covers , which is compact, there must exists a finite subcover which also covers . Call this subcover and the corresponding functions . Consider the function defined as . Since is an ideal, . For every point , there exists an integer between and such that . This implies that . Since is a continuous function on a compact set, it must attain a minimum. By construction of , the value of at its minimum cannot be negative; by what we just showed, it cannot equal zero either. Hence being bounded from below by a positive number, has a continuous inverse. But, if an ideal contains an invertible element, it must be the whole algebra. Hence, we conclude that either there exists a point such that for all or . ∎
Let be a compact Hausdorff topological space. Then an ideal is maximal if and only if it is the ideal of all points which go zero at a given point.
By the previous theorem, every non-trivial ideal must be a subset of an ideal of functions which vanish at a given point. Hence, it only remains to prove that ideals of functions vanishing at a point is maxiamal.
Let be a point of . Assume that the ideal of functions vanishing at is properly contained in ideal . Then there must exist a function such that (otherwise, the inclusion would not be proper). Since is continuous, there will exist an open neighborhood of such that when . By Urysohn’s theorem, there exists a continuous function such that and for all . Since was assumed to contain all functions vanishing at , we must have . Hence, the function defined by must also lie in . By construction, when and when . Because is compact, must attain a minimum somewhere, hence is bounded from below by a positive number. Thus has a continuous inverse, so , hence the ideal of functions vanishing at is maximal. ∎
|Title||characterization of maximal ideals of the algebra of continuous functions on a compact set|
|Date of creation||2013-03-22 17:45:08|
|Last modified on||2013-03-22 17:45:08|
|Last modified by||rspuzio (6075)|