derivations on a ring of continuous functions

Let X be a topological spaceMathworldPlanetmath and denote by the set of reals. Of course the set of all continuous functionsMathworldPlanetmathPlanetmath C(X,) is a -algebra. Let c. By the symbol c¯ we will denote constant function at c, i.e. c¯:X is defined by c¯(x)=c.

PropositionPlanetmathPlanetmath. If D:C(X,)C(X,) is a -derivation, then D(x)=0¯ for any xC(X,).

Proof. Step one. We will prove that D(c¯)=0¯ for any c. Indeed


and thus D(1¯)=0¯. Now from linearity of D we obtain that


Step two. If f:X is continuous and c, then f+c¯ is continuous and obviously D(f)=D(f+c¯). Moreover, if xX then D(f-f(x)¯)=D(f), but (f-f(x)¯)(x)=0. Thus we may assume that f(x)=0 for fixed xX.

Let x0X. Now we will restrict only to such maps f:X that f(x0)=0.

Step three. We now decompose f into sum of two nonnegative functions. Indeed, if f:X is continuous, then define f+,f-:X by the formulaMathworldPlanetmathPlanetmath:


Of course both f- and f+ are continous, nonnegative and f=f+-f-. Thus


so it is enough to show that D(f)=0¯ only for nonnegative and continuous functions.

Step four. Assume that f:X is nonnegative, continuous and f(x0)=0. Then there exists g:X continuous such that g2=f (indeed g=f and it is well defined, continuous map, because f was nonnegative). Then we have


Now we have g(x0)=f(x0)=0 and thus


Now we can take any xX and repeat steps two, three and four to get that for any xX we have


and thus


which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Remark. Note that this proof cannot be repeated if we (for example) consider the set of all smooth functionsMathworldPlanetmath C(M,) on a smooth manifoldMathworldPlanetmath M, because f+, f- and f need not be smooth.

Title derivations on a ring of continuous functions
Canonical name DerivationsOnARingOfContinuousFunctions
Date of creation 2013-03-22 18:37:25
Last modified on 2013-03-22 18:37:25
Owner joking (16130)
Last modified by joking (16130)
Numerical id 14
Author joking (16130)
Entry type Example
Classification msc 17A36
Classification msc 16W25
Classification msc 13N15