What are the projections of a commutative C* algebra?

Question

I am aware that the commutative C* algebra is $C_0(X)$ for some nice space $X$ but I cannot figure out what the projections should be. The natural candidates (indicator functions on nice subsets of $X$ don't work because they are not continuous).

Answer 1

You're on the right track: you want to look at indicator functions. If $f$ is a function such that $f^2=f$, can you prove it must be an indicator function? Now of course, you're right that most indicator functions aren't continuous. Can you describe the indicator functions which are? And of those, which vanish at infinity?

Answers to these questions can be found below:

If $f^2=f$, then $f(x)^2=f(x)$ for all $x$, so $f(x)$ must always be either $0$ or $1$. This means $f$ is the indicator function of the set $S=\{x:f(x)=1\}$. The indicator function of a set $S\subseteq X$ is continuous iff $S$ is both closed and open, and additionally vanishes at infinity iff $S$ is compact. So the projections of $C_0(X)$ are the indicator functions of subsets of $X$ which are both compact and open.