Totally disconnected and Stonean spaces

If $X$ is a compact Hausdorff space, show that $X$ is totally disconnected if and only if $C(X)$ is the closed linear span of its projections ($\equiv$ hermitian idempotents).

Using the terminology of the preceding exercise, show that if $(X, \Omega , \mu )$ is a $\sigma$-finite measure space, the maximal ideal space of $L^\infty (X, \Omega , \mu )$ is totally disconnected.

If $X$ is a compact space, show that $C(X)$ is generated as a $C^*$-algebra by its characteristic functions if and only if $X$ is totally disconnected. If $A$ is as in Exercise 6, show that $\sigma (A)$ is totally disconnected.

If $X$ is any compact metric space, show that there is a totally disconnected compact metric space $Y$ and a continuous surjection $\phi : Y \to X$. (Hint: Start by embedding $C(X)$ into $\mathcal{B}(\mathcal{H})$ and use Exercises 7 and 8.)

Show that every totally disconnected compact metric space is the continuous image of the Cantor ternary set. (Do this directly; do not try to use $C^*$-algebras.) Combine this with Exercise 12 to get that every compact metric space is the continuous image of the Cantor set.

If $\mathcal{A}$ is an abelian von Neumann algebra and $X$ is its maximal ideal space, then $X$ is a Stonean space; that is, if $U$ is open in $X$, then $\operatorname {cl} U$ is open in $X$.