The following is a theorem about self-adjoint subalgebra of $C(X)$ where $X$ is compact Hausdorff and the first half of its proof:
Here are my questions:
- Why $\Gamma:\mathfrak{U}\to C(M_{\mathfrak U})$ being an isometry implies that $\Gamma(\mathfrak{U})$ is closed in $C(M_{\mathfrak U})$?
- Why does $\Gamma(\mathfrak{U})$ separate points?
Here $M_\mathfrak{U}$ denotes the set of non zero complex multiplicative linear functions on $\mathfrak{U}$. For the second question, I think the book means $\Gamma(\mathfrak{U})$ separate points of $M_{\mathfrak U}$, which means for any two distinct points $x,y\in M_{\mathfrak U}$, $f(x)\neq f(y)$ for some $f\in \Gamma(\mathfrak{U})$. But I don't see how to give an argument.