realizing/ understanding $C^*(\phi_g(C([0,1])))$ and "support projection of an element of a $C^*$-algebra

Question

someone asked me the following question but I don't know the answer, but and I'm interested in it too. Let $C([0,1])=\{f:[0,1]\to\mathbb{C}; \text{f is continuous}\}$, $g\in C([0,1])$ be a positive function, $\phi_g:C([0,1])\to C([0,1]),\; f\mapsto gf$. Consider $C^*(\phi_g(C([0,1])))$ the $C^*$-algebra generated by $\phi_g(C([0,1]))$. How can you realize $C^*(\phi_g(C([0,1])))$? You can identify it with $C_0(X)$ (continuos functions vanishing at infinity), $X$ is a locally compact Hausdorff space. Is it possible to determine $X$, or is it possible to say what $C^*(\phi_g(C([0,1])))$ is exactly?
Now let $\phi(1_{C([0,1])})=g$. What is meant by the "support projection of g", we write $1_H$? (because I read something like "the support projection of g can be expressed as $\lim\limits_{n\to\infty}(g+\frac{1}{n}\cdot 1_H)^{-1}g=1_H$ (this is a strong operator-topology limit). $1_H$ has to be an element in $B(H)$, where $H$ is a Hilbert space..
I would appreciate your help. I'm sorry if the question is wrong here, I'm not sure if it is a mathoverflow-level-question. Bye.

Answer 1

This answer is concerned with the first question. The $C^\ast$-subalgebra $B$ you are considering is in fact a closed $^\ast$-ideal and hence given by $$ \{f\in C[0,1]\mid f_C=0\} $$ for some closed subset $C\subseteq [0,1]$. Clearly all functions in $B$ vanish at $x$ when $g(x)=0$. Conversely, if $g(x)\neq 0$, then we have already found a function in $B$ that does not vanish at $x$. Hence $$ B=\{f\in C[0,1]\mid f(x)=0\;\text{if}\;g(x)=0\}. $$ Thus $B\cong C_0(X)$ where $X=\{x\in [0,1]\mid g(x)\neq 0\}$.