Fibres of $C_0(X)$-algebra. Clarification of Basics (Williams "Crossed Products of C*-Algebras") Definition. Let A be a $C^*$-algebra, $X$ - locally compact Hausdorff space. Then A is a $C_0(X)$-algebra if there is a homomorphism $Ф_A$ from 
$C_0(X)$ into the center $ZM(A)$ of the multiplier algebra $M(A)$ which is nondegenerate in that the ideal 
$\Phi_A(C_0(X))A = \operatorname{span}\{ \Phi_A(f)a |f \in C_0(X)\}$ and $a \in A$
is dense in A. 
Definition. $A(x):=A/I_x$ is the fibre over x, where $I_x$:=$\overline{Ф_A(J_x)A}$,  $J_x$ is the ideal of functions vanishing at $x \in X$.
1) I can't see why is it so (particularly I'm not sure why we need closure here): 

Note that if J is an ideal in $C_0(X)$ (or even a subalgebra), then
  $I_x$:=$\overline{Ф_A(J)A}$ is an ideal in A.

Moreover, how does $I_x$ look? Is it {$a'a$| $a' \in ZM(A')$, $A' \subset A$, $a \in A$}?
2) How exactly do fibres over x look? Is it a set {$aI_x$|$a \in A$}?  
3) Can one say that $J_x$ = $\overline{C_0(X-\{x\})}$? 
Thank you in advance! 
 A: 1) Remember that $\phi_A(J)$ is in the centre of $M(A)$. So, for $a,b\in A$, 
$$
b\phi_A(f)a=\phi_A(f)ba\in \phi_A(J)A
$$
and it is trivial to extend to the closure. You need the closure because it is not obvious that $\phi_A(J)A$ is closed. And in when talking C$^*$-algebras, unless it is said explicitly "ideal" means "closed bilateral ideal". 
As for the form of $I_x$, you are forgetting that $A'A$ means "the linear span of all the products $a'a$". More importantly, it is not obvious (to me, as least) that a subalgebra of the centre of the multiplier of $A$ is itself the centre of the multiplier of some subalgebra. 
2) Being a quotient, the fibres are classes $a+I_x$. I don't see a way to specify the classes more explicitly (i.e., to say precisely when $b-a\in I_x$). 
3) The set of functions vanishing at infinity is already closed in the uniform norm, so no closure is needed. Technically, there is no inclusion between $C(X)$ and $C_0(X\setminus\{x\})$ (since the latter are not defined at $x$). But if $\gamma: J_x\to C_0(X\setminus \{x\})$ is defined by $\gamma(f)=f|_{X\setminus\{x\}}$, this is clearly well-defined (by continuity), linear, multiplicative, and preserves adjoints. It is one-to-one: if $\gamma(f)=0$, then $f(t)=0$ for all $t\ne x$, and then $f(x)=0$ by continuity so $f=0$. And it is onto: if $g\in C_0(X\setminus\{x\})$, then extend $g$ as $\tilde g$, with $\tilde g(x)=0$. Then $\tilde g\in J_x$ and $\gamma(\tilde g)=g$. 
So $J_x$ and $C_0(X\setminus \{x\})$ are isomorphic C$^*$-algebras. 
