Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a $J\in\mathfrak{I}$ such that $J\subseteq I_1\cap I_2$). Moreover the intersection $I_0=\bigcap_{I\in\mathfrak{I}} I$ is nonempty. To simplify the notation let's write $A_0$ for $A/I_0$ and $A_I$ for $A/I$. Furthermore we will write $p_I:A\twoheadrightarrow A_I$ and $\tilde{p}_I:A_0\twoheadrightarrow A_I$ for the natural quotient maps. Consider the inverse limit $\lim\limits_{\leftarrow} A_I$ of the $A_I$. This inverse limit can be defined by a universal property (see for instance the diagram here: http://mathworld.wolfram.com/InverseLimit.html, which we consider for C$^*$-algebras).
Concretely the inverse limit is given by \begin{equation} \lim\limits_{\leftarrow} A_I = \{ (a_I)_I \in \prod_{I\in\mathfrak{I}} A_I \mid \sup\|a_I\|<\infty, p_{IJ}(a_J)=a_I \text{ if } J\subseteq I\} \end{equation} where $p_{IJ}$ is the natural quotient map $A_J\twoheadrightarrow A_I$ when $J\subseteq I$. My questions are the following:
(1) Is it true that $A_0\cong \lim\limits_{\leftarrow} A_I$? It is easy to see that the map \begin{equation} P:A_0\rightarrow \lim\limits_{\leftarrow} A_I: a\mapsto (\tilde{p}_I(a))_I \end{equation} is an injection. It seems logical that $A_0\cong \lim\limits_{\leftarrow} A_I$, but I have not been able to show that the above map is surjective as well.
(2) Given any other C$^*$-algebra $B$, is it true that $B\otimes_{min}\lim\limits_{\leftarrow} A_I \cong \lim\limits_{\leftarrow} (B\otimes_{min} A_I)$? Maybe this is even true in general, but if not, it maybe is in this specific situation. Together with (1) this would imply that $B\otimes_{min} A_0 \cong \lim\limits_{\leftarrow} (B\otimes_{min} A_I)$. Again we have a canonical map \begin{equation} \Psi:B\otimes_{min} A_0\rightarrow \lim\limits_{\leftarrow} (B\otimes_{min} A_I): x\mapsto ((id\otimes\tilde{p}_I)(x))_I \end{equation} but I have not been able to prove that it is an isomorphism (assumed it really should be, which I am not sure of either).