decomposition of a $C^*$ algebra

Question

If $A$ is a $C^*$ algebra ,$I$ is a closed ideal in $A$ ,can we decompose A as following?

$A=I\oplus A/I$

Answer 1

As Aweygan's answer already shows, the answer is no. I would like to add an Abelian counterexample. Let $A = C(X)$, with $X$ compact and $I = \{ f : f{|}_K = 0 \} \leq A$ an ideal, where $K \subset X$ is closed. Then $$ A/I \, \oplus \, I = C(K) \oplus C_0(X \setminus K) \cong C_0(K \sqcup X \setminus K ). $$ The spectra will be equivalent to $X$ iff $K$ and $X \setminus K$ are separated. That is a topological restriction that disappears if you take the duals or double duals (intuitively they are "measurable" objects).

Answer 2

No, we can not. A counterexample is $A=B(H)$ and $I=K(H)$ for $H$ an infinite-dimensional Hilbert space. Then $A$ is unital, but $I$ is not unital, hence $I\oplus A/I$ is not unital.