C* algebra exact sequences and ideals if you have C* algebras $A,B$ and $C$ and $\exists$ a short exact sequence as follows $0\rightarrow A\rightarrow B \rightarrow C \rightarrow 0 $ where the functions are $\phi$ and $\psi$ respectively, can you assure that $\phi(A)$ is ideal in B and if so why? How would multiplying $\phi(a)$ with an arbitrary $b\in B$ assure you still land in $\phi(A)$ ?

By exactness, you know $\psi(\phi(a)b) = 0$, hence $\phi(a)b \in A$ for all $a \in A$ and $b \in B$.