I am trying to prove the following result:
Let $(A_{i})_{i\in I}$ be an inductive system of non unital $C^{\ast}$-algebras with connecting homomorphisms $f_{ij}: A_{i} \to A_{j}$ and let's denote the limit by $A$. If $B$ is another non unital $C^{\ast}$-algebra, we would like to show that the inductive limit of $(A_{i}\otimes_{\rm{max}} B)_{i\in I}$ is isomorphic to $A\otimes_{\rm{max}} B$
I know the proof of this in case of unital $C^{\ast}$-algebras but i cant see the proof for non unital case. Any ideas?