*-isomorphism on C* algebra

Question

In case of unital $C^*$ algebras $A$ and $B$ we have that any $^*$-isomorphism preserves norm. Now my question is that what if we take non-unital $C^*$ algebras does $^*$-isomorphism preserves norm i.e $$ \phi:C \rightarrow D$$ *-isomorphism . $C^*$ algebrasDoes $\||\phi(c)\||=\||c\||$ for $c\in C.$ In case of unital we use the spectral radius to prove the result.