I am trying to show that the unitization of a non-unital AF $C^\*$-algebra, $A$ , is again an AF $C^\*$-algebra. 
In order to do so, I tried to claim the following: 

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> Let $A, B$ be $C^\*$ algebras and suppose that $A$ is an ideal in $B$ and $B/A$ is isomorphic to $\Bbb{C}$.   
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>  In these conditions, is it true that $B$ and $A^+$ (the unitization of $A$) are isomorphic as $C^\*$ algebras?
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**Edit:** I think I should require $B$ to be with multiplicative unit.