I have a small question about unitization of (unital) $C^\*$-algebras. I first asked on math.stackexchange because it is basic theory, but I still have no suitable answer, the link <https://math.stackexchange.com/questions/1336297/unitization-of-a-unital-c-algebra> .

I will describe my problem: Let $A$ be a $C^\*$-algebra, which is non-unital. Than
you can prove that $$\|(a,\lambda)\|=\|L\_{(a,\lambda)}\|,$$ with $L\_{(a,\lambda)}:A\to A,\; b\mapsto ab+\lambda b$, defines a norm on the vector space $A\_1=A\oplus\mathbb{C}$. But you need, that $A$ is non-unital to prove: if $\|(a,\lambda)\|=\|L\_{(a,\lambda)}\|=0, \Rightarrow (a,\lambda)=(0,0)$.

In literature frequently it is assumed that $A$ is non-unital and then you unitize it, for example here <http://ncatlab.org/nlab/show/unitization+of+a+C-star-algebra>. But there is not much about the unital case in literature.

The process seems to fail, if $A$ is unital and you want to take the norm above on $A\_1$. In literature, I read that you can take $\|(a,\lambda)\|=max\{\|a\|,|\lambda|\}$ in this case. But is seems curious that you have to take a different norm in this case. 

My question is, is it correct to take an other norm if $A$ is unital ? But it's strange that the unitization-process depends on if $A$ is unital or not.
Regards

**Edit**. I try to precise my question: is there a uniform construction of the norm on A\_1 and a uniform proof that it really is a C\*-norm which does not use the case distinction unital/non-unital?