The following is a remark in Murphy's $C^*$-Algebras and Operator Theory:
If an algebra is non-unital we can adjoin a unit to it. This is very helpful in many cases, and we shall frequently make use of it, but it does not reduce the theory to the unital case. There are situations where adjoining a unit is unnatural, such as when one is studying the group algebra $L^1(G)$ of a locally compact group $G$.
I'm wondering in what sense adjoining a unit $e$ to $L^1(G)$ is unnatural. I know that $L^1(G)$ has a unit iff $G$ is discrete, so we may assume that $G$ is not discrete. Then the unitization of $L^1(G)$ is given by $L^1(G)\oplus\mathbb C$ and the unit is given by $e=(0,1)$. If we were to think of $e$ as being a measurable function on $G$, then we would have that $e$ is the constant function $1$. This seems pretty natural to me.
Is it the correspondence between characters on $G$ and non-zero morphisms $L^1(G)\to\mathbb C$ which is less clean if we considered the unitization?
Or what is a good reason for calling this unitization unnatural?