## Universal Property of the Multiplier Algerba

We call an injective $*$-isomorphism $i:A\to B$ between two C*-algebra an *ideal homomorphism* if $i(A)$ is an closed ideal of $B$. The universal property of the multiplier algebra $M(I)$ of a C*-algebra $I$ is: for any C*-algebra $A$ and an ideal homomorphism $i:I\to A$ there is a unique $*$-homomorphism $f$ from $A$ to $M(I)$ such that the following diagram is commutative:
\begin{xy}
\xymatrix{
A\ar[r]^f & M(I)\\
I\ar[u]^i\ar[ru]_c & ,
}
\end{xy}
where $c$ is the canonical ideal homomorphism.

*An application*. [If $I$ is a unital C*-algebra, then it cannot be an essential ideal of any other C*-algebra.][1]

[1]: https://math.stackexchange.com/q/3046342