unitization of an essential ideal

Question

Suppose $I$ is a non-unital eesential ideal of a non-unital $C^*$ algebra $B$,can we conclude that the unitization $\tilde{I}=I\bigoplus \Bbb C$ is an essential ideal of unitization $B\bigoplus \Bbb C$ of $B$?

Answer 1

If $I$ is an essential ideal of a C*-algebra $A$, then there is a unique injective $*$-homomorphism $\tilde{n}$ extending the natural mapping $n$ from $I$ to $M(I)$, i.e., the following diagram is commutative:

(see Theorem 3.1.8, [1])

When $I$ is unital, the $n$ is an *-isomorphism, which implies that $\tilde{n}$ is an $*$-isomorphism, and thus the embedded mapping $i$ is also an *-isomorphism, i.e., $A=I$.

Back to your question, if $\tilde{I}$ is an essential ideal of $B\oplus \mathbb{C}$, then $\tilde{I}=B\oplus\mathbb{C}$, a contradiction.

[1] Gerald J Murphy. C*-algebras and operator theory. Academic press, 2014.