comparison of multiplier algebras

Question

Suppose $I$ is an essential ideal of a nonunital $C^*$ algebra $A$, can we compare $M(I)$ and $M(A)$,is $M(A)\subset M(I)$,where $M()$ denotes the multiplier algebra.

Answer 1

No. In general $I \subset A$ an essential ideal does not ensure that $M(I) \subset M(A)$.

Take $A = C(K)$ the continuous functions over the compact set $K = \{0\} \cup \{n^{-1}\}_{n=1}^\infty$. Since $A$ is unital $M(A) = A$. Take $I = c_0(\mathbb{N})$ the functions on $K \setminus \{0\}$ vanishing at $0$. Then $M(I)$ is $C(\beta \mathbb N)$, which is not a subset of $A$, where $\beta \mathbb N$ denotes the Stone-Cech compactification.