multiplier algebra and intersection

Question

Let $M(A)$ be the multiplier algebra of non-unital C* algebra $A$. So $A$ is essential ideal of $M(A)$. Does there exists a dense $\ast$-subalgebra $B$ of $M(A)$, such that $B\not\subset A$, and $B\cap A=\{0\}$?

Answer 1

Not always. For example, in I. Farah's book there are references for $C^*$-algebras that have the property that $M(A)\cong\tilde{A}$, the unitization. As MaoWao proves in this post, if $B\subset\tilde{A}$ is a dense subspace in $\tilde{A}$, then we cannot have $B\cap A=\{0\}$.