Recall that a Banach algebra is faithful when $aA=(0)$ implies $a=0$ and $Aa=(0)$ implies $a=0$. A faithful (not necessarily unital) Banach algebra $A$ embeds in its multiplier algebra $\mathit{Mult}(A)$ under $a\mapsto (L_a,R_a)$, where $L_a(b)=ab$ and $R_a(b)=ba$. The multiplier algebra is complete in the strict topology, that is, the topology defined by the seminorms $p_a(L,R)=\|L(a)\|+\|R(a)\|$, $a\in A$.
Is there a nice (i.e. explicit) description of the closure of $A$ in the strict topology on $\mathit{Mult}(A)$? (Of course, here I implicitly exclude Banach algebras with an approximate identity, and thus in particular C*-algebras.)