Let $A$ be a $C^*$-algebra, and $R:A\to A$ its right multilplier. Is it true that
$$
\exists b\in A\quad \forall a\in A \quad R(a)b=a\qquad
$$
implies $A$ is unital. I know this is true if A is a weak$^*$ dense ideal of $W^*$-algebra. But what about the general case? 

This question on [MSE][1].


  [1]: https://math.stackexchange.com/questions/895050/r-mbox-is-a-right-multiplier-and-rab-a-overset-implies-a-mbox-is-unit