A relative bicommutant theorem: the stable case of Pedersen's question

In 1976, D. Voiculescu proved that every separable unital sub-C*-algebra of the Calkin algebra is equal to its (relative) bicommutant. In his minicourse (see reference), G. Pedersen asked in 1988 if Voiculescu's theorem can be extended to a simple corona algebra of a $\sigma$-unital C*-algebra. In this note, we answer Pedersen's question for a stable $\sigma$-unital C*-algebra.