Let $A$ be a separable C*-algebra and $J\subseteq A$ a closed two-sided ideal. Does this make $J$ into a complemented subspace of $A$? In other words, does the quotient map $A\to A/J$ have a continuous linear section?
In 1973, Andersen showed that the answer is positive when $A/J$ has the metric approximation property and noted that separability is a necessary assumption. The general case was left open. What progress has there been since then?