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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?

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    $\begingroup$ This was my thesis problem, which I failed to solve (see MR1897155; but I still got my Ph.D.). In 1973, T. Ando also proved a better result that $X/M$ has a lift whenever $M\subset X$ is an $M$-ideal and $X/M$ is a separable BAP Banach space. Every C*-ideal is an $M$-ideal. Not much has been done since. The issue is that Szankowski's theorem (Acta 1981) is the only source of a C*-algebra without BAP, but the proof is super-difficult and nobody ever succeeded in finding an easier proof of the existence of a C*-algebra without BAP. $\endgroup$ Jun 4, 2017 at 3:16
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    $\begingroup$ @NarutakaOZAWA: Okay, thank you! Perhaps you want to make this into an answer? $\endgroup$ Jun 4, 2017 at 18:15

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