TITLE: Almost idempotent approximate units in C*-algebras
QUESTION [13 upvotes]: As in Blackadar's "Operator Algebras" Definition II.4.1.1., call an approximate unit $(a_\lambda)$ in the positive unit ball of a C*-algebra almost idempotent if $a_\lambda a_\gamma=a_\lambda$ whenever $\lambda<\gamma$.

Does every C*-algebra have an almost idempotent approximate unit?

REPLY [6 votes]: Akemann has constructed a C*-algebra that does not contain an approximate unit of commuting elements. See Example 2.1 in
Akemann. Approximate units and maximal abelian C*-subalgebras. Pacific J. Math. 33 (1970)
As pointed out by Tristan, a C*-algebra might have no approximate unit of commuting elements but nevertheless have an almost idempotent approximate unit. (Thanks for pointing out the mistake in my original answer.) So the original question remains open.