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Assume $A$ is a C*-algebra and $p,q\in A^{**}$ are compact projections.

Can we always find $a,b\in A^1_+$ with $p\leq a$, $q\leq b$ and $||pq||=||ab||$?

Note if $||pq||=1$ this is immediate, while if $||pq||=0$ this follows from Akemann's non-commutative Urysohn Lemma - see "A Gelfand representation theory for C*-algebras" Lemma III.1 (Pacific J. Math. 39 (1971), no. 1, 1--11).

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  • $\begingroup$ There was a big investigation on projections in operator algebras made by D. Blecher and his students. This might help arxiv.org/pdf/1109.5347.pdf, arxiv.org/pdf/1109.5171.pdf $\endgroup$
    – Norbert
    Commented Nov 4, 2015 at 20:21
  • $\begingroup$ Blecher et al have certainly done some great work extending the non-commutative topological theory of C*-algebras to more general Banach algebras. But as far as I can tell, their extensions of Akemann's non-commutative Urysohn lemma (like Theorem 2.1 of the first article you mention) still require the given projections to commute. $\endgroup$
    – Tristan Bice
    Commented Nov 5, 2015 at 0:35

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