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As observed by Arveson and Akemann+Pedersen, if $J$ is an ideal in a ${\rm C}^\ast$-algebra $B$, then one can always find a contractive approximate identity for $J$, call it $(e_\lambda)_{\lambda\in\Lambda}$, such that $0\leq e_\lambda$ for all $\lambda$ and $\Vert be_\lambda-e_\lambda b\Vert \to 0$ for each $b\in B$.

A proof can be found in e.g. the Higson–Roe book on analytic K-homology. The proof actually shows that such an approximate identity can always be found in the closed convex hull of a given b.a.i. for $J$. So in the case where $J=K(H)$ inside $B=B(H)$, it should be possible in theory to work through the proof and extract an explicit construction of a b.a.i. with these additional properties. However, I wondered if there is simply a reference in the literature that already does this. Does anyone know of such a reference?

Note that the obvious approximate identity for $K(H)$, where we fix an o.n. basis for $H$ and then take $e_n={\rm diag}(1,\dots, 1, 0,\dots)$, does not give something quasi-central for $K(H)$ inside $B(H)$: see this note by J. L. Orr (Proc. Amer. Math. Soc. 105 (1989), 149–50) for a proof of a stronger result. Indeed, if it were quasi-central I think we'd end up proving that $B(H)$ is quasi-diagonal ${\rm C}^*$-algebra, which is certainly not the case.

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What about this. Consider the set $S$ of all diagonal matrices ${\rm diag}(c_n)$ with the properties (1) $c_0 = 1$, (2) $0 \leq c_n \leq 1$ for all $n$ and (3) $c_n$ decreases to $0$ as $n \to \infty$. Order $S$ by setting ${\rm diag}(c_n) \preceq {\rm diag}(d_n)$ if $c_{n+1} - c_n \geq d_{n+1} - d_n$ for all $n$. So as we move up the partial order, the diagonal entries decrease more slowly. It seems to me that Arveson's theorem about every operator being approximated by something like block diagonal operators (from his Duke Math J. paper) would imply that the commutator with any operator would go to zero.

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  • $\begingroup$ Thanks Nik: this looks plausible. I'll have to come back and check the details later $\endgroup$
    – Yemon Choi
    Commented Aug 3, 2016 at 14:13
  • $\begingroup$ One thing that worries me slightly: a quick look at Arveson's paper seems to indicate that the theorem you refer to is proved using some constructions with a q.c. b.a.i. in K(H). Now logically there is no circularity, it is reasonable that we could start with an abstract result saying "there exists a q.c.b.a.i." and bootstrap it to say "this particular net is a q.c.b.a.i.", but that seems slightly unsatisfactory from an aesthetic point of view $\endgroup$
    – Yemon Choi
    Commented Aug 3, 2016 at 14:17
  • $\begingroup$ Hmm ... try this: for any operator $A$ and fixed orthonormal basis there is an increasing sequence $n_1 < n_2 < \cdots$ such that deleting the entries $(a_{i.j})$ from the matrix of $A$ with $j > n_i$ or $i > n_j$ changes its norm by at most $\epsilon$. I think you can prove this from scratch, and it can be used instead of Arveson's theorem (which also can be deduced from it, I think). $\endgroup$
    – Nik Weaver
    Commented Aug 3, 2016 at 16:53
  • $\begingroup$ (Choose the $n_i$ sequentially so that removing entries from the $i$th row and column changes the norm by at most $\epsilon/2^i$.) $\endgroup$
    – Nik Weaver
    Commented Aug 3, 2016 at 16:55
  • $\begingroup$ Thanks again Nik. I've got my hands full with other work right now, so apologies if I don't get round to checking this for a while! $\endgroup$
    – Yemon Choi
    Commented Aug 4, 2016 at 19:33

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