Why do the projections in the Calkin algebra not form a lattice?

Maybe it's clearer in our original email correspondence!

From me to Ilijas, 12/31/06:

The poset of projections in any von Neumann algebra is a lattice. Is this true of C(H)? The answer seems to be no. I can give a pair of projections that looks like a counterexample, though proving this might take some work.

From Ilijas to me, 1/4/07:

Right, I can see this now. There is an example that is not too bad.

Enumerate a basis of H as e(i,j,k), i and j in omega and k in {0,1}. Let P be the projection to the closed span of all e(i,j,0). To define Q, let f(i,j) be a linear combination of e(i,j,0) and e(i,j,1) such that ||f(i,j)-e(i,j,0)||=2^{-i}. Let Q be the projection to the span of all f(i,j). Now with pi the quotient map into the Calkin algebra C(H), I claim that pi(P)/\pi(Q) does not exist in C(H). Proof: Fix a projection R' which is below both pi(P) and pi(Q). We can find R below P in the lattice of projections of B(H) such that pi(R)=R'. For each i, there is n(i) such that for all j>=n(i) the distance between e(i,j,0) and R is at least 1/2. Let S be the projection to the span of e(i,n(i),0). Then pi(R/S) is below both pi(P) and pi(Q), but it is strictly above pi(R).

Is that easier to follow?

No doubt this is also the proof that Don had in mind. (And it's also more or less what I was thinking, but this proof is Ilijas's, not mine. So I certainly don't deserve the credit here.)

I can't add a comment, although I'd like to clarify one point: both in Nik Weaver's Set theory and C*-algebras and Eric Wofsey's $P(\omega)/\mathrm{fin}$ and projections in the Calkin algebra it is claimed that projections in Calkin algebra do give rise to the lattice structure (see e.g. Corollary 3.4 and Wofsey's abstract, respectively). How is it possible?

Both proofs (Proposition 5.26 in Farah's notes and email corresponence in this topic) seem to be general in the separable case.