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From: Rotwang on 22 Jul 2010 11:39
Hi all

Let H be a Hilbert space, and suppose that P is a commuting set of
self-adjoint projections on H, with the additional two properties:

1) P is closed under complements, i.e. if p is in P then so is 1 - p.
2) P is closed under suprema of arbitrary subsets, i.e. if S is a subset
of P then sup S is in P (here the projections on H are ordered by
defining p <= q whenever the range of p is contained in the range of q).

Now let V denote the smallest von Neumann algebra containing P;
equivalently V is the closure relative to the weak operator topology of
the set of linear combinations of elements of P. Suppose that p is a
self-adjoint projection in V. Is p in P?
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