When does an orthomodular lattice $L$ of projections onto a given Hilbert space have a non-trivial centre $Z(L)$ and what can we generally say about the cardinality of $Z(L)$?
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$\begingroup$ What do you mean by "projections onto a given Hilbert space"? I inferred from your question that you probably mean "all orthogonal projections on a given Hilbert space". Am I right? $\endgroup$– Gejza JenčaCommented Oct 9, 2014 at 18:41
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$\begingroup$ oh yes, sorry, that is what i meant. $\endgroup$– King KongCommented Oct 14, 2014 at 12:47
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$ Z (L) $ contains the subspaces that are orthogonal or comparable to all the other subspaces in $ L $. So if the Hilbert space has finite dimension $ d $ you can get $2^d $ many elements in $ Z (L) $ with $ L $ of size $2^d $ also. For instance take $ L $ to consist of the coordinate planes in $ R^3$.
answered Oct 6, 2014 at 13:22
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$\begingroup$ It is not true that the coordinate planes in $R^3$ are orthogonal or comparable to all other subspaces. Proof: take any plane that is not among the coordinate planes. $\endgroup$ Commented Oct 9, 2014 at 18:05
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$\begingroup$ @GejzaJenča That's not what I was claiming. The question is a bit hard to read but I took it to mean that $L$ is a lattice of projections but not necessarily the lattice of all projections. So $L$ corresponds to a set of subspaces closed under intersection and join. $\endgroup$ Commented Oct 9, 2014 at 18:34