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 $Z(L)$?



When does an orthomodular projection lattice have a non-trivial centre?
$ 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$.


