Atomicity of blocks in a Hilbert lattice Where can I find the proof that any block (maximal boolean subalgebra) $\mathbf{B}$ of the orthomodular lattice $\mathcal{L}$ of closed subspaces of a separable Hilbert space $\mathcal{H}$ is atomic?
 A: I don't think this is true. 
Firstly, remark that the lattice $L$ of closed subspaces of a Hilbert space $H$ is isomorphic with the lattice $P$ of projections on $H$ via the map $P\mapsto L$ given by $p\mapsto pH$. The order on $P$ then satisfies $p\leq q$ if and only if $pH\subseteq qH$. 
Next, the algebra $B(H)$ of bounded operators (which contains $P$ as subset) is a von Neumann algebra on $H$. Let $M$ be a maximal commutative *-subalgebra of $B(H)$. Then $M''$ is a commutative von Neumann algebra on $H$ containing $M$, and by maximality of $M$, we find $M=M''$, so $M$ is a commutative von Neumann algebra. It follows that $M$ is generated as von Neumann algebra by its projections $B$, i.e., $B''=M$. 
$B$ must be a block. Indeed, if there is some Boolean subalgebra $A\subseteq P$ containing $B$ as a subset, then all elements of $A$ commute as projections on $H$, hence $A''$ is a commutative von Neumann algebra on $H$ containing $M=B''\subseteq A''$. By maximality of $M$ it follows that $M=A''$, which means that $A$ is a set of projections in $M$. Since $B$ is the set of all projections in $M$, it follows that $B=A$. 
Now, in the book Foundation of Quantum Theory by Klaas Landsman, which is open source (follow the link), we find on on page 601 a classification of maximal commutative *-subalgebras of $B(H)$, as well as the assertion that there are no atomic projections (which are precisely the atoms in the block corresponding to the maximal commutative *-subalgebra) in one type of maximal commutative *-subalgebras, namely the one of algebras isomorphic to $L^\infty(0,1)$.
Hence $P$, and so also $L$ has a block that is not atomic.
