Are lattice operations in set of orthogonal projections in Hilbert space continous?

Question

Let $H$ be Hilbert space and denote set of all orthogonal projections in $H$ by $\Pi$. Then $\Pi$ can be given structure of a lattice. We partially order it by declaring $P \leq Q$ if $Q-P$ is nonnegative operator. Then it becomes complete, orthomodular lattice. Some further properties can be found in http://planetmath.org/latticeofprojections

Question is: are the join and meet operations continuous? The answer might depend on choice of one of standard operator topologies (obviously). If that is the case I am interested in all three: norm, strong and weak.

Answer 1

On $H=\mathbb C^2$, for each $\lambda\in \mathbb C$ let $P_\lambda$ be the projection onto the span of $\{(1,\lambda)\}$.

Then $P_\lambda\to P_0$ as $\lambda\to 0$, but for $\lambda\neq 0$, $P_0\land P_\lambda =0$, while $P_0\land P_0=P_0\neq 0$.

Also, for $\lambda\neq 0$, $P_0\lor P_\lambda = I$, while $P_0\lor P_0 = P_0\neq I$.