Weakly dense C*-algebra in a commutative von Neumann algebra and order convergence

Let $ H$ be a Hilbert space and $ \mathscr{A}$ a commutative norm-closed $ *$ -subalgebra of $ \mathcal{B}(H)$ . Let $ \mathscr{M}$ be the weak operator closure of $ \mathscr{A}$ .

Question: For given a projection $ P\in\mathscr{M}$ , is the following true? $ $ P=\inf\left\{A\in\mathscr{A}:P\leq A\leq 1\right\}$ $

It seems that the infimum must exist and is a projection, but I am not able to show that the resulting projection cannot be strictly bigger than $ P$ . Also, if the above is true, what happens if $ \mathscr{A}$ is non-commutative?