# 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?