Let $H$ be a Hilbert space and $\mathscr{A}$ a commutative norm-closed unital $*$-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?
I think I found a counterexample.
Take $\mathscr{M}=L^{\infty}[0,1]$ and $\mathscr{A}=C[0,1]$.
Let $\mathbb{Q}\cap[0,1]=\left\{r_{n}\right\}_{n=1}^{\infty}$ be an enumeration of rationals in $[0,1]$, and define $$E:=\bigcup_{n=1}^{\infty}\left(r_{n}-\frac{\epsilon}{2^{n}},r_{n}+\frac{\epsilon}{2^{n}}\right)\cap[0,1]$$ for some small $\epsilon>0$ so that $m(E)$ is strictly less than $1$. Clearly, the only continuous function $f:[0,1]\rightarrow[0,1]$ satisfying $\mathbb{1}_{E}\leq f\leq 1$ is $f\equiv1$, but the projection $\mathbb{1}_{E}$ is not equal to $1$.