Let $\omega$ be a state on a $C^*$-algebra $C$ and let $A$ and $B$ be commuting unital subalgebras of $C$ (e.g. if $C$ is a $C^*$-tensor product of $A$ and $B$) and let $(\pi,H,\Omega)$ be the GNS-representation of $\omega$. Let $P_A$ be the orthogonal projection onto $[\pi(A)\Omega]$ and let $P_B$ be the projection onto $[\pi(A)\Omega]$ (where $[V]$ is the closed linear hull of a subset $V \subset H$).
Is it true that $P_A$ and $P_B$ commute? Does it help if $\omega$ is pure?
I'm interested in $P_A$ because $(P_A(\pi \upharpoonright A),P_A H,\Omega)$ is unitarily equivalent to the GNS representation of the restricted state $\omega \upharpoonright A$. One can see that $P_A$ commutes with $\pi(A)$.
The answer is no.
Consider $C={\mathbb C}^3$, with $$ A= \{(\lambda , \lambda , \mu ): \lambda , \mu \in {\mathbb C}\}, $$ and $$ B= \{(\lambda , \mu , \mu ): \lambda , \mu \in {\mathbb C}\}. $$ Taking $$ \varphi (\lambda , \mu , \nu ) = (\lambda + \mu +\nu )/3, $$ we have that $H={\mathbb C}^3$, with coordinatewise multiplication for the representation $\pi $, while the cyclic vector is $\Omega =(1,1,1)/\sqrt 3$.
We then have that $$ [\pi (A)\Omega ] = \{(\lambda , \lambda , \mu ): \lambda , \mu \in {\mathbb C}\}, $$ and $$ [\pi (B)\Omega ] = \{(\lambda , \mu , \mu ): \lambda , \mu \in {\mathbb C}\}, $$ while $$ P_A =\pmatrix {1/2 & 1/2 & 0 \cr 1/2 & 1/2 & 0 \cr 0 & 0 & 1}, $$ and $$ P_B =\pmatrix { 1& 0 & 0 \cr 0 & 1/2 & 1/2 \cr 0 & 1/2 & 1/2 }, $$ which are not commuting matrices.