Let $\mathcal{A}$ be a subalgebra of $M_n(\mathbb{C})$. Is there a characterization of (or at least a name for) orthogonal projections $P \in M_n(\mathbb{C})$ with the property that $PABP = PAPBP$ for all $A,B \in \mathcal{A}$? In other words, for which the compression map $A \mapsto PAP$ is a homomorphism.

The orthogonal projection onto any invariant or (orthogonally) coinvariant subspace for $\mathcal{A}$ has this property.

Let $\mathcal{A}$ be a subalgebra of $M_n(\mathbb{C})$. Is there a characterization of (or at least a name for) orthogonal projections $P \in M_n(\mathbb{C})$ with the property that $PABP = PAPBP$ for all $A,B \in \mathcal{A}$? In other words, for which the compression map $A \mapsto PAP$ is a homomorphism.

The orthogonal projection onto any invariant or (orthogonally) coinvariant subspace for $\mathcal{A}$ has this property.

Come to think of it, same question but not assuming $P$ is orthogonal, just a projection ($P^2 = P$).