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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 ($P^2 = P$).

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Since you are refering to subalgebras of $M_n(\mathbb{C})$, i am not sure if this is what you are looking for, but if $S$ is a semigroup of operators on a Hilbert space $\mathcal{H}$, $\mathcal{V}\subseteq\mathcal{H}$ is a subspace of $\mathcal{H}$ and $P$ an orthogonal projection onto $\mathcal{V}$, satisfying $PABP=PAPBP$ for all $A,B\in S$, then $\mathcal{V}$ is called semi-invariant under $S$.

See: p. 290 at: On spectral sets having connected complement. Acta scientiarum mathematicarum, (26) 3-4. pp. 289-299. (1965), by D. Sarason. (Lemma 0 also contains a characterization of such semi-invariant subspaces).

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