This is a follow up on an earlier question.
Let $A$ be a non-separable unital (not necessarily self-adjoint) operator algebra, which is reflexive as a Banach space. Let $W$ be a $1$-complemented separable unital subalgebra, and $P:A\to A$ be a contractive projection onto $W$.
Question: Do we have $P(ux) = uP(x)$ for all $u\in W$, $x\in A$? If not, does there exist another $1$-complemented separable subalgebra with this property, which contains $W$?
PS: Could you please share general references in the comments about projections on Banach algebras, specifically about contractive projections whose range is a subalgebra?