Let $A$ be a non-separable reflexive Banach algebra. Every separable subspace of $A$ is contained in a separable 1-complemented subspace [[Lindenstrauss,1966][1]]. It is straightforward to show that every *separable subalgebra* is contained in a separable subalgebra $W$ of the form $$W=\bigcup_{n=1}^{\infty} E_n$$ where $(E_n)$ is an increasing sequence of separable 1-complemented subspaces of $A$.

**Question:** Is it true that every separable subalgebra of $A$ is contained in a separable complemented subalgebra?


  [1]: https://doi.org/10.1090/S0002-9904-1966-11606-3