Let $A$ be a non-separable reflexive Banach algebra. Every separable subspace of $A$ is contained in a separable 1-complemented subspace [[Lindenstrauss,1966](https://doi.org/10.1090/S0002-9904-1966-11606-3)]. 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?



Separable subalgebras of non-separable reflexive Banach algebras
I am inclined to say *yes*. This is because reflexive Banach spaces have [projectional resolutions of the identity](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/80C18D3F91097CD50DE5EA763C6BEE08/S0004972700013344a.pdf/on-projectional-resolution-of-identity-on-the-duals-of-certain-banach-spaces.pdf), which are increasing ordinal-indexed nets of contractive, commuting projections that exhaust the whole space and satisfy certain compatibility conditions. From this, it follows that separable subspaces of reflexive Banach space are contained in 1-complemented separable subspaces as you mentioned, but we got actually more.


Now, let $A$ be a separable subalgebra of a reflexive Banach algebra $B$. Denote by $\langle S \rangle$ denote the closed subalgebra generated by $S\subset B$. Take $A^1$ to be a separable subspace of $B$ containing $A$ that is 1-complemented by a projection $P^1$ from some fixed PRI and set $A\_1 = \langle A^1 \rangle$. Let $A^2$ be a separable subspace of $B$ containing $A\_1$ that is 1-complemented by a projection $P^2$ from the same PRI and set $A\_2 = \langle A^2 \rangle$. Continue this process recursively so that you get intertwined sequences $A^1 \subset A\_1 \subset A^2 \subset A\_2 \ldots$. In particular, the unions of $(A\_n)$ and $(A^n)$ have the same closure, call it $A\_\omega$. Readily, $A\_\omega$ is a closed subalgebra being the closure of an increasing chain of subalgebras. Since projections from a PRI commute, $(P^n)$ converge in the weak operator topology to a projection, which looks like a projection onto $A\_\omega$.


