This is a follow up on an [earlier question][1].

In [[Lau&Loy, 2008][2]] a Banach algebra $\mathcal{U}$ was called to have the *Tomiyama property* if any contractive projection $P:\mathcal{U}\to \mathcal{U}$, whose range $P(\mathcal{U})$ is a subalgebra, is a conditional expectation. 

Analogously, let's say that $\mathcal{U}$ has the *"separable Tomiyama property"* if any contractive projection, whose range is a *separable* subalgebra, is a conditional expectation. Let's also say that $\mathcal{U}$ has property ($T_\omega$) if every $1$-complemented separable subalgebra of $\mathcal{U}$ is contained in a unital separable subalgebra, which is a range of a conditional expectation. Similarly, $\mathcal{U}$ has property *left* ($T_\omega$) ( resp. *right* ($T_\omega$) ) if every $1$-complemented separable subalgebra of $\mathcal{U}$ is contained in a unital separable subalgebra $W$, which is a range of a contractive projection $P$ satisfying $P(ux) = uP(x)$ ( resp. $P(xu) = P(x)u$ ) for all $u\in W$, $x\in A$. 


**Question:** Let $A$ be a reflexive, non-separable, simple, unital (not necessarily self-adjoint) operator algebra, which possesses no minimal idempotents, i.e., $socle(A)=\{0\}$. Does $A$ have property ($T_\omega$)?

It would be great if there were an answer for the other properties 
instead of (or in addition to) property ($T_\omega$). 

**PS:** Could you please share general references in the comments about projections on Banach algebras, specifically about contractive projections whose range is a subalgebra?


  [1]: https://mathoverflow.net/questions/409491/separable-subalgebras-of-non-separable-reflexive-banach-algebras
  [2]: https://doi.org/10.1016/j.jfa.2008.02.008