A Banach algebra $A$ is a *dual Banach algebra* if it is a dual Banach space with a (not necessarily unique) predual $A\_{\ast}$, and the multiplication on $A$ is separately weak\*-continuous. Dual Banach algebras are naturally analogous to von Neumann algebras [[Daws2007](https://doi.org/10.4064/sm178-3-3), [Daws2011](https://doi.org/10.3318/PRIA.2011.111.1.3)]; thus, one may expect a given dual Banach algebra to possess a rich structured subset of idempotents. (e.g., structured as explained in [[Laustsen2003](https://doi.org/10.1017/S0017089502008947), [DawsHorvath2021](https://doi.org/10.4153/S0008414X20000565)] )


**Question:** Let $E$ be a reflexive Banach space and $A\subseteq B(E)$ be a dual Banach algebra. Does $A$ contain *projectional skeletons* (e.g. [[Kubis2009](https://doi.org/10.1016/j.jmaa.2008.07.006)] ) or *projectional resolutions of identity* ?



Projectional skeletons in dual Banach algebras
No. $A = \ell\_\infty$ is a dual Banach algebra. Every separable complemented subspace of $A$ is finite-dimensional, so there is no way to exhaust $A$ by nicely complemented separable subspaces.


