@Yemon, do you have by chance any reference for this theorem about homomorphisms into $B(X)$?

Thank you. Let me restate my question then. Is there a Banach space $E$ and an ultrafilter $U$ over some cardinal $\lambda>\mathfrak{c}$ such that the above holds and $|E_U|\geqslant \lambda$?

What is the easiest way to prove the existence of such a net? Do we order by inclusion all the finite subfields of the domain of $\mu$ ($L_\infty = L_\infty(\mu)$) and consider subspaces of functions from L∞ which are measurable with respect to these fields?

That's interesting if it can be done any better than SOT. Are there any reasonable assumptions for $X$ to consider uniform limits?

Thank you, Professor. Just to clarify, strongly means in the sense of SOT?

Thanks, any hint for the family $F$? Can it have cardinality $2^{\mathfrak{c}}$?