Pure infiniteness and primary factorisation

We show that there is no real or complex indecomposable Banach space with the primary factorisation property (PFP).

We relate the PFP of a Banach space $E$ to ring-theoretic infiniteness of $\mathcal{B}(E)$ and of $\mathcal{B}(E)/\mathcal{M}_E$, where $\mathcal{M}_E$ denotes the set of operators not factoring the identity on $E$, in the case it is the unique maximal ideal of $\mathcal{B}(E)$.

For complex $E$ with the PFP, this quotient is purely infinite exactly when it is not scalar.

We isolate the quantitative gap relevant to ultrapowers, identify classical sequence spaces as positive non-scalar cases, and show that Read's space $E_{\operatorname{R}}$ does not have the uniform PFP.