No indecomposable Banach space has the primary factorisation property; for complex spaces with PFP the quotient B(E)/M_E is purely infinite precisely when non-scalar.
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Pure infiniteness and primary factorisation
No indecomposable Banach space has the primary factorisation property; for complex spaces with PFP the quotient B(E)/M_E is purely infinite precisely when non-scalar.