Extremely non-complex spaces

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces X such that the norm equality ‖ Id + T 2 ‖ = 1 + ‖ T 2 ‖ holds for every bounded linear operator T : X → X . This answers in the positive Question 4.11 of [V. Kadets, M. Martín, J. Merí, Norm equalities for operators on Banach spaces, Indiana Univ. Math. J. 56 (2007) 2385–2411]. More concretely, we show that this is the case of some C ( K ) spaces with few operators constructed in [P. Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004) 151–183] and [G. Plebanek, A construction of a Banach space C ( K ) with few operators, Topology Appl. 143 (2004) 217–239]. We also construct compact spaces K 1 and K 2 such that C ( K 1 ) and C ( K 2 ) are extremely non-complex, C ( K 1 ) contains a complemented copy of C ( 2 ω ) and C ( K 2 ) contains a (1-complemented) isometric copy of ℓ ∞ .