Extremely non-complex C(K) spaces

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces X such that the norm equality ‖Id+T2‖=1+‖T2‖ holds for every bounded linear operator . 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 K1 and K2 such that C(K1) and C(K2) are extremely non-complex, C(K1) contains a complemented copy of C(ω2) and C(K2) contains a (1-complemented) isometric copy of ℓ∞.