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+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(2^ω) and C(K₂) contains a (1-complemented) isometric copy of ℓ_∞.