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 $|Id + T^2|=1 + |T^2|$ holds for every bounded linear operator $T:Xlongrightarrow X$. This answers in the positive Question 4.11 of [Kadets, Martin, Meri, Norm equalities for operators, emph{Indiana U. Math. J.} textbf{56} (2007), 2385--2411]. More concretely, we show that this is the case of some $C(K)$ spaces with few operators constructed in [Koszmider, Banach spaces of continuous functions with few operators, emph{Math. Ann.} textbf{330} (2004), 151--183] and [Plebanek, A construction of a Banach space $C(K)$ with few operators, emph{Topology Appl.} textbf{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 $C(2^omega)$ and $C(K_2)$ contains a (1-complemented) isometric copy of $ell_infty$.