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 $\mathrm{\parallel }Id+T^2\mathrm{\parallel }=1+\mathrm{\parallel }{T}^2\mathrm{\parallel }$ holds for every bounded linear operator $T:X⟶X$. This answers in the positive Question 4.11 of [Kadets, Martin, Meri, Norm equalities for operators, \emph{Indiana U. Math. J.} $\mathbf{\text{56}}$ (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.} $\mathbf{\text{330}}$ (2004), 151–183] and [Plebanek, A construction of a Banach space $C(K)$ with few operators, \emph{Topology Appl.} $\mathbf{\text{143}}$ (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 ${\mathrm{\ell }}_{\mathrm{\infty }}$.